# Sprague-Grundy values and complexity for LCTR††thanks: This work is
supported in part by the Slovenian Research Agency (research program P1-0383,
research projects N1-0160, N1-0209 and J3-3003, and bilateral project BI-
US/22-24-164).

Eric Gottlieb
Rhodes College,
Memphis,
Tennessee, U.S.A.
<EMAIL_ADDRESS>
Matjaž Krnc Peter Muršič
Faculty of Mathematics, Natural Sciences and Information Technologies
University of Primorska
Koper, Slovenia
<EMAIL_ADDRESS>
###### Abstract

Given a Young diagram on $n$ boxes as a non-increasing sequence of integers,
we consider the impartial combinatorial game LCTR in which moves consist of
removing either the left column or top row of boxes. We show that for both
normal and misère play, the optimal strategy can consist mostly of mirroring
the opponent’s moves. This allows for computing the Sprague-Grundy value of
the given game in $O(\log(n))$ time units, where time unit allows for reading
an integer, or performing a basic arithmetic operation. This improves on the
previous bound of $O(n)$ time units, due to by Ilić (2019), which can be
obtained by an improvement of the Sprague-Grundy recursion.

##  1 Introduction and background

Combinatorial game theory is a large and growing field that includes in its
scope a wide range of game types, generally focusing on two-player games in
which both players have perfect information and there are no moves of chance.
We restrict ourselves to two-player combinatorial games that end in finitely
many moves. Every game ends in one of the two possible outcomes: one of the
players wins, while the other loses. Under _normal play_ , the winner is the
player who makes the last move, leaving their opponent with no move. Under
_misère_ rules, the win condition is reversed.

The main question in combinatorial game theory is, given some position and
optimal play, which player has a winning strategy? An $\mathcal{N}$-position
(winning position) is one in which the player whose turn it is can guarantee a
win. The complementary positions are $\mathcal{P}$-positions (losing
positions) in which the player that goes second can guarantee a win. The game
is effectively solved if we can classify all of its positions as either
$\mathcal{N}$ or $\mathcal{P}$. Sprague [10] and Grundy [6] introduced a
method of quantifying game positions for normal play _impartial_ games, i.e.,
those in which both players have the same possible moves in each position.
These _Sprague-Grundy values_ are a generalization of winning and losing
positions. Furthermore, the Sprague-Grundy values are instrumental in the
theory of _disjunctive sums of impartial games_ ; see [3, 4, 6, 9, 10, 11].
This is why recent research on impartial games is predominantly focused on the
Sprague-Grundy values values rather than on
$\mathcal{P}/\mathcal{N}$-positions. A more detailed introduction to
combinatorial game theory, including impartial games, can be found in [3, 4,
9].

In this paper we study the impartial finite game introduced in [5], where two
players take turns removing from a given Young diagram $\lambda$ either
leftmost column or topmost row – hence the name LCTR. We measure the time
complexity of this problems in terms of $n$, where $\lambda$ is a partition of
$n$. In [5], the authors studied the behaviour of some specifically structured
sub-families of LCTR games (see Lemma 3) and devised a procedure deciding
which player has the winning strategy on a given Young diagram and determining
the Sprague-Grundy value associated to the Young diagram in $O(n)$ time units
[7], where time unit allows performing a basic arithmetic operation (see
Section 6.1).

In Section 2 we cover the basic definitions needed for the study of LCTR and
related games. In Section 3 we establish the properties that are common to
both normal and misère play of LCTR. In Sections 4 and 5 we take the study of
the normal form of LCTR, as well as the normal form of a game called
_Downright_ that is closely related to the misère form of LCTR, and show that
for both games, the optimal strategy can consist mostly of mirroring the
opponent’s moves. This allows for computing the exact Sprague-Grundy values
for both games in $O(\log n)$ time units, as shown in Section 6. In the same
section we also cover some other complexity measures of both games, such as
state-space complexity, and number of leaves and nodes in the corresponding
game tree.

##  2 Preliminaries

In this section, we provide background information on partitions and Young
diagrams, impartial games, Sprague-Grundy theory, and complexity theory.

###  2.1 Young diagrams and Partitions

A _partition_ of a nonnegative integer $n$ is sequence
$\lambda=\llbracket\lambda_{1},\dots,\lambda_{r}\rrbracket$ of integers, where
$\lambda_{1}\geq\cdots\geq\lambda_{r}>0$ and
$\lambda_{1}+\cdots+\lambda_{r}=n$. We also write $\lambda\vdash n$ and we say
that $\lambda$ is a partition of $n$. The $\lambda_{j}$’s are called the
_parts_ of $\lambda$. When some of the parts of a partition are equal, we may
abbreviate
$\llbracket\lambda_{1}^{m_{1}},\dots,\lambda_{k}^{m_{k}}\rrbracket$, where
$\lambda_{j}$ appears $m_{j}\geq 0$ times. The _Young diagram_ of $\lambda$ is
a left-justified array of boxes in which the $j$th row from the top contains
$\lambda_{j}$ boxes. Partitions are drawn as Young diagrams, and no
distinction is made between a partition and its Young diagram. The Young
diagram of $\llbracket 6,4,3,3,1,1\rrbracket=\llbracket
6,4,3^{2},1^{2}\rrbracket$ is shown in Figure 1. The only partition of 0 is
the empty sum, which we write as $\llbracket\rrbracket$. We define the
conjugate partition of $\lambda$ as
$\lambda^{\prime}=\llbracket\lambda^{\prime}_{1},\ldots,\lambda^{\prime}_{\lambda_{1}}\rrbracket$
where $\lambda^{\prime}_{i}$ is the largest integer such that
$\lambda_{\lambda^{\prime}_{i}}\geq i$.

Figure 1: The Young diagram of $\llbracket 6,4^{2},2,1^{2}\rrbracket$

If $i$ and $j$ are such that $\lambda_{i}>0$ and $0<j\leq\lambda_{i}$ then the
box in the $i$th row and $j$th column of the Young diagram is called the
$(i,j)$th box of the Young diagram. The _subpartition_ of $\lambda$
corresponding to the Young diagram rooted at the $(i,j)$th box of $\lambda$ is
$\lambda[i-1,j-1]=\llbracket\lambda_{i}-j+1,\lambda_{i+1}-j+1,\dots\rrbracket$,
where trailing nonpositive entries are removed. We define a _corner_ of a
partition $\lambda$ to be any subpartition $\lambda[i,j]$ (i.e. rooted at
$(i,j)$th box) which corresponds to the partition $\llbracket 1\rrbracket$. If
$i$ and $j$ are positive integers not satisfying the above conditions, then we
define $\lambda[i-1,j-1]$ to be the empty partition $\llbracket\rrbracket$.
Thus $\lambda[0,0]=\lambda$ for any partition. A key property of the
subpartitions is that for all non-negative integers $i_{1},i_{2},j_{1},j_{2}$
we have the identity

$\displaystyle(\lambda[i_{1},j_{1}])[i_{2},j_{2}]=\lambda[i_{1}+i_{2},j_{1}+j_{2}].$
(1)

For a given partition
$\lambda=\llbracket\lambda_{1},\lambda_{2},\dots\rrbracket$ we make use the
notion of _Durfee length_ which is defined as

$d(\lambda)=\max\\{\ell\mid\lambda_{\ell}\geq\ell\\}.$

Let $r$ and $c$ be positive integers. The _staircase_ , _rectangle_ , and
_gamma_ families of partitions are defined as

$\displaystyle\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r}$
$\displaystyle=\llbracket r,r-1,r-2,\ldots,1\rrbracket$
$\displaystyle\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{} {}{} {}{}{}{{}}{} {}{} {}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}$
$\displaystyle=\llbracket c^{r}\rrbracket$ $\displaystyle\leavevmode\hbox
to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}$
$\displaystyle=\llbracket c,1^{r-1}\rrbracket,$

respectively.

Figure 2: Partitions corresponding to $\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
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{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{6}$ and
$\leavevmode\hbox to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{6,4}$ and
$\leavevmode\hbox to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{6,4}$

###  2.2 Impartial games and Sprague-Grundy values

In this paper our focus will be exclusively on impartial finite combinatorial
games, which we will refer to simply as _games_. The positions in the games we
study are partitions or Young diagrams, which we describe in the following
subsection. A _subposition_ of a game is a position that can be reached after
a nonnegative number of moves. A combinatorial game is _short_ , if no
position can be repeated, and if there exist finitely many reachable
subpositions.

A _terminal position_ of a game is a position from which no further moves are
possible. Under normal play of a combinatorial game, a player who reaches a
terminal position is the winner. Under misère play, a player who reaches a
terminal position is the loser. Throughout this paper, we assume normal play
unless otherwise indicated.

Let $G$ be a game under normal (misère) play. We define the set of its winning
positions $\mathcal{N}_{G}$, and the set of its losing positions
$\mathcal{P}_{G}$, as follows:

  1. 1.

Terminal positions are in $\mathcal{P}_{G}$ ($\mathcal{N}_{G}$).

  2. 2.

A position is in $\mathcal{P}_{G}$, if none of its moves leads to a position
in $\mathcal{P}_{G}$.

  3. 3.

A position is in $\mathcal{N}_{G}$, if at least one of its moves leads to a
position in $\mathcal{P}_{G}$.

We denote the above sets simply as $\mathcal{P}$ and $\mathcal{N}$, whenever
the game in question is clear from the context.

A short two player combinatorial game is _impartial_ if both players have the
same possible moves in each position. Let $A$ be a position of such a game
$G$. If there is a move from $A$ to position $B$, we write $A\to B$. For a
finite set $S$ of nonnegative integers the minimum excluded value of $S$ is
defined as the smallest nonnegative integer that is not in $S$ and is denoted
by $\operatorname{mex}(S)$. The _Sprague-Grundy_ value ($\mathbb{SG}$ value)
of $A$ is defined recursively by

$\displaystyle\mathbb{SG}(A)=\operatorname{mex}(\\{\mathbb{SG}(B)\mid A\to
B\\}).$ (2)

A position of $\mathbb{SG}$ value $t$ is called a $t$-position. For the case
of terminal positions we have $\operatorname{mex}(\emptyset)=0$, hence such
positions have $\mathbb{SG}$ value $0$.

$\mathbb{SG}$ values provide a useful mechanism for determining winning and
losing positions for short impartial games under normal play. In particular,
the $\mathcal{P}$-positions are precisely those that have $\mathbb{SG}$ value
$0$ while the $\mathcal{N}$-positions are those with nonzero $\mathbb{SG}$
values. The Sprague-Grundy theorem [9] shows that given the sum of impartial
games under normal play, we can replace every game with a pile of NIM, with
the size corresponding to the $\mathbb{SG}$ value of the game. Doing so will
not change the $\mathcal{P}/\mathcal{N}$-positions of the sum.

For normal play, one can partition the positions to equivalence classes with
respect to their $\mathbb{SG}$ value. For misère play such equivalence
classes, for which we can replace one game with another within the sum, are
different and far more difficult to characterize in general. The equivalence
classes for misère are called misère quotients and form commutative monoid
[8]. Hence while one can study misère quotients, one can also focus on just
$\mathcal{P}/\mathcal{N}$-positions.

We make use of the following corollary, which follows from the definition of
$\mathbb{SG}$ value. For detailed coverage of combinatorial game theory we
refer the reader to [9].

######  Corollary 1.

Let $A,B$ be positions such that $A\to B$. Then
$\mathbb{SG}(A)\neq\mathbb{SG}(B)$.

###  2.3 Complexity theory

We provide some technical definitions regarding complexity in the context of
impartial games that will be used later. Let $G$ be an impartial game and let
$p$ be a fixed starting position, which we sometimes refer to as $G(p)$. The
corresponding _game tree_ $T_{p}$ is a rooted tree obtained by the following
procedure.

  * •

The original position $p$ is the root of $T_{p}$.

  * •

For each subposition $\tilde{p}$ with $p\to\tilde{p}$, there is a directed
edge from $p$ to the root of $T_{\tilde{p}}$.

We distinguish several parameters related to $G$ and $p$:

  1. 1.

Number of leaves of the game tree $T_{p}$ is denoted by
$\text{\scriptsize\faLeaf}(T_{p})$, while number of nodes of the game tree
$T_{p}$ is denoted by $\leavevmode\hbox to6.49pt{\vbox
to8pt{\pgfpicture\makeatletter\hbox{\hskip 3.24544pt\lower-3.24544pt\hbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(T_{p})$. We extend
those notations by defining
$\text{\scriptsize\faLeaf}(G(p))=\text{\scriptsize\faLeaf}(T_{p})$, and
similarly $\leavevmode\hbox to6.49pt{\vbox
to8pt{\pgfpicture\makeatletter\hbox{\hskip 3.24544pt\lower-3.24544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-0.56908pt}{4.55273pt}\pgfsys@lineto{2.27637pt}{3.9836pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-1.28297pt}{-1.28297pt}\pgfsys@moveto{1.28297pt}{1.28297pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.28297pt}{-1.28297pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(G(p))=\leavevmode\hbox
to6.49pt{\vbox to8pt{\pgfpicture\makeatletter\hbox{\hskip
3.24544pt\lower-3.24544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-0.56908pt}{4.55273pt}\pgfsys@lineto{2.27637pt}{3.9836pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-1.28297pt}{-1.28297pt}\pgfsys@moveto{1.28297pt}{1.28297pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.28297pt}{-1.28297pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(T_{p})$.

  2. 2.

A _height_ of $T_{p}$ is the length of the longest path from $p$. Provided
that $T_{p}$ has finite height, the height of any proper subtree of $T_{p}$ is
at least one less than the height of $T_{p}$.

  3. 3.

We say that subpositions $\tilde{p}$ and $\bar{p}$ of $p$ are _distinct_ if
$T_{\tilde{p}}$ is not isomorphic to $T_{\bar{p}}$. The _state-space
complexity_ of a game is the number of pairwise distinct game positions
reachable from the initial position $p$ of the game.

A game $G$ is said to be _finite_ if $T_{p}$ has finite height for any
position $p$ of $G$. Observe that $G$ is short, if and only if
$\leavevmode\hbox to6.49pt{\vbox to8pt{\pgfpicture\makeatletter\hbox{\hskip
3.24544pt\lower-3.24544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-0.56908pt}{4.55273pt}\pgfsys@lineto{2.27637pt}{3.9836pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-1.28297pt}{-1.28297pt}\pgfsys@moveto{1.28297pt}{1.28297pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.28297pt}{-1.28297pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(T_{p})$ is finite for
all positions $p$. _Computational complexity_ of a game is the time required
to decide whether a given initial position is winning.

##  3 LCTR and Downright

We now describe the core objects of this paper, games LCTR and Downright.
After providing definitions in Sections 3.1 and 3.2, we mention several
important properties of both games in Section 3.3.

###  3.1 LCTR

A position in the _leftmost column—topmost row_ , or LCTR game (see [5]), is
denoted by $\mathcal{L}(\lambda)$, where $\lambda$ is a partition, i.e., a
non-increasing list of $r$ positive integers
$\lambda=\llbracket\lambda_{1},\lambda_{2},\ldots,\lambda_{r}\rrbracket$ with
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{r}$.

The _terminal position_ occurs when $r=0$, which we denote by
$\mathcal{L}(\llbracket\rrbracket)$. Let
$A\neq\mathcal{L}(\llbracket\rrbracket)$ be a non-terminal position. There are
two possible moves a player can make, leading to a new position $\tilde{A}$,
namely:

  * •

$\tilde{A}=\mathcal{L}(\lambda[1,0])$, or

  * •

$\tilde{A}=\mathcal{L}(\lambda[0,1])$.

Given two non-negative integers $i<r$ and $j\leq\lambda_{i+1}$, or
$(i,j)=(r,0)$, we denote the subposition reached from $A$ by $i$ top row moves
and $j$ left column moves as $A[i,j]$, formally
$A[i,j]=\mathcal{L}(\lambda[i,j]).$

###  3.2 Downright

We now introduce a new game named _Downright_ , which is simpler than LCTR and
is a good introduction to the study of LCTR. Normal play in Downright is
equivalent to misère play in LCTR in a way that is made more precise in
Section 4. In Downright, a “restricted” rook chess piece begins in the top-
left box of a Young diagram. The rook can move a single adjacent box down or
right. Under normal play, then, the player who gets the rook to a corner,
where no moves remain, is the winner. There are similarities with Impartial
chess played with a King [1, 2], but in Downright we cannot move diagonally
and the board is a (not necessarily rectangular) partition.

Let us define the game of Downright $\mathcal{D}(\lambda)$, for a given
partition
$\lambda=\llbracket\lambda_{1},\lambda_{2},\ldots,\lambda_{r}\rrbracket$.
There are at most two moves a player can make. In particular, the move
resulting in the subposition $\mathcal{D}(\lambda[1,0])$ is allowed whenever
$r>1$, the move resulting in subposition $\mathcal{D}(\lambda[0,1])$ is
allowed whenever $\lambda_{1}>1$. Given two non-negative integers $i<r$ and
$j<\lambda_{i+1}$, we denote the subposition reached from $A$ by the rook
moving $j$ boxes to the right and $i$ boxes down as $A[i,j]$, formally
$A[i,j]=\mathcal{D}(\lambda[i,j])$.

$\lambda\to\lambda[0,1]$$\lambda\to\lambda[1,0]$ Figure 3: The partition
$\lambda=\llbracket 8,7,6,5^{2},2,1\rrbracket$ along with two subpartitions
$\lambda[1,0]=\llbracket 7,6,5^{2},2,1\rrbracket$ and $\lambda[0,1]=\llbracket
7,6,5,4^{2},1\rrbracket$ that correspond to moves in LCTR or Downright.

###  3.3 Observations regarding LCTR and Downright

Let $\lambda$ be a partition, let $i,j$ be nonnegative integers such that
$\lambda[i,j]$ is defined and non-empty. Within the game tree of either
$\mathcal{L}(\lambda)$ or $\mathcal{D}(\lambda)$, a position corresponding to
$\lambda[i,j]$ occurs several times111At least $\binom{i+j}{j}$ times.  but
its $\mathbb{SG}$ value need only be computed once. Furthermore, in either
game we can capture the $\mathbb{SG}$ values of every position by only storing
the values $\mathbb{SG}(A[i,j])$. To show this graphically we write the value
$\mathbb{SG}(A[i,j])$ inside the box of the Young diagram for $A$, located on
coordinates $(i+1)$-th row and $(j+1)$-th column (see e.g. Figure 4).

0002101011021010102110102010211010110000000000000210101102101010211010201021101011000000000000000000000000111111111112221000110000000000000011111111111222100011000000000000001111111111122210001100000000000000111111111112221000110000000000000011111111111222100011000000000000001111111111122210001100000000000000111111111112221000110000000000000011111111111222100011
Figure 4: $\mathbb{SG}$ values for subpositions in games of LCTR (left) and
Downright (right), played on $\lambda=\llbracket 8,7,6,5^{2},2,1\rrbracket$.
The zeros that appear outside of the Young diagram for $\mathcal{L}(\lambda)$
correspond to terminal positions.

Since each position admits at most two moves, we simplify the notation for the
minimum excluded value and also use $a\,\triangledown\,b$ instead of
$\operatorname{mex}(\\{a,b\\})$. According to Equation 2, we can compute the
$\mathbb{SG}$ value of $A=\mathcal{L}(\lambda)$ as follows:

$\mathbb{SG}(A[i,j])=\begin{cases}\mathbb{SG}(A[i+1,j])\,\triangledown\,\mathbb{SG}(A[i,j+1])&\text{if
}A[i,j]\neq\llbracket\rrbracket\\\ 0&\text{otherwise}\end{cases}.$ (3)

Similarly, we can compute the $\mathbb{SG}$ value of $A=\mathcal{D}(\lambda)$
as follows:

$\displaystyle\mathbb{SG}(A[i,j])=\begin{cases}0&\text{if
}\lambda[i,j]=\llbracket 1\rrbracket\\\
\operatorname{mex}\\{\mathbb{SG}(A[i,j+1])\\}&\text{if
}\lambda[i,j]=\llbracket c\rrbracket\text{ for }c\geq 2\\\
\operatorname{mex}\\{\mathbb{SG}(A[i+1,j])\\}&\text{if
}\lambda[i,j]=\llbracket 1^{r}\rrbracket\text{ for }r\geq 2\\\
\mathbb{SG}(A[i,j+1])\,\triangledown\,\mathbb{SG}(A[i,j+1])&\text{otherwise}\end{cases}.$
(4)

See Figure 3 for an example.

We now describe $\mathbb{SG}$ values for some families of games introduced in
Section 2.1.

######  Lemma 2.

For a positive integer $n$ we have that

$\mathbb{SG}(\mathcal{L}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{n}))=n\bmod
2\quad\text{and}\quad\mathbb{SG}(\mathcal{D}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{n}))=(n-1)\bmod 2.$

###### Proof.

Let $A_{n}=\mathbb{SG}(\mathcal{L}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{n}))$ or
$A_{n}=\mathbb{SG}(\mathcal{D}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{n}))$. Due to
Equations 3 and 4, we have

$\displaystyle\mathbb{SG}(A_{n})$
$\displaystyle=\mathbb{SG}(A_{n-1})\,\triangledown\,\mathbb{SG}(A_{n-1}).$

Hence in both cases the $\mathbb{SG}$ values must alternate between 0 and 1 as
$n$ increments by 1. The difference is that $\llbracket\rrbracket$ is a
terminal position for LCTR and has $\mathbb{SG}$ value $0$, while
$\leavevmode\hbox to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}=\llbracket
1\rrbracket$ is a terminal position for Downright with
$\mathbb{SG}(\mathcal{D}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-4.55254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{1}))=0$. ∎

######  Lemma 3.

For positive integers $r$ and $c$ we have

$\displaystyle\mathbb{SG}(\mathcal{L}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}))$
$\displaystyle=\begin{cases}0&c>1\mbox{ and }r>1\\\ 1&r=1\mbox{ and }c\mbox{
is odd, or }c=1\mbox{ and }r\mbox{ is odd}\\\ 2&\text{otherwise}\end{cases}$
(5)

and

$\displaystyle\mathbb{SG}(\mathcal{D}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}))$
$\displaystyle=\begin{cases}0&\text{if both }c\text{ and }r\text{ are odd}\\\
2&\text{if they are of different parity and }c>1\text{ and }r>1\\\
1&\text{otherwise}\\\ \end{cases}.$ (6)

###### Proof.

As Equation 5 is already shown in Ilić [5] we only consider the case of
Downright. If $c=1$ or $r=1$, then induction shows that $\mathbb{SG}(A)$ is 0
if $cr$ is odd and 1 otherwise. Hence $\mathbb{SG}(A)$ and $cr$ are of
distinct parity. If $c>1$ and $r>1$, then by definition,

$\mathbb{SG}(A)=\mathbb{SG}(A[1,0])\,\triangledown\,\mathbb{SG}(A[0,1])=\begin{cases}0&\text{if
}r\text{ and }c\text{ are odd}\\\ 1&\text{if }r\text{ and }c\text{ are
even}\\\ 2&\text{otherwise}\end{cases}.\qed$

######  Observation 4.

Let $A$ be a position in any impartial game under normal play. Then
$\mathbb{SG}(A)$ does not exceed the number of allowable moves from position
$A$.

Due to Equation 1, every permissible sequence with the same number of left
column and top row moves leads to the same position.

We observe that the $\mathbb{SG}$ values for LCTR and Downright are
conjugation-invariant, as made precise in the following lemma.

######  Lemma 5.

Let $\lambda$ be a partition. Then in LCTR or Downright, we have

  1. 1.

$\lambda^{\prime}[i,j]=\lambda[j,i]^{\prime}$ and also

  2. 2.

$\mathbb{SG}(\mathcal{L}(\lambda^{\prime}))=\mathbb{SG}(\mathcal{L}(\lambda))$
and
$\mathbb{SG}(\mathcal{D}(\lambda^{\prime}))=\mathbb{SG}(\mathcal{D}(\lambda))$
.

###### Proof.

Let $\lambda=\llbracket\lambda_{1},\lambda_{2},\dots,\lambda_{r}\rrbracket$ be
a partition and let
$\lambda^{\prime}=\llbracket\lambda^{\prime}_{1},\dots,\lambda^{\prime}_{\lambda_{1}}\rrbracket$
be its conjugate. We first show Item 1. It is clear that the claim holds for
values of $i$ and $j$ that result in an empty partition. Let us assume, then,
that $i\in\\{0,1,\ldots,r-1\\}$ and $j\in\\{0,1,\ldots,\lambda_{i+1}-1\\}$.

On the left side we have the positive prefix of
$\llbracket\lambda^{\prime}_{i+1}-j,\lambda^{\prime}_{i+2}-j,\ldots,\lambda^{\prime}_{\lambda_{1}}-j\rrbracket$
which corresponds to

$\llbracket\lambda^{\prime}_{i+1}-j,\lambda^{\prime}_{i+2}-j,\ldots,\lambda^{\prime}_{\lambda_{j+1}}-j\rrbracket.$

On the right side we have the conjugate of the positive prefix of
$\llbracket\lambda_{j+1}-i,\lambda_{j+2}-i,\ldots,\lambda_{r}-i\rrbracket$
which corresponds to

$\llbracket\lambda_{j+1}-i,\lambda_{j+2}-i,\ldots,\lambda_{\lambda^{\prime}_{i+1}}-i\rrbracket^{\prime}=\llbracket\lambda^{\prime}_{i+1}-j,\lambda^{\prime}_{i+2}-j,\ldots,\lambda^{\prime}_{\lambda_{j+1}}-j\rrbracket,$

as desired.

We now show Item 2. Let $A$ be either $\mathcal{L}(\lambda)$, or
$\mathcal{D}(\lambda)$. We divide the proof into two cases.

  1. 1.

If $\min(r,\lambda_{1})=1$: By the symmetry of the roles of $r$ and $c$ in
Lemma 3, we have

$\displaystyle\mathbb{SG}(\mathcal{D}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}))$
$\displaystyle=\mathbb{SG}(\mathcal{D}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{c,r}))=\mathbb{SG}(\mathcal{D}(\leavevmode\hbox
to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}^{\prime}))\text{
and }$ $\displaystyle\mathbb{SG}(\mathcal{L}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}))$
$\displaystyle=\mathbb{SG}(\mathcal{L}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{c,r}))=\mathbb{SG}(\mathcal{L}(\leavevmode\hbox
to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}^{\prime})).$

The claim holds for this case, since this corresponds to either $c=1$ or
$r=1$.

  2. 2.

($\min(r,\lambda_{1})\geq 2$): We show the claim for $\min(r,\lambda_{1})\geq
2$ by induction on $\tau=r+\lambda_{1}$, i.e. the total number of rows and
columns. This, together with Equations 3 and 4 implies $\tau\geq 4$ and also

$\displaystyle\mathbb{SG}(A)=\mathbb{SG}(A[1,0])\,\triangledown\,\mathbb{SG}(A[0,1]).$
(7)

Now suppose that the claim holds for all values less than $\tau$. Observe that

$\displaystyle\mathbb{SG}(A^{\prime})=\mathbb{SG}(A^{\prime}[1,0])\,\triangledown\,\mathbb{SG}(A^{\prime}[0,1])=\mathbb{SG}(A[0,1]^{\prime})\,\triangledown\,\mathbb{SG}(A[1,0]^{\prime}),$
(8)

where the former equality follows from Equation 7 while the latter equality is
obtained from Item 1 of Lemma 5 above. Since both
$\mathbb{SG}(A[0,1])^{\prime}$ as well as $\mathbb{SG}(A[1,0])^{\prime}$ have
the sum of columns and rows strictly less than $\tau$ it follows from the
induction hypothesis or from Item 1 of this proof, that

$\mathbb{SG}(A^{\prime})=\mathbb{SG}(A[0,1]^{\prime})\,\triangledown\,\mathbb{SG}(A[1,0]^{\prime})=\mathbb{SG}(A[0,1])\,\triangledown\,\mathbb{SG}(A[1,0])=\mathbb{SG}(A).\qed$

##  4 LCTR misère play and Downright

As discussed in Section 2.2, it is in general difficult to analyze misère play
of a given combinatorial game. Nevertheless, in this section we show that
misère LCTR can be analyzed using Sprague-Grundy theory. In fact, Downright is
misère LCTR without its terminal positions.

######  Lemma 6.

Let $\lambda\neq\llbracket\rrbracket$ be a partition. Then $\lambda$
corresponds to a $\mathcal{P}$-position in misère LCTR if and only if it
corresponds to a $\mathcal{P}$-position in Downright (under normal play).

###### Proof.

By the definition of $\mathcal{P}$ and $\mathcal{N}$ positions (Section 2.2),
all empty subpositions $\llbracket\rrbracket$ in misère LCTR belong to
$\mathcal{N}$ while all corners are in $\mathcal{P}$. For Downright, we have
that all corners are in $\mathcal{P}$. Recall that empty partition
$\llbracket\rrbracket$ is not valid input partition for the game Downright.

Now consider either of the games (misère LCTR or Downright) played on a row
$\llbracket c\rrbracket$, $c>1$, and observe that such a game belongs to
$\mathcal{N}$ if and only if the same game played on $(c-1)$ belongs to
$\mathcal{P}$. The same arguments hold for the case of the column game, when
either game is played on the partition $\llbracket 1^{r}\rrbracket,r>1$. We
conclude that the membership in $\mathcal{N}$ or $\mathcal{P}$ coincides for
both games, whenever played on a column, or on a row partition.

Consider now an arbitrary partition $\lambda\neq\llbracket\rrbracket$ and
consider both games played on $\lambda$. We already established that in the
cases of subpartitions $\llbracket c\rrbracket$ or $\llbracket
1^{r}\rrbracket$, both belong to $\mathcal{P}$ or $\mathcal{N}$ in the same
way. Since the definition of $\mathcal{P}$ and $\mathcal{N}$ (Section 2.2)
computes membership of positions from both games recursively in the same way,
this concludes the proof of the claim. ∎

Lemma 6 motivates the following theorem, from which it can be derived as a
corollary.

######  Definition 7 (truncation).

Let $G$ be a finite impartial game starting on position $p$. The _truncation_
of $G$ is an impartial game $\textnormal{trunc}(G)$ obtained from $G$ as
follows:

  * •

The starting position of $\textnormal{trunc}(G)$ corresponds to $p$.

  * •

$\textnormal{trunc}(G)$ has the same moves as $G$ except moves $A\to B$ where
$B$ is a terminal position of $G$ no longer exist.

We will assume $\textnormal{trunc}(G)$ is played under normal play, unless
stated otherwise.

######  Theorem 8.

The $\mathcal{P}$-positions of any finite impartial misère game exactly
correspond to those of its truncation.

###### Proof.

Let $G$ be a finite impartial misère game, let $H=\textnormal{trunc}(G)$, and
denote the sets of their $\mathcal{P}$-positions by
$\mathcal{P}_{G},\mathcal{P}_{H}$, respectively. Note that the deleted
$G$-terminal (i.e. terminal in $G$) positions are not $\mathcal{P}_{G}$, by
definition of misère, which supports our claim. All the remaining positions
exist in both $G$ and $H$. Since $G$ is a finite game, for every position $A$,
all of its subpositions have smaller height than $A$. Hence we can establish
$\mathcal{P}_{G}=\mathcal{P}_{H}$ by induction on the height of the game tree
of $G$.

First we show that the claim holds for all subpositions $A$ such that $A$ is
of height $1$ in $G$. The only moves from $A$ are to $G$-terminal position
which are all $\mathcal{N}_{G}$, thus $A\in\mathcal{P}_{G}$ follows. Since $A$
is $H$-terminal and $H$ is under normal play, it holds trivially that
$A\in\mathcal{P}_{H}$.

Now assume that the claim holds for all subpositions reachable from $A$. Let
$\mathcal{T}$ denote the terminal positions to which we can move in $G$ from
$A$, and let $\mathcal{B}$ correspond to the set of non-terminal positions
reachable from $A$ in one move. Recall that
$\mathcal{T}\subseteq\mathcal{N}_{G}$. We consider two cases depending on
whether $A$ is $\mathcal{P}_{G}$ or $\mathcal{N}_{G}$.

  * •

Let $A\in\mathcal{P}_{G}$. Then we have $\mathcal{B}\subseteq\mathcal{N}_{G}$
by the definition of $\mathcal{P}$ and $\mathcal{N}$ positions (Section 2.2).
By induction hypothesis we have for all $\mathcal{B}\subseteq\mathcal{N}_{H}$.
Since $A$ can only move to positions $\mathcal{B}$ in $H$, we have that
$A\in\mathcal{P}_{H}$.

  * •

Let $A\in\mathcal{N}_{G}$. Then there exists $B\in\mathcal{B}$ s.t.
$B\in\mathcal{P}_{G}$ by the definition of $\mathcal{P}$ and $\mathcal{N}$. By
induction hypothesis we have that $B\in\mathcal{P}_{H}$. Since position $A$
can move to position $B$ in $H$, we have that $A\in\mathcal{N}_{H}$.∎

The above theorem directly implies that $\mathcal{N}$-positions also coincide
except on the deleted terminal positions of the original game, as they do not
exist in the truncation. We find it noteworthy that the truncation of LCTR is
played on the same Young diagram under the rules of Downright. Indeed, since
the terminal positions of any LCTR game $G$ are “located” outside its Young
diagram, removing them does not alter the diagram for $\textnormal{trunc}(G)$.

One might also consider the misère variant of Downright in which case its
terminal positions are the corners of the corresponding Young diagram. In this
setting, truncated Downright turns out to be equivalent to normal play
Downright on an altered Young diagram where all corners are removed.

Lemma 6 further motivates the study of the $\mathbb{SG}$ values of Downright,
which is the focus of this section. We now show how $\mathbb{SG}$ values
propagate along diagonals, as seen in Figure 4 (right). To simplify the
notation further, we write $\bcancel{b}$ for an arbitrary integer from the set
$\\{0,1,2\\}\setminus\\{b\\}$. This allows us to simplify some computations
such as $\bcancel{0}\,\triangledown\,\bcancel{0}=0$, or
$0\,\triangledown\,\bcancel{1}=1$.

######  Lemma 9.

Let $\lambda$ be a partition and let $i,j>1$ be integers such that
$A=\mathcal{D}(\lambda)$ and $A[i,j]$ are positions in Downright. Then
$\mathbb{SG}(A[i-1,j-1])=\mathbb{SG}(A[i,j])$.

###### Proof.

Due to 4 it is enough to consider three cases depending on the value of
$\mathbb{SG}(A[i-1,j-1])$.

  1. 1.

Suppose $\mathbb{SG}(A[i,j])=0$. Then $A[i-1,j]$ and $A[i,j-1]$ must both be
$\mathcal{N}$-positions since we can calculate them via the formula
$0\,\triangledown\,b\neq 0$ where $b\in\\{0,1,2\\}$ and therefore
$\mathbb{SG}(A[i-1,j-1])=A[i-1,j]\,\triangledown\,A[i,j-1]=\bcancel{0}\,\triangledown\,\bcancel{0}=0.$

  2. 2.

Suppose $\mathbb{SG}(A[i,j])=1$. Then
$0\in\\{\mathbb{SG}(A[i,j+1]),\mathbb{SG}(A[i+1,j])\\}$. By Item 1 we have
that $0\in\\{\mathbb{SG}(A[i-1,j]),\mathbb{SG}(A[i,j-1])\\}$. Due to Corollary
1 we have that $1\not\in\\{\mathbb{SG}(A[i-1,j]),\mathbb{SG}(A[i,j-1])\\}$ and
thus $\mathbb{SG}(A[i-1,j-1])=0\,\triangledown\,\bcancel{1}=1.$

  3. 3.

Suppose $\mathbb{SG}(A[i,j])=2$. Then it follows
$\\{\mathbb{SG}(A[i+1,j]),\mathbb{SG}(A[i,j+1])\\}=\\{0,1\\}$. By Items 1 and
2 it follows $\\{\mathbb{SG}(A[i,j-1]),\mathbb{SG}(A[i-1,j])\\}=\\{0,1\\}$ and
therefore $\mathbb{SG}(A[i-1,j-1])=1\,\triangledown\,0=2$.∎

Due to Lemma 9, we can now state the main theorem of this section.

######  Theorem 10.

Let $\lambda$ be a partition, let $A=\mathcal{D}(\lambda)$ be a game of
Downright and let $\ell=d(\lambda)$. Then

$\mathbb{SG}(A)=\mathbb{SG}(A[1,1])=\mathbb{SG}(A[2,2])=\dots=\mathbb{SG}(A[\ell-1,\ell-1]).$

To put things in perspective, to calculate the $\mathbb{SG}$ value of a game
of Downright, by Theorem 10 we can calculate the subpartition at the end of
the diagonal, which turns out to be always be a gamma partition, which we know
how to calculate by Lemma 3.

##  5 LCTR normal play

In this section we focus on LCTR as described in Section 3.1. We start by
showing how to obtain $\mathbb{SG}$ values of small games, namely for LCTR
games of Durfee length at most three. We postpone the analysis of
computational complexity until Section 6.

###  5.1 LCTR games of Durfee length at most two

Let $\lambda$ be a partition and let $A=\mathcal{L}(\lambda)$. If
$d(\lambda)=0$, then $\lambda=\llbracket\rrbracket$ with $\mathbb{SG}(A)=0$.
The case when $d(\lambda)=1$ is handled by Lemma 3; see [5].

If $d(\lambda)=2$, then we can write
$\lambda=\llbracket\lambda_{1},\lambda_{2},2^{a},1^{b}\rrbracket$ where
$\lambda_{1}\geq\lambda_{2}\geq 2$, for some non-negative integers $a,b$. We
resolve $\mathbb{SG}(A)$ in two phases (also see Figure 6):

  1. 1.

First we obtain values
$\mathbb{SG}(A[2,0]),\mathbb{SG}(A[2,1]),\mathbb{SG}(A[0,2]),\mathbb{SG}(A[1,2])$.

  2. 2.

Given those boundary values we resolve the remaining $\mathbb{SG}$ values,
namely $\mathbb{SG}(A[1,0])$, $\mathbb{SG}(A[0,1])$, $\mathbb{SG}(A[1,1])$,
and finally $\mathbb{SG}(A[0,0])=\mathbb{SG}(A)$.

To resolve the $\mathbb{SG}$ values mentioned in Item 1, we use the following
lemma from [5].

######  Lemma 11 (one-row game).

Let $A=\mathcal{L}(\lambda)$ with $\lambda=\llbracket\lambda_{1}\rrbracket$.
Then

$\mathbb{SG}(A)=\begin{cases}0&\text{if $\lambda_{1}=0$}\\\ 1&\text{if
$\lambda_{1}$ is odd}\\\ 2&\text{otherwise}\end{cases}.$

Clearly one may obtain $\mathbb{SG}(A[2,1])$ and $\mathbb{SG}(A[1,2])$ by
using Lemma 11 together with input partition $\llbracket a\rrbracket$, or
$\llbracket\lambda_{2}-2\rrbracket$, respectively. In the former case we used
the fact that $\mathbb{SG}(\mathcal{L}(\llbracket
a\rrbracket))=\mathbb{SG}(\mathcal{L}(\llbracket 1^{a}\rrbracket))$, due to
Lemma 5.

######  Lemma 12 (two-row game).

Let $A=\mathcal{L}(\lambda)$ with
$\lambda=\llbracket\lambda_{1},\lambda_{2}\rrbracket$.

  1. 1.

If $\lambda_{1}=\lambda_{2}$ then $\mathbb{SG}(A)=\begin{cases}0&\text{if
}\lambda_{1}\text{ is even}\\\ 2&\text{if }\lambda_{1}\text{ is
odd}\end{cases}.$

  2. 2.

If $\lambda_{1}>\lambda_{2}>0$ then $\mathbb{SG}(A)=\begin{cases}0&\text{if
}\lambda_{2}\text{ is odd}\\\ 1&\text{if }\lambda_{2}\text{ is
even}\end{cases}.$

###### Proof.

Let $0\leq i\leq\lambda_{2}$. Throughout this proof keep in mind that
$\mathbb{SG}(A[1,i])=\mathbb{SG}(\mathcal{L}(\llbracket i\rrbracket))$ can be
obtained due to Lemma 11.

Now recall $\mathbb{SG}(A[0,\lambda_{1}])=0$ and consider the case
$\lambda_{1}=\lambda_{2}$. It thus follows that
$\mathbb{SG}(A[0,\lambda_{2}-1])=1\,\triangledown\,0=2$, and similarly
$\mathbb{SG}(A[0,\lambda_{2}-2])=2\,\triangledown\,2=0$. In particular, those
two $\mathbb{SG}$ values alternate through subgames $A[0,j]$ where $0\leq
j<\lambda_{2}$, yielding the desired formula.

Finally consider the case $\lambda_{1}>\lambda_{2}>0$. It follows from Lemma
11 that $\mathbb{SG}(A[0,\lambda_{2}])=\bcancel{0}$ and thus
$\mathbb{SG}(A[0,\lambda_{2}-1])=1\,\triangledown\,\bcancel{0}=0$, and
similarly $\mathbb{SG}(A[0,\lambda_{2}-2])=2\,\triangledown\,0=1$. In
particular, those two $\mathbb{SG}$ values alternate through subgames $A[0,j]$
where $0\leq j<\lambda_{2}$, yielding the desired formula. ∎

Lemmas 11 and 12 can now be used to obtain the remaining values
$\mathbb{SG}(A[2,0])$ and $\mathbb{SG}(A[0,2])$, together with input partition
$\llbracket a+b,b\rrbracket$, or
$\llbracket\lambda_{1}-2,\lambda_{2}-2\rrbracket$, respectively. In the former
case we again used Lemma 5 to obtain $\mathbb{SG}(\mathcal{L}(\llbracket
a+b,b\rrbracket))=\mathbb{SG}(\mathcal{L}(\llbracket 2^{a},1^{b}\rrbracket))$.

We now resolve the remaining $\mathbb{SG}$ values as announced in Item 2. One
may now use Equation 3 to express those $\mathbb{SG}$ values as

$\displaystyle\mathbb{SG}(A[1,1])$
$\displaystyle=\mathbb{SG}(A[2,1])\,\triangledown\,\mathbb{SG}(A[1,2])$
$\displaystyle\mathbb{SG}(A[0,1])$
$\displaystyle=\mathbb{SG}(A[0,2])\,\triangledown\,\mathbb{SG}(A[1,1])$
$\displaystyle\mathbb{SG}(A[1,0])$
$\displaystyle=\mathbb{SG}(A[2,0])\,\triangledown\,\mathbb{SG}(A[1,1])$
$\displaystyle\mathbb{SG}(A)$
$\displaystyle=\mathbb{SG}(A[0,1])\,\triangledown\,\mathbb{SG}(A[1,0])$ (9)

For simplicity, call the $\mathbb{SG}$ values of the subgames $A[2,0],A[2,1]$
by $\alpha_{20},\alpha_{21}$ and the subgames $A[0,2],A[1,2]$ by
$\alpha_{02},\alpha_{12}$, respectively. By definition, we may express the
$\mathbb{SG}$ value of game $A$ as

$\displaystyle\mathbb{SG}(A)$
$\displaystyle=\left(\left(\alpha_{21}\,\triangledown\,\alpha_{12}\right)\,\triangledown\,\alpha_{02}\right)\,\triangledown\,\left(\alpha_{20}\,\triangledown\,\left(\alpha_{21}\,\triangledown\,\alpha_{12}\right)\right).$
(10)

Lemma 11
---
12121$\cdots$ Lemma 12
---
(case 1)
01212$\cdots$ Lemma 12
---
(case 2)
$\bcancel{0}$121$\cdots$ Lemma 13
---
(case 1)
02020$\cdots$ Lemma 13
---
(case 2)
$\bcancel{0}$202$\cdots$ Lemma 13
---
(case 3)
$\bcancel{1}$010$\cdots$ Lemma 13
---
(case 4)
010$\cdots$0 Lemma 13
---
(case 5)
0$\bcancel{0}$010$\cdots$ Figure 5: Cases from Lemmas 11, 12 and 13. The
symbol $\cdots$ denotes alternating values from the two boxes right from it.

###  5.2 LCTR games of Durfee length three

Let $\lambda$ be a partition such that $d(\lambda)=3$, and let
$A=\mathcal{L}(\lambda)$. We write
$\lambda=\llbracket\lambda_{1},\lambda_{2},\lambda_{3},3^{a},2^{b},1^{c}\rrbracket$
where $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq 3$. We again resolve
$\mathbb{SG}(A)$ in the following two phases:

  1. 1.

First we obtain values $\mathbb{SG}(A[3,0])$, $\mathbb{SG}(A[3,1])$,
$\mathbb{SG}(A[3,2])$, $\mathbb{SG}(A[0,3])$, $\mathbb{SG}(A[1,3])$,
$\mathbb{SG}(A[2,3])$.

  2. 2.

Given the values from Item 1 we resolve the remaining nine $\mathbb{SG}$
values, namely

$\displaystyle\mathbb{SG}(A[2,2]),$ $\displaystyle\mathbb{SG}(A[1,2]),$
$\displaystyle\mathbb{SG}(A[0,2]),$ $\displaystyle\mathbb{SG}(A[2,1]),$
$\displaystyle\mathbb{SG}(A[1,1]),$ $\displaystyle\mathbb{SG}(A[0,1]),$ (11)
$\displaystyle\mathbb{SG}(A[2,0]),$ $\displaystyle\mathbb{SG}(A[1,0]),$
$\displaystyle\mathbb{SG}(A[0,0])=\mathbb{SG}(A).$

To resolve the $\mathbb{SG}$ values mentioned in Item 1 notice that values
$\mathbb{SG}(A[3,1])$, $\mathbb{SG}(A[3,2])$, $\mathbb{SG}(A[1,3])$, and
$\mathbb{SG}(A[2,3])$ may be obtained by using Lemmas 11 and 12, together with
partitions $\llbracket b+c,c\rrbracket,$ $\llbracket c\rrbracket,$
$\llbracket\lambda_{2}-3,\lambda_{3}-3\rrbracket$, and
$\llbracket\lambda_{3}-3\rrbracket$, respectively.

To obtain $\mathbb{SG}(A[3,0])$ and $\mathbb{SG}(A[0,3])$ we introduce the
following lemma.

######  Lemma 13 (three-row game).

Let $A=\mathcal{L}(\lambda)$ with
$\lambda=\llbracket\lambda_{1},\lambda_{2},\lambda_{3}\rrbracket$.

  1. 1.

If $\lambda_{1}=\lambda_{2}=\lambda_{3}>0$ then
$\mathbb{SG}(A)=\begin{cases}0&\text{if }\lambda_{3}\geq 3\text{ and odd}\\\
1&\text{if }\lambda_{3}=1\text{ or }(\lambda_{3}\geq 4\text{ and even})\\\
2&\text{if }\lambda_{3}=2\\\ \end{cases}.$

  2. 2.

If $\lambda_{1}>\lambda_{2}=\lambda_{3}>0$ then
$\mathbb{SG}(A)=\begin{cases}0&\lambda_{3}\text{ is odd}\\\
1&\lambda_{3}\text{ is even}\\\ \end{cases}.$

  3. 3.

If $\lambda_{1}=\lambda_{2}>\lambda_{3}>0$ then
$\mathbb{SG}(A)=\begin{cases}0&\lambda_{3}\text{ is even}\\\
1&\lambda_{3}\text{ is odd}\\\ \end{cases}.$

  4. 4.

if $\lambda_{1}>\lambda_{2}>\lambda_{3}=1$ then
$\mathbb{SG}(A)=\begin{cases}1&\lambda_{2}\text{ is even}\\\
2&\lambda_{2}\text{ is odd}\\\ \end{cases}.$

  5. 5.

if $\lambda_{1}>\lambda_{2}>\lambda_{3}>1$ then
$\mathbb{SG}(A)=\begin{cases}0&\lambda_{3}\text{ is even}\\\
1&\lambda_{3}\text{ is odd}\\\ \end{cases}.$

###### Proof.

We resolve the above-mentioned five cases separately, relying heavily on Lemma
12. Recall from Lemma 12 that the $\mathbb{SG}$ values of positions $A[1,j]$
with $0\leq j<\lambda_{3}$ alternate between $0$ and $2$ whenever
$\lambda_{2}=\lambda_{3}$, or between $1$ and $2$ otherwise. Also observe that
the value $\mathbb{SG}(A[0,\lambda_{3}])$ can be obtained from either Lemma 11
or Lemma 12.

  1. 1.

($\lambda_{1}=\lambda_{2}=\lambda_{3}>0$): We have
$\mathbb{SG}(A[0,\lambda_{3}])=0$, while the $\mathbb{SG}$ values of subgames
$A[1,j]$ with $0\leq j<\lambda_{3}$ alternate between $2$ and $0$, starting
with $\mathbb{SG}(A[1,\lambda_{3}-1])=2$.

  2. 2.

($\lambda_{1}>\lambda_{2}=\lambda_{3}>0$): We have
$\mathbb{SG}(A[0,\lambda_{3}])=\bcancel{0}$ while the $\mathbb{SG}$ values of
subgames $A[1,j]$ with $0\leq j<\lambda_{3}$ alternate between $2$ and $0$,
starting with $\mathbb{SG}(A[1,\lambda_{3}-1])=2$.

  3. 3.

($\lambda_{1}=\lambda_{2}>\lambda_{3}>0$): We have
$\mathbb{SG}(A[0,\lambda_{3}])=\bcancel{1}$, while the $\mathbb{SG}$ values of
subgames $A[1,j]$ with $0\leq j<\lambda_{3}$ alternate between $1$ and $0$,
starting with $\mathbb{SG}(A[1,\lambda_{3}-1])=0$.

  4. 4.

($\lambda_{1}>\lambda_{2}>\lambda_{3}=1$): We have
$\mathbb{SG}(A[0,1])=\mathbb{SG}(\mathcal{L}(\llbracket\lambda_{1}-1,\lambda_{2}-1\rrbracket))=\lambda_{2}\bmod
2$, while the $\mathbb{SG}$ value of subgame $\mathbb{SG}(A[1,0])=0$.

  5. 5.

($\lambda_{1}>\lambda_{2}>\lambda_{3}>1$): Due to Item 2 of Lemma 12 and
Corollary 1 we have $\mathbb{SG}(A[1,\lambda_{3}-1])=0$,
$\mathbb{SG}(A[1,\lambda_{3}-2])=1$ and also
$\mathbb{SG}(A[0,\lambda_{3}-1])=\bcancel{0}$. This gives
$\mathbb{SG}(A[0,\lambda_{3}-2])=0$. The $\mathbb{SG}$ values of subgames
$A[1,j]$ with $0\leq j\leq\lambda_{3}-2$ alternate between $1$ and $0$,
starting with $\mathbb{SG}(A[0,\lambda_{3}-2])=0$.

From the $\mathbb{SG}$ values of subgames $A[1,j]$ with $0\leq
j\leq\lambda_{3}-1$, we derive $\mathbb{SG}$ values of subgames $A[0,j]$ with
$0\leq j\leq\lambda_{3}-1$ and obtain the desired result for each case. ∎

Lemmas 13, 12 and 11, together with partitions
$\llbracket\lambda_{1}-3,\lambda_{2}-3,\lambda_{3}-3\rrbracket$ and
$\llbracket a+b+c,b+c,c\rrbracket$ resolve $\mathbb{SG}(A[0,3])$ and
$\mathbb{SG}(A[3,0])$, respectively. The remaining nine Sprage-Grundy values
mentioned in Equation 11 can, similarly as in Equation 9, be expressed as

$\displaystyle\mathbb{SG}(A[2,2])$
$\displaystyle=\mathbb{SG}(A[2,3])\,\triangledown\,\mathbb{SG}(A[3,2]),$
$\displaystyle\mathbb{SG}(A[2,1])$
$\displaystyle=\mathbb{SG}(A[2,2])\,\triangledown\,\mathbb{SG}(A[3,1]),$
$\displaystyle\mathbb{SG}(A[2,0])$
$\displaystyle=\mathbb{SG}(A[2,1])\,\triangledown\,\mathbb{SG}(A[3,0]),$
$\displaystyle\mathbb{SG}(A[1,2])$
$\displaystyle=\mathbb{SG}(A[1,3])\,\triangledown\,\mathbb{SG}(A[2,2]),$
$\displaystyle\mathbb{SG}(A[1,1])$
$\displaystyle=\mathbb{SG}(A[1,2])\,\triangledown\,\mathbb{SG}(A[2,1]),$
$\displaystyle\mathbb{SG}(A[1,0])$
$\displaystyle=\mathbb{SG}(A[1,1])\,\triangledown\,\mathbb{SG}(A[2,0]),$
$\displaystyle\mathbb{SG}(A[0,2])$
$\displaystyle=\mathbb{SG}(A[0,3])\,\triangledown\,\mathbb{SG}(A[1,2]),$
$\displaystyle\mathbb{SG}(A[0,1])$
$\displaystyle=\mathbb{SG}(A[0,2])\,\triangledown\,\mathbb{SG}(A[1,1]),$

and finally

$\mathbb{SG}(A)=\mathbb{SG}(A[0,1])\,\triangledown\,\mathbb{SG}(A[1,0]).$ (12)

$\scriptstyle\alpha_{03}$$\scriptstyle\alpha_{13}$$\scriptstyle\alpha_{23}$$\scriptstyle\alpha_{30}$$\scriptstyle\alpha_{31}$$\scriptstyle\alpha_{32}$
Figure 6: Resolving games played on $\llbracket 3,1^{4}\rrbracket$,
$\llbracket 5,3,2^{2},1\rrbracket$, and $\llbracket 6,5,3^{2},2\rrbracket$.

For simplicity, call the $\mathbb{SG}$ values of the subgames
$A[3,0],A[3,1],A[3,2]$ by $\alpha_{30},\alpha_{31},\alpha_{32}$ and the
subgames $A[0,3],A[1,3],A[2,3]$ by $\alpha_{03},\alpha_{13},\alpha_{23}$,
respectively. As in Equation 10, we express the $\mathbb{SG}$ value of game
$A$ as

$\displaystyle{\left(\left(\alpha_{30}\,\triangledown\,\left(\alpha_{31}\,\triangledown\,\left(\alpha_{32}\,\triangledown\,\alpha_{23}\right)\right)\right)\,\triangledown\,\left(\left(\alpha_{31}\,\triangledown\,\left(\alpha_{32}\,\triangledown\,\alpha_{23}\right)\right)\,\triangledown\,\left(\left(\alpha_{32}\,\triangledown\,\alpha_{23}\right)\,\triangledown\,\alpha_{13}\right)\right)\right)}\phantom{.}\quad\triangledown$
$\displaystyle{\left(\left(\left(\alpha_{31}\,\triangledown\,\left(\alpha_{32}\,\triangledown\,\alpha_{23}\right)\right)\,\triangledown\,\left(\left(\alpha_{32}\,\triangledown\,\alpha_{23}\right)\,\triangledown\,\alpha_{13}\right)\right)\,\triangledown\,\left(\left(\left(\alpha_{32}\,\triangledown\,\alpha_{23}\right)\,\triangledown\,\alpha_{13}\right)\,\triangledown\,\alpha_{03}\right)\right)}.\quad\phantom{\star}$
(13)

###  5.3 LCTR games of Durfee length at least four

We now give a bit more complicated version of the propagation from Lemma 9,
which holds for LCTR normal play.

######  Lemma 14.

Let $\lambda$ be a partition, let $A=\mathcal{L}(\lambda)$ be a game of LCTR,
and let $i,j\geq 1$ be such that $\lambda[i,j]\neq\llbracket\rrbracket$.

  1. 1.

If $\mathbb{SG}(A[i,j])=0$, then $d(\lambda[i,j])\geq 1$ and
$\mathbb{SG}(A[i-1,j-1])=0$.

  2. 2.

If $\mathbb{SG}(A[i,j])=1$ and $d(\lambda[i,j])\geq 2$, then
$\mathbb{SG}(A[i-1,j-1])=1$.

  3. 3.

If $\mathbb{SG}(A[i,j])=2$ and $d(\lambda[i,j])\geq 3$, then
$\mathbb{SG}(A[i-1,j-1])=2$.

###### Proof.

We have that $\lambda[i^{\prime},j^{\prime}]\neq\llbracket\rrbracket$ for any
choice of $0\leq i^{\prime}<d(\lambda)$ and $0\leq j^{\prime}<d(\lambda)$. In
particular, for any corresponding subgame $A[i^{\prime},j^{\prime}]$ we may
use the first case of recursion in Equation 3. We consider three cases
depending on the value of $\mathbb{SG}(A[i,j])$.

  1. 1.

Suppose $\mathbb{SG}(A[i,j])=0$. Due to Corollary 1 we have

$\displaystyle\mathbb{SG}(A[i-1,j])\neq 0\neq\mathbb{SG}(A[i,j-1]).$

Therefore
$\mathbb{SG}(A[i-1,j-1])=A[i-1,j]\,\triangledown\,A[i,j-1]=\bcancel{0}\,\triangledown\,\bcancel{0}=0$.
Since $\lambda[i,j]\neq\llbracket\rrbracket$, clearly $d(\lambda[i,j])\geq 1$.

  2. 2.

Suppose $\mathbb{SG}(A[i,j])=1$ and $d(\lambda[i,j])\geq 2$, in particular
neither $\lambda[i,j+1]$ nor $\lambda[i+1,j]$ is empty. First notice that by
Corollary 1 we have

$\displaystyle\mathbb{SG}(A[i,j-1])$ $\displaystyle\neq
1\neq\mathbb{SG}(A[i-1,j]).$

This implies
$\mathbb{SG}(A[i-1,j-1])=\bcancel{1}\,\triangledown\,\bcancel{1}\leq 1$. Now
assume $\mathbb{SG}(A[i-1,j-1])=0$. From Corollaries 1 and 4 it follows that

$\displaystyle\mathbb{SG}(A[i,j-1])=\mathbb{SG}(A[i-1,j])=2,\text{ and}$
$\displaystyle\mathbb{SG}(A[i+1,j-1])=\mathbb{SG}(A[i-1,j+1])=0.$

Since neither $\lambda[i,j+1]$ nor $\lambda[i+1,j]$ is empty, by Corollary 1
we have

$\mathbb{SG}(A[i+1,j])=\mathbb{SG}(A[i,j+1])=2.$

This leads to $\mathbb{SG}(A[i,j])=2\,\triangledown\,2\neq 1,$ a
contradiction.

  3. 3.

Suppose $\mathbb{SG}(A[i,j])=2$ and $d(\lambda[i,j])\geq 3$. Due to
$2\,\triangledown\,0=1$ it follows that

$\\{\mathbb{SG}(A[i,j+1]),\mathbb{SG}(A[i+1,j])\\}=\\{0,1\\}.$

Since $d(A[i,j])\geq 3$ we have that $d(A[i,j+1])\geq 2$ and $d(A[i+1,j])\geq
2$. In addition, Items 1 and 2 imply that

$\\{\mathbb{SG}(A[i-1,j]),\mathbb{SG}(A[i,j-1])\\}=\\{0,1\\}.$

Hence $\mathbb{SG}(A[i-1,j-1])=0\,\triangledown\,1=2$. ∎

The following corollary will be used in Section 6, for efficient computation
of $\mathbb{SG}$ values.

$\scriptstyle\alpha_{03}$$\scriptstyle\alpha_{13}$$\scriptstyle\alpha_{23}$$\scriptstyle\alpha_{30}$$\scriptstyle\alpha_{31}$$\scriptstyle\alpha_{32}$
Figure 7: Resolving the games LCTR (left) and Downright (right), played on
$\lambda=\llbracket 12,11,9,7,6,5,3,1\rrbracket$. The gray boxes are filled
with the same $\mathbb{SG}$ values.

######  Corollary 15.

Let $\lambda$ be a partition, let $\ell=d(\lambda)-3\geq 0$, and let
$A=\mathcal{L}(\lambda)$. Then

$\mathbb{SG}(A)=\mathbb{SG}(A[\ell,\ell]).$

###### Proof.

Let $\bar{A}=A[\ell,\ell]$. Since $\ell\geq 0$ it follows that the position
$\bar{A}$ is properly defined and we have $d(\bar{A})=3$. By Lemmas 14 and 4,
regardless of the value of $\mathbb{SG}(\bar{A})$ we infer

$\mathbb{SG}(\bar{A})=\mathbb{SG}(A[\ell-1,\ell-1])=\cdots=\mathbb{SG}(A[\ell-\ell,\ell-\ell])=\mathbb{SG}(A).\qed$

In view of this corollary we restrict ourselves to the study of games with
Durfee length at most 3. As an application of this corollary we now compute
$\mathbb{SG}(\llbracket c^{r}\rrbracket)$.

######  Lemma 16.

For positive integers $c$ and $r$, we have

$\mathbb{SG}(\mathcal{L}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{} {}{} {}{}{}{{}}{} {}{} {}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}))=\begin{cases}0&\text{if
}c>1\mbox{ and }r>1\mbox{ and }c+r\mbox{ is even}\\\ 2&\text{if }c\leq 2\mbox{
or }r\leq 2,\mbox{ and }c+r\mbox{ is odd}\\\ 1&\text{otherwise}\end{cases}.$

###### Proof.

By Lemma 5 we can assume $c\geq r$, otherwise we can take its conjugate which
has this property. If $r=1$, then the formula holds by Lemma 11. If $r=2$,
then the result holds by Item 1 of Lemma 12. If $r\geq 3$ then by Corollary 15
the game $\mathbb{SG}(\mathcal{L}(\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{} {}{} {}{}{}{{}}{} {}{} {}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{c-r+3,3}))=\mathbb{SG}(\mathcal{L}(\leavevmode\hbox
to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{} {}{} {}{}{}{{}}{} {}{} {}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{c,r}))$ and this
game can be solved with Item 1 of Lemma 13. ∎

##  6 The complexity of LCTR and Downright

In this section we discuss computational aspects of both LCTR and Downright.
Let $n=\sum_{i=1}^{r}\lambda_{i}$, where
$\lambda=\llbracket\lambda_{1},\dots,\lambda_{r}\rrbracket$ is the input
partition for either game. Complexity results will be expressed in terms of
$\lambda_{1}$, $r$, and $n$. In Section 6.1, we discuss the time complexity
required to compute $\mathbb{SG}$-values of LCTR and Downright. In Section
6.2, we study the number of nodes and leaves in their game trees. In Section
6.3, we examine the state-space complexity of these games.

###  6.1 Time complexity of computing $\mathbb{SG}$ values

The results of this section will be expressed in terms of the number of time
units needed. We consider the _time unit_ to be any of the following:

  1. 1.

Reading an integer. Note that our input partition
$\lambda=\llbracket\lambda_{1},\dots,\lambda_{r}\rrbracket$ consists of a
sequence of $r$ integers, so reading the input takes $r$ time units.

  2. 2.

Basic arithmetic operations. Here we mean integer multiplication or division
by $2$, addition or subtraction of the given integers, or computing the
parity.

  3. 3.

Minimum excluded value: We consider computing a minimal excluded value to be a
basic operation which takes one time unit. Note that mex computations will
only be performed with integers from the set $\\{0,1,2\\}$.

For a given partition
$\lambda=\llbracket\lambda_{1},\lambda_{2},\dots,\lambda_{r}\rrbracket$ we
show that the $\mathbb{SG}$ value of both LCTR and Downright, played on
$\lambda$, can be determined $O(\log(r))\leq O(\log(n))$ time units. For
comparison, the dynamic programming approach by [5] required $\Theta(n)$ time
units for the same task.

######  Lemma 17.

Given a partition
$\lambda=\llbracket\lambda_{1},\ldots,\lambda_{r}\rrbracket\vdash n$ we can
find its Durfee length $d(\lambda)$ in $O(\log(r))\leq O(\log(n))$ time units.

###### Proof.

Recall that the Durfee length is the largest $i\in\\{1,\ldots,r\\}$ such that
$\lambda_{i}\geq i$. We can find such a value in $\log(r)$ time units by using
binary search. ∎

######  Lemma 18.

Given a partition
$\lambda=\llbracket\lambda_{1},\ldots,\lambda_{r}\rrbracket\vdash n$ we can
compute its conjugate $\lambda^{\prime}$ in $O(\lambda_{1}\log(r))\leq
O(n\log(n))$ time units.

###### Proof.

For every $i\in\\{1,\ldots,\lambda_{1}\\}$ we find $\lambda^{\prime}_{i}$ via
a binary search on the list
$\llbracket\lambda_{1},\ldots,\lambda_{r}\rrbracket$ which takes $O(\log(r))$
time units each. Hence, in total we need $O(\lambda_{1}\log(r))$ time units. ∎

######  Theorem 19.

Let $\lambda=\llbracket\lambda_{1},\lambda_{2},\dots,\lambda_{r}\rrbracket$ be
a partition of $n$. The $\mathbb{SG}$ value of $\mathcal{D}(\lambda)$ can be
computed in $O(\log(r))\leq O(\log(n))$ time units.

###### Proof.

We can compute the Durfee length $d(\lambda)$ in $O(\log(r))$ time units due
to Lemma 17. Let
$\bar{\lambda}=\llbracket\bar{\lambda}_{1},\bar{\lambda}_{2},\ldots,\bar{\lambda}_{\bar{r}}\rrbracket$
be a partition corresponding to the smallest “diagonal subgame,” i.e.
$A[d(\lambda)-1,d(\lambda)-1]=\mathcal{D}(\llbracket\bar{\lambda}_{1},\bar{\lambda}_{2},\ldots,\bar{\lambda}_{\bar{r}}\rrbracket)$.
Due to Theorem 10 it is enough to compute $\mathbb{SG}$ value of
$\mathcal{D}(\bar{\lambda})$, as it corresponds of that of $K(\lambda)$.

To this end observe that, by construction, $\bar{\lambda}$ is of shape
$\leavevmode\hbox to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{a,b}$. Here
$a=\lambda_{d(\lambda)}-d(\lambda)+1$, while integer $b$ can be computed via
binary search in $O(\log r)$ time units.222Note that we never need to
explicitly compute $\bar{\lambda}$, as this would cause the computational
complexity to grow. It remains to verify, by Lemma 3, that the $\mathbb{SG}$
value of $\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{a,b}$ can be
computed in $O(1)$ time units. Summarizing the time complexity, the
computation of $\mathbb{SG}$ value of the game of Downright requires
$O(\log(r))$ time units. ∎

######  Theorem 20.

Let $\lambda=\llbracket\lambda_{1},\lambda_{2},\dots,\lambda_{r}\rrbracket$ be
a partition of $n$. The Sprague-Grundy value of $\mathcal{L}(\lambda)$ can be
computed in $O(\log(r))\leq O(\log(n))$ time units.

###### Proof.

Let $A=\mathcal{L}(\lambda)$. Due to Lemma 17, we can compute the Durfee
length $d(\lambda)$ in $O(\log(r))$ time units. We distinguish three cases,
depending on the Durfee length of $\lambda$:

  * •

Case $d(\lambda)=1$. Notice that $\lambda$ is of gamma shape, and the
corresponding $\mathbb{SG}$ value can be computed in $O(1)$ time units due to
Lemma 3.

  * •

Case $d(\lambda)=2$. Denote the lengths of the first two columns by $r$ and
$j$, i.e. $j$ is the maximal integer such that $\lambda_{j}\geq 2$. We need
$O(\log(r))$ time units to compute $j$ via binary search.

We express $\mathbb{SG}(A)$ as a function of $\alpha_{02}$, $\alpha_{20}$,
$\alpha_{12}$, $\alpha_{21}$, where $\alpha_{ij}=\mathbb{SG}(A[i,j])$. We
calculate those four values as a function of $\lambda_{1},\lambda_{2},r$ and
$j$, by using Equation 10 where

$\displaystyle\alpha_{02}$
$\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket\lambda_{1}-2,\lambda_{2}-2\rrbracket))$
$\displaystyle\alpha_{20}$ $\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket
r-2,j-2\rrbracket))$ $\displaystyle\alpha_{12}$
$\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket\lambda_{2}-2\rrbracket))$
$\displaystyle\alpha_{21}$ $\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket
j-2\rrbracket)).$

The computation of $\alpha_{02},\alpha_{20},\alpha_{12},\alpha_{21}$ can be
done in $O(1)$ time units; see Lemmas 11 and 12. Afterwards the application of
Equation 9 can be performed in constantly many time units, so the overall
computation time is $O(\log(r))$ time units.

  * •

Case $d(\lambda)\geq 3$. Finally, let
$\bar{\lambda}=\llbracket\bar{\lambda}_{1},\bar{\lambda}_{2},\ldots,\bar{\lambda}_{\bar{r}}\rrbracket$
be a partition so that $\bar{\lambda}=\lambda[d(\lambda)-3,d(\lambda)-3]$. We
do not compute $\bar{\lambda}$ but whenever we check a value
$\bar{\lambda}_{\ell}$ we instead use
$\bar{\lambda}_{\ell}=\lambda_{d(\lambda)+\ell-3}-d(\lambda)+3$.

It is enough to compute $\mathbb{SG}$ value of $\mathcal{L}(\bar{\lambda})$ as
it corresponds of that of $\mathcal{L}(\lambda)$, due to Corollary 15.

Let $\bar{r},j$ and $i$ be the (possibly zero) lengths of the first three
columns of $\bar{\lambda}$, respectively. Formally, let $\bar{r},j$ and $i$ be
maximum integers such that $\bar{\lambda}_{\bar{r}}\geq 1$, and
$\bar{\lambda}_{j}\geq 2$, and $\bar{\lambda}_{i}\geq 3$, respectively. We
need $O(\log(r))$ time units to compute $\bar{r},j$ and $i$. The value of
$\mathbb{SG}(A)$ can now be computed in $O(1)$ time units by using Equation
13, where

$\displaystyle\alpha_{03}$
$\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket\bar{\lambda}_{1}-3,\bar{\lambda}_{2}-3,\bar{\lambda}_{3}-3\rrbracket))$
$\displaystyle\alpha_{30}$
$\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket\bar{r}-3,j-3,i-3\rrbracket))$
$\displaystyle\alpha_{13}$
$\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket\bar{\lambda}_{2}-3,\bar{\lambda}_{3}-3\rrbracket))$
$\displaystyle\alpha_{31}$ $\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket
j-3,i-3\rrbracket))$ $\displaystyle\alpha_{23}$
$\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket\bar{\lambda}_{3}-3\rrbracket))$
$\displaystyle\alpha_{32}$ $\displaystyle=\mathbb{SG}(\mathcal{L}(\llbracket
i-3\rrbracket)).\qed$

###  6.2 The game tree of LCTR and Downright

In this section we describe the number of nodes and leaves that the game tree
of LCTR, or Downright, might have.

#### The number of nodes in the game tree

Recall that $\leavevmode\hbox to6.49pt{\vbox
to8pt{\pgfpicture\makeatletter\hbox{\hskip 3.24544pt\lower-3.24544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-0.56908pt}{4.55273pt}\pgfsys@lineto{2.27637pt}{3.9836pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-1.28297pt}{-1.28297pt}\pgfsys@moveto{1.28297pt}{1.28297pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.28297pt}{-1.28297pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(A)$, and
$\text{\scriptsize\faLeaf}(A)$, denote the number of nodes, and leaves, in the
game tree of the game $A$, respectively. The following corollary connects
those notions, and follows immediately from Definition 7.

######  Corollary 21.

Let $\lambda$ be a partition. Then $\leavevmode\hbox to6.49pt{\vbox
to8pt{\pgfpicture\makeatletter\hbox{\hskip 3.24544pt\lower-3.24544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-0.56908pt}{4.55273pt}\pgfsys@lineto{2.27637pt}{3.9836pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-1.28297pt}{-1.28297pt}\pgfsys@moveto{1.28297pt}{1.28297pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.28297pt}{-1.28297pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(\mathcal{L}(\lambda))=\leavevmode\hbox
to6.49pt{\vbox to8pt{\pgfpicture\makeatletter\hbox{\hskip
3.24544pt\lower-3.24544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-0.56908pt}{4.55273pt}\pgfsys@lineto{2.27637pt}{3.9836pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-1.28297pt}{-1.28297pt}\pgfsys@moveto{1.28297pt}{1.28297pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.28297pt}{-1.28297pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(\mathcal{D}(\lambda))+\text{\scriptsize\faLeaf}(\mathcal{L}(\lambda))$.

Given a game $A$ of either type, recall that a position may occur several
times within the game tree of $A$. More precisely, if
$A[i,j]\neq\llbracket\rrbracket$ then the pair $(i,j)$ gives rise to exactly
$\binom{i+j}{i}$ nodes within the game tree of $A$.

######  Lemma 22.

For positive integers $r,c$ we have $\leavevmode\hbox to6.49pt{\vbox
to8pt{\pgfpicture\makeatletter\hbox{\hskip 3.24544pt\lower-3.24544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-0.56908pt}{4.55273pt}\pgfsys@lineto{2.27637pt}{3.9836pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-1.28297pt}{-1.28297pt}\pgfsys@moveto{1.28297pt}{1.28297pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.28297pt}{-1.28297pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(\mathcal{D}(\leavevmode\hbox
to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{} {}{} {}{}{}{{}}{} {}{} {}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}))=\binom{r+c}{r}-1$.

###### Proof.

First observe that if $r=1$ or $c=1$, then $\leavevmode\hbox to7.23pt{\vbox
to7.23pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-7.02881pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{} {}{} {}{}{}{{}}{} {}{} {}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}=\leavevmode\hbox
to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{2.27626pt}{-4.55254pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-2.27626pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}$, thus the
claim holds (also see Table 1). The proof is by induction on $r+c$. Since the
base case was already covered, assume that the claim holds for all partitions
$\leavevmode\hbox to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{} {}{} {}{}{}{{}}{} {}{} {}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r^{\prime},c^{\prime}}$
with $r^{\prime}+c^{\prime}<\ell$ and let $r+c=\ell$. By the principle of
inclusion-exclusion first observe the following recurrence of
$\leavevmode\hbox to6.49pt{\vbox to8pt{\pgfpicture\makeatletter\hbox{\hskip
3.24544pt\lower-3.24544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4143pt}{0.0pt}\pgfsys@curveto{1.4143pt}{0.7811pt}{0.7811pt}{1.4143pt}{0.0pt}{1.4143pt}\pgfsys@curveto{-0.7811pt}{1.4143pt}{-1.4143pt}{0.7811pt}{-1.4143pt}{0.0pt}\pgfsys@curveto{-1.4143pt}{-0.7811pt}{-0.7811pt}{-1.4143pt}{0.0pt}{-1.4143pt}\pgfsys@curveto{0.7811pt}{-1.4143pt}{1.4143pt}{-0.7811pt}{1.4143pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}{}\pgfsys@moveto{-0.56908pt}{4.55273pt}\pgfsys@lineto{2.27637pt}{3.9836pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{-1.28297pt}{-1.28297pt}\pgfsys@moveto{1.28297pt}{1.28297pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.28297pt}{-1.28297pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\par
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(\mathcal{D}(\leavevmode\hbox
to7.23pt{\vbox to7.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-7.02881pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{} {}{} {}{} {}{}{}{{}}{}
{}{} {}{} {}{}{}{{}}{} {}{} {}{} {}{}{}{{}}{} {}{} {}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{-6.82881pt}\pgfsys@lineto{2.27626pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@lineto{6.82881pt}{-2.27626pt}\pgfsys@lineto{0.0pt}{-2.27626pt}\pgfsys@moveto{0.0pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-4.55254pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{0.0pt}{-6.82881pt}\pgfsys@moveto{4.55254pt}{0.0pt}\pgfsys@lineto{4.55254pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{-6.82881pt}\pgfsys@lineto{6.82881pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}_{r,c}))$, whenever
$\min(r,c)>1$:

$\displaystyle\leavevmode\hbox to6.49pt{\vbox
to8pt{\pgfpicture\makeatletter\hbox{\hskip 3.24544pt\lower-3.24544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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