# Combinatorics of integer partitions with
prescribed perimeter

Zhicong Lin  Research Center for Mathematics and Interdisciplinary Sciences,
Shandong University, Qingdao 266237, P.R. China<EMAIL_ADDRESS>,  Huan Xiong
Institute for Advanced Study in Mathematics, Harbin Institute of Technology,
Harbin 150001, P.R. China<EMAIL_ADDRESS>and  Sherry H.F. Yan
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, P.R.
China<EMAIL_ADDRESS>
###### Abstract.

We prove that the number of even parts and the number of times that parts are
repeated have the same distribution over integer partitions with a fixed
perimeter. This refines Straub’s analog of Euler’s Odd-Distinct partition
theorem. We generalize the two concerned statistics to these of the part-
difference less than $d$ and the parts not congruent to $1$ modulo $d+1$ and
prove a distribution inequality, that has a similar flavor as Alder’s ex-
conjecture, over partitions with a prescribed perimeter. Both of our results
are proved analytically and combinatorially.

###### Key words and phrases:

Integer partitions; Perimeter; Repeated parts; Even parts; Euler’s partition
theorem

##  1\. Introduction

Integer partitions [1] play important roles in combinatorics, number theory
and other related mathematical branches. For a positive integer $n$, a
sequence $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$ of weakly
decreasing positive integers such that $\sum_{i}\lambda_{i}=n$ is called a
partition of $n$. The $\lambda_{i}$’s are called the parts of $\lambda$ and
$k$ is the number of parts that we denote $\ell(\lambda)$. The perimeter of
$\lambda$ is defined to be

$\Gamma(\lambda)=\lambda_{1}+\ell(\lambda)-1$

and the minimal part-difference of $\lambda$ is

$\min\\{\lambda_{i}-\lambda_{i+1}:1\leq i<\ell(\lambda)\\}.$

One of the most famous results in partition theory is Euler’s Odd-Distinct
partition theorem, which asserts that the number of partitions of $n$ with odd
parts is equal to the number with distinct parts. Two beautiful bijective
proofs (see [4, pp. 63-65]) of Euler’s partition theorem with different
refinements were constructed respectively by Sylvester and Glaisher. Recently,
Straub [5, Thereom 1.4] considered the set $\mathcal{H}_{n}$ of partitions
with perimeter $n$, rather than the set of all partitions of $n$, and proved
the following analog of Euler’s partition theorem.

######  Theorem 1 (Straub).

There are as many partitions in $\mathcal{H}_{n}$ with odd parts as with
distinct parts.

For a fixed positive integer $d$, Alder’s ex-conjecture (see [1, Sec. 4.3] and
[2]) states that there are not more partitions of $n$ into parts congruent to
$\pm 1$ modulo $d+3$ than into parts with minimal part-difference at least
$d$. Indeed, Alder’s ex-conjecture is Euler’s partition theorem (resp.
Rogers–Ramanujan identity) for $d=1$ (resp. $d=2$). Inspired by Alder’s ex-
conjecture, Fu and Tang [2, Theorem 2.15] proved the following $d$-extension
of Straub’s result.

######  Theorem 2 (Fu–Tang).

There are as many partitions in $\mathcal{H}_{n}$ with parts congruent to $1$
modulo $d+1$ as with minimal part-difference at least $d$.

The purpose of this paper is to study further the combinatorics of integer
partitions with a prescribed perimeter after the aforementioned works by
Straub and Fu–Tang.

We are interested in two natural statistics on partitions. Define for a
partition $\lambda=(\lambda_{1},\ldots,\lambda_{k})$ the statistics

$\mathrm{rep}(\lambda)=|\\{1\leq
i\leq\ell(\lambda)-1:\lambda_{i}=\lambda_{i+1}\\}|\quad\text{and}\quad\mathrm{even}(\lambda)=|\\{1\leq
i\leq\ell(\lambda):\lambda_{i}\text{ is even}\\}|,$

called the number of repeated parts and the number of even parts,
respectively. Our first result refines Theorem 1 by the above two statistics.

######  Theorem 3.

For any positive integer $n$, we have

(1.1)
$\sum_{\lambda\in\mathcal{H}_{n}}t^{\mathrm{rep}(\lambda)}=\sum_{\lambda\in\mathcal{H}_{n}}t^{\mathrm{even}(\lambda)}.$

$(0,0)$$(0,1)$$(0,2)$$(0,1)$$(1,2)$$(1,0)$$(0,1)$$(1,2)$$(1,0)$$(1,1)$$(2,0)$$(2,1)$$(2,2)$$(2,3)$$(3,4)$$(4,0)$
Figure 1. Partitions in $\mathcal{H}_{5}$ (represented as Young diagrams) with
their pair of statistics $(\mathrm{rep},\mathrm{even})$ on the top.

See Fig. 1 for an example of (1.1) for $n=5$, which shows

$\sum_{\lambda\in\mathcal{H}_{5}}t^{\mathrm{rep}(\lambda)}=5+5t+4t^{2}+t^{3}+t^{4}=\sum_{\lambda\in\mathcal{H}_{5}}t^{\mathrm{even}(\lambda)}.$

Setting $t=0$ in (1.1) recovers Theorem 1. We will investigate the
combinatorics of Theorem 3 by providing three different proofs:

  1. (1)

a recurrence relation proof with connection to extraordinary subsets studied
by Grimaldi [3];

  2. (2)

a generating function proof with some related consequences;

  3. (3)

a bijective proof that admits an unexpected extension to general $d$ for our
next result.

Our next result can be considered as an extension of Theorem 2, as will be
seen (see Theorem 19) when exploiting the $d$-extension of our bijective proof
of Theorem 3.

######  Theorem 4.

Fix integers $d\geq 1$ and $n\geq 1$. The total number of indices $i$ with
$1\leq i<\ell(\lambda)$ satisfying $\lambda_{i}-\lambda_{i+1}<d$ in all
partitions $\lambda$ in $\mathcal{H}_{n}$ is not less than the total number of
parts not congruent to $1$ modulo $d+1$ in all partitions in
$\mathcal{H}_{n}$.

Again, three different proofs for Theorem 4 will be provided: a generating
function proof, an injective proof and a bijective proof.

The proofs of Theorems 3 and 4 are given in Sections 2 and 3, respectively.
This paper is concluded with further remarks in Section 4.

##  2\. Three proofs of Theorem 3

###  2.1. A recurrence relation proof

Let $A(n,k)$ (resp. $B(n,k)$) be the number of partitions $\lambda$ in
$\mathcal{H}_{n}$ with $\mathrm{rep}(\lambda)=k$ (resp.
$\mathrm{even}(\lambda)=k$). In the following two lemmas, we prove that
$A(n,k)$ and $B(n,k)$ share the same (binomial-like) recurrence relation,
which proves Theorem 3.

######  Lemma 5.

The number $A(n,k)$ satisfies the recurrence relation

(2.1) $A(n,k)=A(n-1,k)+A(n-1,k-1)+A(n-2,k)-A(n-2,k-1)$

for $n\geq 2$ with the initial value $A(1,0)=1$.

###### Proof.

Suppose that $n\geq 2$ and
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\ldots)$ is a partition in
$\mathcal{H}_{n}$ with $\mathrm{rep}(\lambda)=k$. To obtain the recursion
formula for $A(n,k)$, we consider three cases:

  1. (i)

$\lambda_{1}=\lambda_{2}$. In this case, by deleting the largest part of
$\lambda$, we obtain $\lambda^{\prime}=(\lambda_{2},\lambda_{3},\ldots)$. Then
$\lambda^{\prime}\in\mathcal{H}_{n-1}$ and
$\mathrm{rep}(\lambda^{\prime})=k-1$. Furthermore, $\lambda^{\prime}$ can be
any of the partitions satisfying $\lambda^{\prime}\in\mathcal{H}_{n-1}$ and
$\mathrm{rep}(\lambda^{\prime})=k-1$. Therefore, the number of partitions
$\lambda\in\mathcal{H}_{n}$ satisfying $\lambda_{1}=\lambda_{2}$ and
$\mathrm{rep}(\lambda)=k$ is equal to $A(n-1,k-1)$.

  2. (ii)

$\lambda_{1}=\lambda_{2}+1$. In this case, again by deleting the largest part
of $\lambda$, we obtain $\lambda^{\prime}=(\lambda_{2},\lambda_{3},\ldots)$.
It is clear that $\lambda^{\prime}\in\mathcal{H}_{n-2}$ and
$\mathrm{rep}(\lambda^{\prime})=k$. Furthermore, $\lambda^{\prime}$ can be any
of the partitions satisfying $\lambda^{\prime}\in\mathcal{H}_{n-2}$ and
$\mathrm{rep}(\lambda^{\prime})=k$. Therefore, the number of partitions
$\lambda\in\mathcal{H}_{n}$ satisfying $\lambda_{1}=\lambda_{2}+1$ and
$\mathrm{rep}(\lambda)=k$ is equal to $A(n-2,k)$.

  3. (iii)

$\lambda_{1}\geq\lambda_{2}+2$. In this case, by removing one box in the first
row of the Young diagram of $\lambda$, we obtain
$\lambda^{\prime}=(\lambda_{1}-1,\lambda_{2},\lambda_{3},\ldots)$. It is clear
that $\lambda^{\prime}\in\mathcal{H}_{n-1}$,
$\mathrm{rep}(\lambda^{\prime})=k$ and
${\lambda^{\prime}}_{1}\geq{\lambda^{\prime}}_{2}+1$. By (i), the number of
partitions $\lambda^{\prime}$ satisfying
$\lambda^{\prime}\in\mathcal{H}_{n-1}$, $\mathrm{rep}(\lambda^{\prime})=k$ and
${\lambda^{\prime}}_{1}={\lambda^{\prime}}_{2}$ is equal to $A(n-2,k-1)$.
Thus, the number of partitions $\lambda^{\prime}$ satisfying
$\lambda^{\prime}\in\mathcal{H}_{n-1}$, $\mathrm{rep}(\lambda^{\prime})=k$ and
${\lambda^{\prime}}_{1}\geq{\lambda^{\prime}}_{2}+1$ is equal to
$A(n-1,k)-A(n-2,k-1)$. Finally, this implies that the number of partitions
$\lambda\in\mathcal{H}_{n}$ satisfying $\lambda_{1}\geq\lambda_{2}+2$ and
$\mathrm{rep}(\lambda)=k$ is equal to $A(n-1,k)-A(n-2,k-1)$.

Combining the above three cases gives (2.1). ∎

######  Lemma 6.

The number $B(n,k)$ satisfies the recurrence relation

(2.2) $B(n,k)=B(n-1,k)+B(n-1,k-1)+B(n-2,k)-B(n-2,k-1)$

for $n\geq 2$ with the initial value $B(1,0)=1$.

###### Proof.

Suppose that $n\geq 2$ and
$\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\ldots)$ is a partition in
$\mathcal{H}_{n}$ with $\mathrm{even}(\lambda)=k$. Let $B_{o}(n,k)$ (resp.
$B_{e}(n,k)$) be the number of partitions $\lambda\in\mathcal{H}_{n}$ with
$\lambda_{1}$ odd (resp. even) and $\mathrm{even}(\lambda)=k$. To obtain the
recursion formula for $B(n,k)$ is more involved and we again consider three
cases:

  1. (i)

$\lambda_{1}=\lambda_{2}$. In this case, consider the deletion of the largest
part of $\lambda$, we see that the number of partitions
$\lambda\in\mathcal{H}_{n}$ satisfying $\lambda_{1}$ odd (resp. even) and
$\mathrm{even}(\lambda)=k$ is equal to $B_{o}(n-1,k)$ (resp.
$B_{e}(n-1,k-1)$).

  2. (ii)

$\lambda_{1}=\lambda_{2}+1$. In this case, again by considering the deletion
of the largest part of $\lambda$, we see that the number of partitions
$\lambda\in\mathcal{H}_{n}$ satisfying $\lambda_{1}$ odd (resp. even),
$\lambda_{1}=\lambda_{2}+1$ and $\mathrm{even}(\lambda)=k$ is equal to
$B_{e}(n-2,k)$ (resp. $B_{o}(n-2,k-1)$).

  3. (iii)

$\lambda_{1}\geq\lambda_{2}+2$. In this case, by removing two boxes in the
first row of the Young diagram of $\lambda$, we obtain
$\lambda^{\prime}=(\lambda_{1}-2,\lambda_{2},\lambda_{3},\ldots)$. It is clear
that $\lambda^{\prime}\in\mathcal{H}_{n-2}$ and
$\mathrm{even}(\lambda^{\prime})=k$. Furthermore, $\lambda^{\prime}$ can be
any of the partitions satisfying $\lambda^{\prime}\in\mathcal{H}_{n-2}$ and
$\mathrm{even}(\lambda^{\prime})=k$. Therefore, the number of partitions
$\lambda\in\mathcal{H}_{n}$ satisfying $\lambda_{1}$ odd (resp. even),
$\lambda_{1}\geq\lambda_{2}+2$ and $\mathrm{even}(\lambda)=k$ is equal to
$B_{o}(n-2,k)$ (resp. $B_{e}(n-2,k)$).

Combining the above three cases together, we derive that

(2.3)
$B(n,k)=B(n-2,k)+B_{o}(n-1,k)+B_{o}(n-2,k-1)+B_{e}(n-1,k-1)+B_{e}(n-2,k).$

On the other hand, by deleting the first part of $\lambda$ when
$\lambda_{1}=\lambda_{2}$ or by removing one box in the first row of the Young
diagram of $\lambda$ when $\lambda_{1}\geq\lambda_{2}+1$, we see that

(2.4) $B_{o}(n-1,k-1)=B_{o}(n-2,k-1)+B_{e}(n-2,k)$

and

(2.5) $B_{e}(n-1,k)=B_{e}(n-2,k-1)+B_{o}(n-2,k-1)=B(n-2,k-1).$

Now (2.3), (2.4) and (2.5) together results in (2.2), as desired. ∎

For positive integers $n$ and $k$ with $1\leq k\leq n$, a subset $S$ of
$[n]:=\\{1,2,\ldots,n\\}$ is called a $k$-extraordinary subset if $|S|$ equals
the $k$-th smallest element of $S$. For example, all the $2$-extraordinary
subsets of $[5]$ are

$\\{1,2\\},\,\,\\{1,3,4\\},\,\,\\{1,3,5\\},\,\,\\{2,3,4\\},\,\,\\{2,3,5\\}.$

Let $C(n,k)$ be the number of $k$-extraordinary subsets of $[n]$. In [3],
Grimaldi showed that

$C(n,k)=C(n-1,k)+\sum_{i=1}^{k}C(n-k-2+i,i)$

for $n\geq 2$ and the initial value $C(1,1)=1$. It follows from the above
recursion that

$\displaystyle C(n,k)$
$\displaystyle=C(n-1,k)+C(n-2,k)+\sum_{i=1}^{k-1}C(n-k-2+i,i)$
$\displaystyle=C(n-1,k)+C(n-2,k)+C(n-1,k-1)-C(n-2,k-1).$

Comparing with the recursions for $A(n,k)$ and $B(n,k)$ derived in the above
two lemmas (and their initial values) yields the following result.

######  Proposition 7.

For $n\geq 1$ and $0\leq k\leq n-1$, we have

$A(n,k)=B(n,k)=C(n,k+1).$

###  2.2. A generating function proof

We will compute the generating function for the joint distribution of
$(\mathrm{rep},\mathrm{even})$ on $\mathcal{H}_{n}$. For a partition
$\lambda$, let $\mathrm{dist}(\lambda)$ be the number of distinct parts of
$\lambda$. Then, $\mathrm{dist}(\lambda)=\ell(\lambda)-\mathrm{rep}(\lambda)$.
A partition $\lambda$ can also be represented as
$\lambda=1^{m_{1}}2^{m_{2}}\cdots$, where $m_{i}$ is the number of parts
equating $i$ of $\lambda$. Consider the weight function

$\mathrm{wt}(\lambda)=p^{\mathrm{dist}(\lambda)}q^{\mathrm{even}(\lambda)}x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots.$

Then,

(2.6)
$1+\sum_{\lambda\in\mathcal{P}}\mathrm{wt}(\lambda)=\biggl{(}1+\frac{px_{1}}{1-x_{1}}\biggr{)}\biggl{(}1+\frac{pqx_{2}}{1-qx_{2}}\biggr{)}\biggl{(}1+\frac{px_{3}}{1-x_{3}}\biggr{)}\biggl{(}1+\frac{pqx_{4}}{1-qx_{4}}\biggr{)}\cdots,$

where $\mathcal{P}$ denotes the set of all integer partitions. Let
$\mathcal{P}_{n}$ be the set of partitions
$\lambda=(\lambda_{1},\lambda_{2},\ldots)$ such that $\lambda_{1}=n$. Then,

(2.7)
$\sum_{\lambda\in\mathcal{P}_{2n}}\mathrm{wt}(\lambda)=\biggl{(}1+\frac{px_{1}}{1-x_{1}}\biggr{)}\biggl{(}1+\frac{pqx_{2}}{1-qx_{2}}\biggr{)}\cdots\biggl{(}1+\frac{px_{2n-1}}{1-x_{2n-1}}\biggr{)}\biggl{(}\frac{pqx_{2n}}{1-qx_{2n}}\biggr{)}$

and

(2.8)
$\sum_{\lambda\in\mathcal{P}_{2n+1}}\mathrm{wt}(\lambda)=\biggl{(}1+\frac{px_{1}}{1-x_{1}}\biggr{)}\biggl{(}1+\frac{pqx_{2}}{1-qx_{2}}\biggr{)}\cdots\biggl{(}1+\frac{pqx_{2n}}{1-qx_{2n}}\biggr{)}\biggl{(}\frac{px_{2n+1}}{1-x_{2n+1}}\biggr{)}.$

It follows from (2.7) and (2.8) that

$\displaystyle\quad\sum_{\lambda\in\mathcal{P}}p^{\mathrm{dist}(\lambda)}q^{\mathrm{even}(\lambda)}x^{\ell(\lambda)}y^{\lambda_{1}}$
$\displaystyle=\frac{pxy}{1-x}+\biggl{(}\frac{pqx}{1-qx}+\frac{pxy(1-qx+pqx)}{(1-x)(1-qx)}\biggr{)}\sum_{n\geq
1}\biggl{(}1+\frac{px}{1-x}\biggr{)}^{n}\biggl{(}1+\frac{pqx}{1-qx}\biggr{)}^{n-1}y^{2n}$
$\displaystyle=\frac{pxy}{1-x}+\frac{\biggl{(}\frac{pqx}{1-qx}+\frac{pxy(1-qx+pqx)}{(1-x)(1-qx)}\biggr{)}\biggl{(}1+\frac{px}{1-x}\biggr{)}y^{2}}{1-\biggl{(}1+\frac{px}{1-x}\biggr{)}\biggl{(}1+\frac{pqx}{1-qx}\biggr{)}y^{2}}.$

Therefore, we have

######  Theorem 8.

The generating function for partitions by size of first parts, number of parts
and the pair $(\mathrm{dist},\mathrm{even})$ is

(2.9)
$\sum_{\lambda\in\mathcal{P}}p^{\mathrm{dist}(\lambda)}q^{\mathrm{even}(\lambda)}x^{\ell(\lambda)}y^{\lambda_{1}}=\frac{pxy}{1-x}+\frac{\biggl{(}\frac{pqx}{1-qx}+\frac{pxy(1-qx+pqx)}{(1-x)(1-qx)}\biggr{)}\biggl{(}1+\frac{px}{1-x}\biggr{)}y^{2}}{1-\biggl{(}1+\frac{px}{1-x}\biggr{)}\biggl{(}1+\frac{pqx}{1-qx}\biggr{)}y^{2}}.$

Consequently ($p\leftarrow p^{-1},x\leftarrow px,y\leftarrow x$),

$\sum_{\lambda\in\mathcal{P}}p^{\mathrm{rep}(\lambda)}q^{\mathrm{even}(\lambda)}x^{\Gamma(\lambda)}=\frac{x(1-(p-1)q(x^{2}+x))}{1-p(1+q)x-(1-p^{2}q)x^{2}-(1-p)(1+q)x^{3}-(p-1)^{2}qx^{4}}.$

In particular,

(2.10)
$\sum_{\lambda\in\mathcal{P}}p^{\mathrm{rep}(\lambda)}x^{\Gamma(\lambda)}=\sum_{\lambda\in\mathcal{P}}p^{\mathrm{even}(\lambda)}x^{\Gamma(\lambda)}=\frac{x}{1-(1+p)x-(1-p)x^{2}}.$

######  Remark 9.

Dividing both sides of (2.9) by $x$ and then substituting $p\leftarrow
p^{-1},q\leftarrow 1,x\leftarrow-px,y\leftarrow x$ yields

(2.11)
$\sum_{\lambda\in\mathcal{P}}(-1)^{\ell(\lambda)}p^{\mathrm{rep}(\lambda)}x^{\Gamma(\lambda)}=\frac{x}{(p-1)(x^{2}-x)-1}.$

Thus, if we denote

$h_{n}(p)=\sum_{\Gamma(\lambda)=n}(-1)^{\ell(\lambda)}p^{\mathrm{rep}(\lambda)}=\sum_{\Gamma(\lambda)=n\atop\ell(\lambda)\text{
even}}p^{\mathrm{rep}(\lambda)}-\sum_{\Gamma(\lambda)=n\atop\ell(\lambda)\text{
odd}}p^{\mathrm{rep}(\lambda)},$

then $\sum_{n\geq 1}h_{n}(p)x^{n}=\frac{x}{(p-1)(x^{2}-x)-1}$, which is
equivalent to the recurrence relation

(2.12) $h_{n}(p)=(1-p)(h_{n-1}(p)-h_{n-2}(p))$

for $n\geq 3$ with initial values $h_{1}(p)=-1$ and $h_{2}(p)=p-1$. In
particular, we have

$h_{n}(0)=\begin{cases}(-1)^{m},\quad&\text{if $n=(6m-3\pm 1)/2$};\\\
0,&\text{otherwise}.\end{cases}$

This was first proved by Fu and Tang [2, Theroem 1.4], which is an analog to
Euler’s pentagonal number theorem (see [1, Sec. 3.5]). Two other special
evaluations of $h_{n}(p)$ deserve to be mentioned:

  * •

We have $h_{n}(1)=0$, which means that there are as many partitions in
$\mathcal{H}_{n}$ with odd number of parts as with even number of parts. It is
an interesting exercise to construct an involution proof of this simple fact.

  * •

$|h_{n}(2)|$ is equal to the $n$-th Fibonacci number $F_{n}$, which satisfies
$F_{n}=F_{n-1}+F_{n-2}$ for $n\geq 3$ and initial values $F_{1}=F_{2}=1$.

Let $a_{o}$ (resp. $a_{e}$) be the total number of odd (resp. even) parts in
all partitions in $\mathcal{H}_{n}$. It can be easily deduced from (2.9) the
following closed formulae for $a_{o}$ and $a_{e}$.

######  Corollary 10.

For all positive integer $n\geq 2$, we have

$a_{o}=(n+2)\cdot 2^{n-3}\qquad\text{and}\qquad a_{e}=n\cdot 2^{n-3}.$

The above result means that, the average difference between the number of odd
parts and the number of even parts in all partitions with perimeter $n$ is
equal to $1/2$.

###  2.3. A bijective proof

This subsection is devoted to a bijective proof of the equidistribution on
$\mathcal{H}_{n}$:

$\sum_{\lambda\in\mathcal{H}_{n}}t^{\mathrm{rep}(\lambda)}=\sum_{\lambda\in\mathcal{H}_{n}}t^{\mathrm{even}(\lambda)}.$

######  Theorem 11.

There exists a recursively defined bijection
$\phi:\mathcal{H}_{n}\rightarrow\mathcal{H}_{n}$ satisfying

$\mathrm{rep}(\lambda)=\mathrm{even}(\phi(\lambda))$

for every $\lambda\in\mathcal{H}_{n}$.

It is convenience to use the following representation of partitions as
$01$-sequences.

######  Lemma 12.

There exists a natural bijection $\lambda\mapsto
w(\lambda)=w_{0}w_{1}w_{2}\cdots w_{n}$ between partitions in $\mathcal{H}(n)$
and the set $\Lambda_{n}$ of $01$-sequences such that $w_{0}=0,w_{n}=1$.
Furthermore,

$\ell(\lambda)=|\\{1\leq i\leq n:w_{i}=1\\}|,$
$\mathrm{rep}(\lambda)=|\\{1\leq i\leq n:w_{i}=w_{i-1}=1\\}|$

and

$\mathrm{even}(\lambda)=|\\{1\leq i\leq n:w_{i}=1\text{ and }\\#\\{0\leq
j<i:w_{j}=0\\}\text{ is even}\\}|.$

$0$$0$$0$$1$$1$$1$$1$$0$$0$$0$ Figure 2. The 01-sequence of
$\lambda=(6,3,3,1)$ is $w(\lambda)=0100110001$.

###### Proof.

Visually, the $01$-sequence representation $w(\lambda)$ of a partition
$\lambda\in\mathcal{H}_{n}$ can be obtained as follows. For the edges in the
boundary of the Young diagram of $\lambda$, starting at the left bottom and
ending to the right top, we label the vertical (resp. horizontal) edges with 1
(resp. 0); see Fig. 2 for an example. In this way, we get a 01-sequence
$w(\lambda)\in\Lambda_{n}$. The three desired properties are obvious from this
correspondence. ∎

###### Proof of Theorem 11.

For each $w\in\Lambda_{n}$, denote

$\displaystyle\widetilde{\mathrm{rep}}(w)$ $\displaystyle=|\\{1\leq i\leq
n:w_{i}=w_{i-1}=1\\}|,$ $\displaystyle\widetilde{\mathrm{even}}(w)$
$\displaystyle=|\\{1\leq i\leq n:w_{i}=1\text{ and }\\#\\{0\leq
j<i:w_{j}=0\\}\text{ is even}\\}|.$

In view of Lemma 12, we aim to define recursively a bijection
$\tilde{\phi}:\Lambda_{n}\rightarrow\Lambda_{n}$ satisfying

(2.13)
$\widetilde{\mathrm{rep}}(w)=\widetilde{\mathrm{even}}(\tilde{\phi}(w))$

for each $w\in\Lambda_{n}$ and then set $\phi=w^{-1}\circ\tilde{\phi}\circ w$.
Set $\tilde{\phi}(\emptyset)=\emptyset$ and $\tilde{\phi}(01)=01$. For each
$w\in\Lambda_{n}$ ($n\geq 2$), we distinguish the following two cases:

  * •

If $w=00w_{2}\cdots w_{n}$, then $\tilde{\phi}(w)$ is obtained from
$\tilde{\phi}(0w_{2}\cdots w_{n})$ by inserting $1$ immediately after the
initial $0$.

  * •

If $w=01w_{2}\cdots w_{n}$, then suppose $w_{m}=0$ is the second $0$ in $w$ if
exists, otherwise set $m=n+1$. Define $\tilde{\phi}(w)$ to be the
concatenation $00w_{2}\cdots w_{m-1}\tilde{\phi}(w_{m}\cdots w_{n})$.

For example, we have

$\displaystyle 001\mapsto 011\mapsto 001;\,\,0101\mapsto 0001\mapsto
0111\mapsto 0011\mapsto 0101;$ $\displaystyle 00001\mapsto 01111\mapsto
00111\mapsto 01011\mapsto 00001,\,01001\mapsto 00011\mapsto 01101\mapsto
00101\mapsto 01001$

under the mapping $\tilde{\phi}$. It is routine to check by induction on $n$
that $\tilde{\phi}$ is well-defined and satisfies (2.13). To see that
$\tilde{\phi}$ is a bijection, we define its inverse explicitly. Given
$w\in\Lambda_{n}$ ($n\geq 2$), we consider the following two cases:

  * •

If $w=01w_{2}\cdots w_{n}$, then $\tilde{\phi}^{-1}(w)$ is obtained from
$\tilde{\phi}(0w_{2}\cdots w_{n})$ by inserting $0$ immediately after the
initial $0$.

  * •

If $w=00w_{2}\cdots w_{n}$, then suppose $w_{m}=0$ is the third $0$ in $w$ if
exists, otherwise set $m=n+1$. Define $\tilde{\phi}^{-1}(w)$ to be the
concatenation $01w_{2}\cdots w_{m-1}\tilde{\phi}^{-1}(w_{m}\cdots w_{n})$.

It is easy to check by induction on $n$ that $\tilde{\phi}$ and
$\tilde{\phi}^{-1}$ are inverse to each other. ∎

##  3\. Three proofs of Theorem 4

For a partition $\lambda$, let

$\mathrm{mod}_{d}(\lambda):=|\\{i:\lambda_{i}\equiv
1\text{($\mathrm{mod}\,d+1$)}\\}|\quad\text{and}\quad\mathrm{mod}^{\prime}_{d}(\lambda):=\ell(\lambda)-\mathrm{mod}_{d}(\lambda).$

Then, $\mathrm{mod}^{\prime}_{d}(\lambda)$ is $d$-generalization of
$\mathrm{even}(\lambda)$ as
$\mathrm{mod}^{\prime}_{1}(\lambda)=\mathrm{even}(\lambda)$. Introduce the
$d$-generalization of $\mathrm{rep}(\lambda)$ by

$\mathrm{dif}_{d}(\lambda):=|\\{i:1\leq
i<\ell(\lambda),\lambda_{i}-\lambda_{i+1}<d\\}|.$

It is clear that Theorem 4 is equivalent to

(3.1)
$\sum_{\lambda\in\mathcal{H}_{n}}\mathrm{dif}_{d}(\lambda)\geq\sum_{\lambda\in\mathcal{H}_{n}}\mathrm{mod}^{\prime}_{d}(\lambda).$

###  3.1. A generating function proof

First we compute the generating function for partitions by the perimeter and
the statistic $\mathrm{mod}_{d}$.

######  Lemma 13.

We have

(3.2)
$\sum_{\lambda\in\mathcal{P}}t^{\mathrm{mod}_{d}(\lambda)}x^{\ell(\lambda)}y^{\lambda_{1}}=\frac{txy(1-x)^{d}+\frac{xy}{x+y-1}(y^{d+1}-y(1-x)^{d})}{(1-tx)(1-x)^{d}-y^{d+1}}.$

In particular,

(3.3)
$\sum_{\lambda\in\mathcal{P}}t^{\mathrm{mod}_{d}^{\prime}(\lambda)}x^{\Gamma(\lambda)}=\frac{x(tx^{d+1}+(1-tx)^{d}(x-1))}{(1-x-tx)((1-tx)^{d}(x-1)+x^{d+1})}.$

###### Proof.

Since every integer partition $\lambda$ can be written as
$\lambda=1^{m_{1}}2^{m_{2}}\cdots$, we have

$\displaystyle\sum_{\lambda\in\mathcal{P}}t^{m_{d}(\lambda)}x^{\ell(\lambda)}y^{\lambda_{1}}$
$\displaystyle=tx\sum_{n\equiv
1(\mathrm{mod}\,d+1)}y^{n}\biggl{(}\frac{1}{1-tx}\biggr{)}^{\lceil
n/(d+1)\rceil}\biggl{(}\frac{1}{1-x}\biggr{)}^{n-\lceil n/(d+1)\rceil},$
$\displaystyle\qquad+x\sum_{n\not\equiv
1(\mathrm{mod}\,d+1)}y^{n}\biggl{(}\frac{1}{1-tx}\biggr{)}^{\lceil
n/(d+1)\rceil}\biggl{(}\frac{1}{1-x}\biggr{)}^{n-\lceil n/(d+1)\rceil}$
$\displaystyle=tx\sum_{k\geq
0}\biggl{(}\frac{y}{1-tx}\biggr{)}^{k+1}\biggl{(}\frac{y}{1-x}\biggr{)}^{dk}$
$\displaystyle\qquad+x\sum_{i=1}^{d}\sum_{k\geq
0}\biggl{(}\frac{y}{1-tx}\biggr{)}^{k+1}\biggl{(}\frac{y}{1-x}\biggr{)}^{dk+i},$

which is simplified to (3.2). ∎

Next we compute the generating function for partitions by the perimeter and
the statistic $\mathrm{dif}_{d}$.

######  Lemma 14.

We have

(3.4)
$\sum_{\lambda\in\mathcal{P}}t^{\mathrm{dif}_{d}(\lambda)}x^{\ell(\lambda)}y^{\lambda_{1}}=\frac{xy}{1-y-tx(1-y^{d})-xy^{d}}.$

In particular,

(3.5)
$\sum_{\lambda\in\mathcal{P}}t^{\mathrm{dif}_{d}(\lambda)}x^{\Gamma(\lambda)}=\frac{x}{1-x-tx(1-x^{d})-x^{d+1}}.$

###### Proof.

Notice that in the 01-sequence $w(\lambda)$ of $\lambda$, each maximal segment
$00\cdots 01$ ($k$ 0’s followed by one 1) after the leftmost $1$ contributes a
$txy^{k}$ in the following generating function if $0\leq k\leq d-1$; and
contributes a $xy^{k}$ if $k\geq d$. Therefore, we have

$\displaystyle\sum_{\lambda\in\mathcal{P}}t^{\mathrm{dif}_{d}(\lambda)}x^{\ell(\lambda)}y^{\lambda_{1}}$
$\displaystyle=\frac{xy}{1-y}\cdot\left(1+\biggl{(}\frac{tx(1-y^{d})}{1-y}+\frac{xy^{d}}{1-y}\biggr{)}+\biggl{(}\frac{tx(1-y^{d})}{1-y}+\frac{xy^{d}}{1-y}\biggr{)}^{2}+\cdots\right)$
$\displaystyle=\frac{xy}{1-y-tx(1-y^{d})-xy^{d}},$

as desired. ∎

Take the derivative with respect to $t$ and then set $t=1$ in (3.3) gives

(3.6) $\sum_{n\geq
1}x^{n}\sum_{\lambda\in\mathcal{H}_{n}}\mathrm{mod}^{\prime}_{d}(\lambda)=\frac{x(1-x)(x^{d+1}-x(1-x)^{d})}{(1-2x)^{2}(x^{d+1}-(1-x)^{d+1})}.$

On the other hand, the same operation on (3.5) yields

(3.7) $\sum_{n\geq
1}x^{n}\sum_{\lambda\in\mathcal{H}_{n}}\mathrm{dif}_{d}(\lambda)=\frac{x^{2}(1-x^{d})}{(1-2x)^{2}}.$

To finish the proof of Theorem 4 (or equivalently, inequality (3.1)), it
remains to show that

(3.8)
$\frac{x^{2}(1-x^{d})}{(1-2x)^{2}}-\frac{x(1-x)(x^{d+1}-x(1-x)^{d})}{(1-2x)^{2}(x^{d+1}-(1-x)^{d+1})}=\frac{x^{d+2}(1-2x+x^{d+1}-(1-x)^{d+1})}{(1-2x)^{2}((1-x)^{d+1}-x^{d+1})}$

has nonnegative coefficients for each $d\geq 1$. Since

$\frac{1-2x+x^{d+1}-(1-x)^{d+1}}{(1-2x)^{2}((1-x)^{d+1}-x^{d+1})}=\frac{1}{1-2x}\biggl{(}\frac{1}{(1-x)^{d+1}-x^{d+1}}-\frac{1}{1-2x}\biggr{)},$

Theorem 4 then follows from the following interesting positivity result.

######  Theorem 15.

The rational function

$\frac{1}{(1-x)^{d+1}-x^{d+1}}-\frac{1}{1-2x}=:\Delta_{d}(x)$

has nonnegative coefficients for each $d\geq 1$.

Note that $\Delta_{0}(x)=\Delta_{1}(x)=0$ and

$\displaystyle\Delta_{2}(x)$ $\displaystyle=\sum_{n\geq 1}x^{n}\sum_{i\geq
0}{n\choose 3i+1}=x+2x^{2}+3x^{3}+5x^{4}+10x^{5}+21x^{6}+\cdots,$
$\displaystyle\Delta_{3}(x)$ $\displaystyle=\sum_{n\geq 1}x^{n}\sum_{i\geq
0}2{n+1\choose 4i+2}=2x+6x^{2}+12x^{3}+20x^{4}+32x^{5}+56x^{6}+\cdots,$
$\displaystyle\Delta_{4}(x)$ $\displaystyle=\sum_{n\geq 1}x^{n}\sum_{i\geq
0}\biggl{(}3{n+2\choose 5i+3}-{n\choose
5i+2}\biggr{)}=3x+11x^{2}+27x^{3}+54x^{4}+95x^{5}+156x^{6}+\cdots.$

For two polynomials $f(x),g(x)\in\mathbb{Z}[x]$, we write

$f(x)\geq_{x}g(x)\Leftrightarrow f(x)-g(x)\text{ has nonnegative
coefficients}.$

To prove Theorem 15, we need the following auxiliary lemma.

######  Lemma 16.

For $d\geq 1$, we have

$\frac{x}{(1-x)^{d}}\geq_{x}\frac{x^{d}}{(1-x)^{d}}.$

###### Proof.

This follows from the binomial theorem

$(1-x)^{-d}=\sum_{k\geq 0}{d+k-1\choose k}x^{k}$

and the monotonicity ${d+k\choose k+1}\geq{d+k-1\choose k}$ for $k\geq 0$. ∎

We can now prove Theorem 15.

###### Proof of Theorem 15.

By Lemma 16 and by induction on $d$, we have

$\displaystyle\quad\frac{1}{(1-x)^{d}}-\frac{1}{1-x}\biggl{(}1+\frac{x}{1-x}+(\frac{x}{1-x})^{2}+\cdots+(\frac{x}{1-x})^{d-1}\biggr{)}$
$\displaystyle\geq_{x}\frac{1}{(1-x)^{d}}-\frac{x}{(1-x)^{d}}-\frac{1}{1-x}\biggl{(}1+\frac{x}{1-x}+(\frac{x}{1-x})^{2}+\cdots+(\frac{x}{1-x})^{d-2}\biggr{)}$
$\displaystyle=\frac{1}{(1-x)^{d-1}}-\frac{1}{1-x}\biggl{(}1+\frac{x}{1-x}+(\frac{x}{1-x})^{2}+\cdots+(\frac{x}{1-x})^{d-2}\biggr{)}$
$\displaystyle\geq_{x}0$

for each $d\geq 2$. It then follows from the above assertion that

$\displaystyle\Delta_{d-1}(x)$
$\displaystyle=\frac{1}{(1-x)^{d}-x^{d}}-\frac{1}{1-2x}$
$\displaystyle=\frac{1}{(1-x)^{d}}\frac{1}{1-(\frac{x}{1-x})^{d}}-\frac{1}{1-x}\frac{1}{1-\frac{x}{1-x}}$
$\displaystyle=\frac{1}{(1-x)^{d}}\biggl{(}1+(\frac{x}{1-x})^{d}+(\frac{x}{1-x})^{2d}+(\frac{x}{1-x})^{3d}+\cdots\biggr{)}$
$\displaystyle\quad-\frac{1}{1-x}\biggl{(}1+\frac{x}{1-x}+(\frac{x}{1-x})^{2}+(\frac{x}{1-x})^{3}+\cdots\biggr{)}$
$\displaystyle=\biggl{(}\sum_{k\geq
0}(\frac{x}{1-x})^{kd}\biggr{)}\biggl{(}\frac{1}{(1-x)^{d}}-\frac{1}{1-x}\biggl{(}1+\frac{x}{1-x}+(\frac{x}{1-x})^{2}+\cdots+(\frac{x}{1-x})^{d-1}\biggr{)}\biggr{)}$
$\displaystyle\geq_{x}0,$

which completes the proof of the theorem. ∎

###  3.2. An injective proof

This section is motivated by finding an interpretation for

$\sum_{\lambda\in\mathcal{H}_{n}}\mathrm{dif}_{d}(\lambda)-\sum_{\lambda\in\mathcal{H}_{n}}\mathrm{mod}^{\prime}_{d}(\lambda).$

A partition $\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{i}^{*},\cdots)$
whose $i$-th part, $1\leq i\leq\ell(\lambda)$, receives a star is called a
labeled partition. It is convenience to represent such a labeled partition by
$(\lambda,i)$. Let us consider the two sets of labeled partitions:

$\displaystyle\mathcal{D}_{n,d}:=\\{(\lambda,i):\lambda\in\mathcal{H}_{n},0\leq\lambda_{i-1}-\lambda_{i}<d\\}\quad\text{and
}$
$\displaystyle\mathcal{M}_{n,d}:=\\{(\lambda,i):\lambda\in\mathcal{H}_{n},\lambda_{i}\not\equiv
1(\mathrm{mod}\,d+1)\\}.$

For example,

$\mathcal{D}_{4,2}=\\{32^{*},33^{*},211^{*},21^{*}1,221^{*},22^{*}1,222^{*},22^{*}2,1111^{*},111^{*}1,11^{*}11\\}$

and

$\mathcal{M}_{4,2}=\\{3^{*}1,32^{*},3^{*}2,33^{*},3^{*}3,2^{*}11,22^{*}1,2^{*}21,222^{*},22^{*}2,2^{*}22\\}.$

It is clear that Theorem 4 is equivalent to
$|\mathcal{D}_{n,d}|\geq|\mathcal{M}_{n,d}|$ for $n,d\geq 1$. The purpose of
this section is to provide an injective proof of Theorem 4 that leads to a
partition interpretation of $|\mathcal{D}_{n,d}|-|\mathcal{M}_{n,d}|$.

######  Theorem 17.

For any fixed $n,d\geq 1$, there exists an injection $\xi$ from
$\mathcal{M}_{n,d}$ to $\mathcal{D}_{n,d}$ such that

(3.9)
$\mathcal{D}_{n,d}\setminus\xi(\mathcal{M}_{n,d})=\\{(\lambda,i)\in\mathcal{D}_{n,d}:\lambda_{i-1}\equiv
1(\mathrm{mod}\,d+1),\lambda_{i}\not\equiv 1(\mathrm{mod}\,d+1)\\}.$

###### Proof.

For a labeled partition $(\lambda,i)\in\mathcal{M}_{n,d}$, we define
$\xi(\lambda,i)\in\mathcal{D}_{n,d}$ according to the following two cases.
Suppose that $\lambda_{i}=l(d+1)+k$ for some $2\leq k\leq d+1$ and $l\geq 0$.
Note that $0\leq k-2<d$ and we use the convention
$\lambda_{\ell(\lambda)+1}=0$.

  * •

If $\lambda_{i}-\lambda_{i+1}\leq k-2$, then set
$\xi(\lambda,i)=(\lambda,i+1)$. As $0\leq k-2<d$, it is clear that
$\xi(\lambda,i)\in\mathcal{D}_{n,d}$. In this case, $\lambda_{i+1}\not\equiv
1(\mathrm{mod}\,d+1)$.

  * •

If $\lambda_{i}-\lambda_{i+1}>k-2$, then set
$\xi(\lambda,i)=(\lambda^{\prime},i+1)$, where
$\lambda^{\prime}=\lambda^{\prime}_{1}\lambda^{\prime}_{2}\cdots\lambda^{\prime}_{\ell(\lambda)+1}$
is a partition with one more part than $\lambda$ defined as

$\lambda^{\prime}_{j}=\begin{cases}\lambda_{j}-1,&\quad\text{if $j\leq i$};\\\
l(d+1)+1,&\quad\text{if $j=i+1$};\\\ \lambda_{j-1},&\quad\text{if
$i+1<j\leq\ell(\lambda)+1$}.\end{cases}$

Since $\lambda^{\prime}_{i}-\lambda^{\prime}_{i+1}=l(d+1)+k-1-l(d+1)-1=k-2$,
we see $\xi(\lambda,i)\in\mathcal{D}_{n,d}$. In this case,
$\lambda_{i+1}=l(d+1)+1\equiv 1(\mathrm{mod}\,d+1)$.

For example, if $(\lambda,3)=(14,13,11^{*},10,5,2)\in\mathcal{M}_{19,5}$ then
$\xi(\lambda,3)=(14,13,11,10^{*},5,2)\in\mathcal{D}_{19,5}$ is constructed in
the first case, and if $(\lambda,3)=(14,13,11^{*},5,5,2)\in\mathcal{M}_{19,5}$
then $\xi(\lambda,3)=(13,12,10,7^{*},5,5,2)\in\mathcal{D}_{19,5}$ is
constructed in the second case.

To see that the mapping $\xi$ is an injection, observe that for any
$(\lambda,i+1)\in\mathcal{D}_{n,d}$:

  * •

$(\lambda,i+1)$ is the image under $\xi$ from the first case above if
$\lambda_{i+1}\not\equiv 1(\mathrm{mod}\,d+1)$ and $\lambda_{i}\not\equiv
1(\mathrm{mod}\,d+1)$;

  * •

$(\lambda,i+1)$ is the image under $\xi$ from the second case above if
$\lambda_{i+1}\equiv 1(\mathrm{mod}\,d+1)$;

  * •

$(\lambda,i+1)$ is not an image under $\xi$ if $\lambda_{i+1}\not\equiv
1(\mathrm{mod}\,d+1)$ but $\lambda_{i}\equiv 1(\mathrm{mod}\,d+1)$.

Since the above two cases of $\xi$ are reversible, $\xi$ is an injection and
(3.9) holds. ∎

######  Example 18.

As an example of (3.9), we see that the number
$|\mathcal{D}_{6,2}|-|\mathcal{M}_{6,2}|=4$ counts the labeled partitions
$(4,3^{*},1)$, $(4,3^{*},2)$, $(4,4,3^{*})$ and $(4,3^{*},3)$.

###  3.3. A bijective proof

The following stronger version of Theorem 4 was originally suggested by
numerical computations.

######  Theorem 19.

Fix $d\geq 1$. There exists a bijection
$\phi_{d}:\mathcal{H}_{n}\rightarrow\mathcal{H}_{n}$ such that

$\mathrm{dif}_{d}(\lambda)\geq\mathrm{mod}^{\prime}_{d}(\phi_{d}(\lambda))$

for every $\lambda\in\mathcal{H}_{n}$. Moreover, $\mathrm{dif}_{d}(\lambda)=0$
if and only if $\mathrm{mod}^{\prime}_{d}(\phi_{d}(\lambda))=0$.

######  Remark 20.

Theorem 19 is a common generalization of Theorems 2 and 4. At the beginning,
we attempted to prove Theorem 19 by using generating function but failed. To
find such a proof remains an interesting open problem.

The construction of $\phi_{d}$ is a $d$-extension of $\phi$ defined in Theorem
11.

###### Proof of Theorem 19.

Under the correspondence $\lambda\mapsto w=w(\lambda)$ in Lemma 12, the two
statistics $\mathrm{dif}_{d}(\lambda)$ and
$\mathrm{mod}^{\prime}_{d}(\lambda)$ are transformed to

$\displaystyle\widetilde{\mathrm{dif}}_{d}(w)$
$\displaystyle=|\\{i\in[n-1]:w_{i}=w_{i+k}=1\text{ and
$w_{i+1}=\cdots=w_{i+k-1}=0$ for some $k\in[d]$}\\}|,$
$\displaystyle\widetilde{\mathrm{mod}}^{\prime}(w)$ $\displaystyle=|\\{2\leq
i\leq n:w_{i}=1\text{ and }\\#\\{0\leq j<i:w_{j}=0\\}\not\equiv
1(\mathrm{mod}\,d+1)\\}|.$

So it is sufficient to define recursively a bijection
$\tilde{\phi}_{d}:\Lambda_{n}\rightarrow\Lambda_{n}$ such that

(3.10)
$\widetilde{\mathrm{dif}}_{d}(w)\geq\widetilde{\mathrm{mod}}^{\prime}_{d}(\tilde{\phi}_{d}(w))$

for each $w\in\Lambda_{n}$ and then set
$\phi_{d}=w^{-1}\circ\tilde{\phi}_{d}\circ w$.

Set $\tilde{\phi}_{d}(\emptyset)=\emptyset$ and $\tilde{\phi}_{d}(01)=01$. For
each $w\in\Lambda_{n}$ ($n\geq 2$), we distinguish the following two cases:

  * •

If $w=00w_{2}\cdots w_{n}$, then $\tilde{\phi}_{d}(w)$ is obtained from
$\tilde{\phi}_{d}(0w_{2}\cdots w_{n})$ by inserting $1$ immediately after the
initial $0$.

  * •

If $w=01w_{2}\cdots w_{n}$, then suppose $w_{m}=0$ is the $(d+1)$-th $0$ in
$w$ if exists, otherwise set $m=n+1$. Define $\tilde{\phi}_{d}(w)$ to be the
concatenation $00w_{2}\cdots w_{m-1}\tilde{\phi}_{d}(w_{m}\cdots w_{n})$.

For example, we have

$\displaystyle 001\mapsto 011\mapsto 001;\,\,0101\mapsto 0001\mapsto
0111\mapsto 0011\mapsto 0101;$ $\displaystyle 00001\mapsto 01111\mapsto
00111\mapsto 01011\mapsto 00011\mapsto 01101\mapsto 00101\mapsto 01001\mapsto
00001.$

under the mapping $\tilde{\phi}_{2}$. We need to verify that
$\tilde{\phi}_{d}$ satisfies (3.10) by induction on $n$ depending on two cases
of $\tilde{\phi}_{d}$:

  * •

Since inserting $1$ after the initial $0$ of $\tilde{\phi}_{d}(0w_{2}\cdots
w_{n})$ dose not change $\widetilde{\mathrm{mod}}^{\prime}$, we have

$\widetilde{\mathrm{dif}}_{d}(w)=\widetilde{\mathrm{dif}}_{d}(0w_{2}\cdots
w_{n})\geq\widetilde{\mathrm{mod}}^{\prime}_{d}(\tilde{\phi}_{d}(0w_{2}\cdots
w_{n}))=\widetilde{\mathrm{mod}}^{\prime}_{d}(\tilde{\phi}_{d}(w)).$

  * •

As the two statistics $\widetilde{\mathrm{dif}}_{d}$ and
$\widetilde{\mathrm{mod}}^{\prime}_{d}$ can be extended to any $01$ sequence
and notice that there are exactly $d+1$ $0$’s in the prefix $00w_{2}\cdots
w_{m-1}$ of $\tilde{\phi}_{d}(w)$ whenever $m\neq n+1$, we have

$\displaystyle\widetilde{\mathrm{dif}}_{d}(w)$
$\displaystyle\geq\widetilde{\mathrm{dif}}_{d}(01w_{2}\cdots
w_{m-1})+\widetilde{\mathrm{dif}}_{d}(w_{m}\cdots w_{n})$
$\displaystyle\geq\widetilde{\mathrm{mod}}^{\prime}_{d}(00w_{2}\cdots
w_{m-1})+\widetilde{\mathrm{mod}}^{\prime}_{d}(\tilde{\phi}_{d}(w_{m}\cdots
w_{n}))$ $\displaystyle=\widetilde{\mathrm{mod}}^{\prime}_{d}(\phi_{d}(w)).$

For the case $m=n+1$, we have

$\widetilde{\mathrm{dif}}_{d}(w)=\widetilde{\mathrm{dif}}_{d}(01w_{2}\cdots
w_{n})=\widetilde{\mathrm{mod}}^{\prime}_{d}(00w_{2}\cdots
w_{n})=\widetilde{\mathrm{mod}}^{\prime}_{d}(\tilde{\phi}_{d}(w)).$

To see that $\tilde{\phi}_{d}$ is a bijection, we construct its inverse
$\tilde{\phi}_{d}^{-1}$ explicitly. Given $w\in\Lambda_{n}$ ($n\geq 2$), we
consider the following two cases:

  * •

If $w=01w_{2}\cdots w_{n}$, then $\tilde{\phi}_{d}^{-1}(w)$ is constructed
from $\tilde{\phi}_{d}^{-1}(0w_{2}\cdots w_{n})$ by inserting $0$ immediately
after the initial $0$.

  * •

If $w=00w_{2}\cdots w_{n}$, then suppose $w_{m}=0$ is the $(d+2)$-th $0$ in
$w$ if exists, otherwise set $m=n+1$. Define $\tilde{\phi}_{d}^{-1}(w)$ to be
the concatenation $01w_{2}\cdots w_{m-1}\tilde{\phi}_{d}^{-1}(w_{m}\cdots
w_{n})$.

It can be checked routinely that $\tilde{\phi}_{d}$ and
$\tilde{\phi}_{d}^{-1}$ are inverse to each other, which proves that
$\tilde{\phi}_{d}$ is a bijection.

Finally, we need to verify that whenever $\mathrm{mod}^{\prime}_{d}(w)=0$ then
$\mathrm{dif}_{d}(\tilde{\phi}_{d}^{-1}(w))=0$ by induction on $n$ according
to the two cases of $\tilde{\phi}_{d}^{-1}$:

  * •

In the first case, we have

$0=\mathrm{mod}^{\prime}_{d}(w)=\mathrm{mod}^{\prime}_{d}(0w_{2}\cdots
w_{n})=\mathrm{dif}_{d}(\tilde{\phi}_{d}^{-1}(0w_{2}\cdots
w_{n}))=\mathrm{dif}_{d}(\tilde{\phi}_{d}^{-1}(w)).$

  * •

In the second case, since $\mathrm{mod}^{\prime}_{d}(w)=0$, we must have
$m=d+2$ and the prefix $00w_{2}\cdots w_{m}$ of $w$ are all $0$’s, which leads
to

$\mathrm{dif}_{d}(\tilde{\phi}_{d}^{-1}(w))=\tilde{\phi}_{d}^{-1}(w_{m}\cdots
w_{n})=0.$

The proof of the theorem is now complete. ∎

##  4\. Further remarks

We will conclude this paper with the following three remarks.

  1. (1)

Note that Eq. (3.7) is equivalent to the following closed formula

$\sum_{\lambda\in\mathcal{H}_{n}}\mathrm{dif}_{d}(\lambda)=(n-1)\cdot
2^{n-2}-(n-d-1)\cdot 2^{n-d-2},$

which can also de deduced directly as follows. By the 01-sequence
representation of partitions, we have

$\displaystyle\quad\sum_{\lambda\in\mathcal{H}_{n}}\mathrm{dif}_{d}(\lambda)=\sum_{\lambda\in\Lambda_{n}}\widetilde{\mathrm{dif}}_{d}(\lambda)$
$\displaystyle=\sum_{j=1}^{d}\sum_{w\in\Lambda_{n}}|\\{1\leq i\leq
n-j:w_{i}=w_{i+j}=0,~{}~{}w_{i+1}=w_{i+2}=\ldots=w_{i+j-1}=1\\}|$
$\displaystyle=\sum_{j=1}^{d}\sum_{i=1}^{n-j}|\\{w\in\Lambda_{n}:w_{i}=w_{i+j}=0,~{}~{}w_{i+1}=w_{i+2}=\ldots=w_{i+j-1}=1\\}|$
$\displaystyle=\sum_{j=1}^{d}\left((n-j-1)2^{n-j-2}+2^{n-j-1}\right)$
$\displaystyle=(n-1)\cdot 2^{n-2}-(n-d-1)\cdot 2^{n-d-2}.$

  2. (2)

In Theorem 8, we have computed the generating function for the joint
distribution of $(\mathrm{rep},\mathrm{even})$ over $\mathcal{H}_{n}$, which
is a rational formal power series. The pair
$(\mathrm{dif}_{d},\mathrm{mod}^{\prime}_{d})$ is a $d$-extension of
$(\mathrm{rep},\mathrm{even})$, however, in Lemmas 13 and 14 we have to deal
separably with the distribution of $\mathrm{dif}_{d}$ and
$\mathrm{mod}^{\prime}_{d}$ on $\mathcal{H}_{n}$. It remains an open problem
to compute the generating function for the joint distribution of
$(\mathrm{dif}_{d},\mathrm{mod}^{\prime}_{d})$ over $\mathcal{H}_{n}$ for
general $d$.

  3. (3)

In [6], Wilf proved via the Principle of Inclusion-Exclusion [4, Sec. 2.1] an
interesting refinement of Euler’s Odd-Distinct partition theorem by using two
valued version of our (position) statistics “$\mathrm{rep}$” and
“$\mathrm{even}$”:

$\mathrm{rep}^{*}(\lambda)=|\\{\lambda_{i}:\lambda_{i}=\lambda_{i+1}\\}|\quad\text{and}\quad\mathrm{even}^{*}(\lambda)=|\\{\lambda_{i}:\lambda_{i}\text{
is even}\\}|,$

called the number of repeated part sizes and the number of even part sizes,
respectively. Namely, he proved the equidistribution

(4.1)
$\sum_{\lambda\in\mathrm{Par}_{n}}t^{\mathrm{rep}^{*}(\lambda)}=\sum_{\lambda\in\mathrm{Par}_{n}}t^{\mathrm{even}^{*}(\lambda)},$

where $\mathrm{Par}_{n}$ is the set of all partitions of $n$. Interestingly,
this equidistribution holds true when replacing $\mathrm{Par}_{n}$ by
$\mathcal{H}_{n}$:

(4.2)
$\sum_{\lambda\in\mathcal{H}_{n}}t^{\mathrm{rep}^{*}(\lambda)}=\sum_{\lambda\in\mathcal{H}_{n}}t^{\mathrm{even}^{*}(\lambda)},$

which is a valued version of Theorem 3. See Fig. 3 for an example of (4.2) for
$n=5$. The approach of Wilf via Inclusion-Exclusion doesn’t seem to work for
(4.2).

$(0,0)$$(0,1)$$(0,2)$$(0,1)$$(1,1)$$(1,0)$$(0,1)$$(1,1)$$(1,0)$$(1,1)$$(1,0)$$(1,1)$$(2,1)$$(1,1)$$(1,1)$$(1,0)$
Figure 3. Partitions in $\mathcal{H}_{5}$ (represented as Young diagrams) with
their pair of statistics $(\mathrm{rep}^{*},\mathrm{even}^{*})$ on the top.

However, the bijection $\phi$ constructed in Theorem 11 also proves (4.2). Is
there any unified generalization of both (4.1) and (4.2)?

## Acknowledgments

Authors thank Grimaldi for kindly sending them a version of his paper [3].
This work was supported by the National Science Foundation of China grants
11871247 and 12071440, and the project of Qilu Young Scholars of Shandong
University.

## References

  * [1] G.E. Andrews, K. Eriksson, Integer Partitions, Cambridge University Press, 2004.
  * [2] S. Fu and D. Tang, Partitions with fixed largest hook length, Ramanujan J., 45 (2018), 375–390.
  * [3] R.P. Grimaldi, Extraordinary subsets: a generalization, Fibonacci Quart., 55 (2017), 114–122.
  * [4] R.P. Stanley, Enumerative Combinatorics, Vol. 1, Second edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 2012.
  * [5] A. Straub, Core partitions into distinct parts and an analog of Euler’s theorem, European J. Combin., 57 (2016), 40–49.
  * [6] H.S. Wilf, Identically distributed pairs of partition statistics, Sém. Lothar. Combin., 44 (2000), Art. B44c, 3 pp.


# Borromean supercounterfluidity

Emil Blomquist  Department of Physics, KTH Royal Institute of Technology,
Stockholm SE-10691, Sweden  Andrzej Syrwid  Department of Physics, KTH Royal
Institute of Technology, Stockholm SE-10691, Sweden  Egor Babaev  Department
of Physics, KTH Royal Institute of Technology, Stockholm SE-10691, Sweden

###### Abstract

We demonstrate microscopically the existence of a new superfluid state of
matter in a three-component Bose mixture trapped in an optical lattice. The
superfluid transport involving co-flow of all three components is arrested in
that state, while counterflows between any pair of components are
dissipationless. The presence of three components allows for three different
types of counterflows with only two independent superfluid degrees of freedom.

The advent of optical lattices Jaksch  _et al._ (1998); Jaksch and Zoller
(2005); Greiner  _et al._ (2002); Bloch (2005); Bloch  _et al._ (2008);
Lewenstein  _et al._ (2007) allowed for highly controllable access to
strongly-correlated quantum many-body systems and opened up a way to realize
various phases of matter. One of the theoretical predictions was that bosons
in optical lattices could host a new type of transport phenomenon called
supercounterfluidity Kuklov and Svistunov (2003); Kuklov  _et al._ (2004a, b,
2006); Capogrosso-Sansone  _et al._ (2008); Dahl  _et al._ (2008a, b, c);
Capogrosso-Sansone and Kuklov (2011); Sellin and Babaev (2018). That is,
having two bosonic fields $\psi_{1,2}$, in an ordinary case, one finds a
superfluid mixture when $\langle\psi_{1,2}\rangle$ reveals a (quasi) long-
range order. This state is predicted to appear Kuklov and Svistunov (2003);
Kuklov  _et al._ (2004a, b, 2006); Capogrosso-Sansone  _et al._ (2008); Dahl
_et al._ (2008a, b, c); Capogrosso-Sansone and Kuklov (2011); Sellin and
Babaev (2018) when the averages of individual fields vanish,
$\langle\psi_{1,2}\rangle=0$, but there is a (quasi) long-range order in the
composite field $\langle\psi_{1}\psi_{2}^{*}\rangle\neq 0$. Therefore, in a
supercounterfluid phase, individual bosonic species do not exhibit
superfluidity, but the transport of particle-hole composites is
dissipationless. A similar type of order was predicted in superconducting
systems arising from a different microscopic origin Babaev ; Babaev  _et al._
(2004); Smiseth  _et al._ (2005); Herland  _et al._ (2010); Kuklov  _et al._
(2008); Agterberg and Tsunetsugu (2008); Berg  _et al._ (2009); Weston and
Babaev (2021). For a general discussion, see Svistunov  _et al._ (2015).
Recently an experimental observation of a discrete-symmetry counterpart of
this type of order was reported in a three-component superconductor Grinenko
_et al._ (2021). In the two-component case, the order parameter for the
supercounterfluid is partially similar to a condensate of bound particle-hole
pairs between two different components. However, the three-component situation
is more subtle. In this Letter, we demonstrate microscopically the existence
of a new “super” state in a three-component Bose mixture.

At the superfluid hydrodynamic level, $N$-component isotropic superfluid
systems can in general be described by the free-energy density
$f=\frac{1}{2}\sum_{\alpha,\beta}\rho_{\alpha\beta}\mathbf{v}_{\alpha}\cdot\mathbf{v}_{\beta}$.
Here $\mathbf{v}_{\alpha}=\nabla\theta_{\alpha}/m_{\alpha}$ ($\hbar=1$)
represents the superfluid velocity of component $\alpha$, where
$\theta_{\alpha}\in[0,2\pi)$ and $m_{\alpha}$ denote superfluid phase and
particle mass of the $\alpha$th component, respectively. In the presence of
inter-component interactions, the superfluid stiffness tensor $\rho$ will in
addition to the diagonal superfluid density elements, $\rho_{\alpha\alpha}$,
also contain off-diagonal elements, $\rho_{\alpha\beta}$ ($\alpha\neq\beta$),
coupling the different components Andreev and Bashkin (1975). This in turns
has principal consequences for the corresponding superflows
$\mathbf{j}_{\alpha}=\partial
f/\partial\mathbf{v}_{\alpha}=\rho_{\alpha\alpha}\mathbf{v}_{\alpha}+\sum_{\beta\neq\alpha}\rho_{\alpha\beta}\mathbf{v}_{\beta}$.
Namely, the superflow of one component can be induced by the flow of a
different component. This fundamental phenomenon is referred to as
Andreev–Bashkin drag Andreev and Bashkin (1975). The Andreev–Bashkin effect
was studied in various regimes in optical lattices for binary Kuklov and
Svistunov (2003); Kuklov  _et al._ (2004a, a); Capogrosso-Sansone  _et al._
(2008); Linder and Sudbø (2009); Hofer  _et al._ (2012); Contessi  _et al._
(2021); Sellin and Babaev (2018), and trinary Hartman  _et al._ (2018)
mixtures. Note that in certain asymmetrical optical lattices there are
additional terms responsible for transverse entrainment Syrwid  _et al._
(2021). However, in this Letter, we will restrict ourselves to square lattices
where only the Andreev–Bashkin drag effect is present.

In the simplest case of two components with identical masses, the free-energy
density can be cast into the form associated with co- and counter-flows
$f=\uprho_{2}(\nabla\theta_{1}+\nabla\theta_{2})^{2}/4+\uprho_{0}(\nabla\theta_{1}-\nabla\theta_{2})^{2}/4$.
Here $\uprho_{\xi}=\rho_{\text{k}}+(\xi-1)\rho_{\text{d}}\geq 0$ where
$\rho_{\text{k}}>0$ describe the prefactor of the standard gradient term, and
$\rho_{\text{d}}$—either positive or negative—denotes the drag strength. When
the drag is sufficiently strong and negative, the cheapest topological
excitations become co-circulating composite vortices, i.e., vortices where
both phases $\theta_{1,2}$ wind by $2\pi$ around the core Svistunov  _et al._
(2015). Thermal or quantum fluctuations can then lead to the proliferation of
these composite vortices—but not elementary ones—resulting in a phase
transition to a supercounterfluid (for a detailed discussion of the principle,
see, e.g., Chapter 6 in Svistunov  _et al._ (2015)). The composite vortices do
not induce gradients in the phase difference and thus do not disorder the
phase difference part of the free energy. The free-energy density of the
resulting state therefore only involves the phase stiffness corresponding to
the phase difference, i.e.,
$f=\uprho_{0}(\nabla\theta_{1}-\nabla\theta_{2})^{2}/4$. This term can be
interpreted as the kinetic free-energy contribution related to the composite
particle-hole order parameter $\psi_{1}\psi_{2}^{*}$. Consequently, only
counter-flow dissipationless transport can take place.

In a system with more than two components, states may arise with no direct
counterpart among two-component superfluids. Let us therefore consider a two-
dimensional $N$-component symmetric quantum system, i.e., components with
identical masses and densities, and equal Andreev–Bashkin drag strength
$\rho_{\text{d}}$ between each pair. We start with a phase-only approximation
assuming identical and homogeneous densities of unit mass particles
($m_{\alpha}=1$) in each superfluid component. The corresponding free-energy
density reads

$\begin{split}f&=\frac{\rho_{\text{k}}}{2}\sum_{\alpha}(\nabla\theta_{\alpha})^{2}+\frac{\rho_{\text{d}}}{2}\sum_{\alpha,\beta\neq\alpha}\nabla\theta_{\alpha}\cdot\nabla\theta_{\beta}\\\
&=\frac{\uprho_{N}}{2N}\bigg{(}\sum_{\alpha}\nabla\theta_{\alpha}\bigg{)}^{2}+\frac{\uprho_{0}}{4N}\sum_{\alpha,\beta}(\nabla\theta_{\alpha}-\nabla\theta_{\beta})^{2}\,,\end{split}$
(1)

where again $\uprho_{\xi}=\rho_{\text{k}}+(\xi-1)\rho_{\text{d}}\geq 0$. Now,
when the drag is strong and negative, i.e., when $\uprho_{N}\ll\uprho_{0}$,
the cheapest topological excitations are composite vortices where all $N$
phases wind by $2\pi$ with the same orientation. Consequently, the
proliferation of the three-component topological defects leads to a state
where the sum of three phases is disordered. However, these composite vortices
are unable to disorder phase differences and the system retains $N-1$
superfluid modes. Equation (1) therefore reduces to
$f\propto\sum_{\alpha,\beta}(\nabla\theta_{\alpha}-\nabla\theta_{\beta})^{2}$
and the corresponding phase is characterized by zero net superflow, i.e.,
$\sum_{\alpha}\mathbf{j}_{\alpha}\propto\sum_{\alpha}\partial
f/\partial(\nabla\theta_{\alpha})=\mathbf{0}$.

Consequently, for $N>2$, the transport properties of the new phase can be
understood as a counterflow of two components where the presence of the third
symmetric component allows for fluctuations in the type of counterpropagating
companions. One would anticipate that this should be reflected in the world
lines of the microscopic path-integral formulation. That is, in different
regions of the system, one should find different types of particle-hole paired
world lines. Moreover, there is no superfluid co-flow of bound $N$ particle
states, while counter-propagation of any two different components is
dissipationless. Specifically, for $N=3$ where
$\alpha\in\\{\text{r},\text{g},\text{b}\\}$, there are three types of
counterflows for which there are only two independent degrees of freedom. That
implies that we can have a superfluid co-flow of two components as long as
their combined flow is counteracted by the flow of the third component, e.g.,
$\mathbf{j}_{\text{r}}=\mathbf{j}_{\text{g}}=\mathbf{j}$ and
$\mathbf{j}_{\text{b}}=-2\mathbf{j}$. Here we can draw a distant analogy to
the Borromean rings where three rings are confined while each pair of rings is
deconfined, see Fig. 1. Hence we coin this phenomenon Borromean
supercounterfluidity.

Figure 1:  The Borromean rings. If a single ring is removed, the two remaining
ones will become unlinked.

Below we microscopically demonstrate that such superfluid state exists in a
three-component ($N=3$) Bose–Hubbard model Gersch and Knollman (1963)

$\begin{split}\hat{H}=&-t\sum_{\alpha}\sum_{\langle
ij\rangle}\hat{b}^{\dagger}_{i\alpha}\hat{b}_{j\alpha}+\frac{U}{2}\sum_{\alpha}\sum_{i}\hat{n}_{i\alpha}\big{(}\hat{n}_{i\alpha}-1\big{)}\\\
&+\frac{U^{\prime}}{2}\sum_{\alpha,\beta\neq\alpha}\sum_{i}\hat{n}_{i\alpha}\hat{n}_{i\beta}\,.\end{split}$
(2)

Here $\hat{b}_{i\alpha}$ ($\hat{b}^{\dagger}_{i\alpha}$) is the bosonic
annihilation (creation) operator of component $\alpha$ at site $i$, and
$\hat{n}_{i\alpha}=\hat{b}^{\dagger}_{i\alpha}\hat{b}_{i\alpha}$ is the
corresponding particle number operator. Greek subscripts label the component
type, i.e., r (red), g (green), and b (blue). The parameter $t$ represents the
hopping amplitude, while $U$ and $U^{\prime}$, respectively, are the intra-
component and inter-component on-site interaction strengths. We will consider
a $L\times L$ square lattice with unit lattice constant and periodic boundary
conditions. We further analyze the two separate cases where either the
individual particle-number densities are fixed, i.e.,
$n_{\alpha}\coloneqq\langle\sum_{i}\hat{n}_{i\alpha}\rangle/L^{2}=1/3$, or the
total particle-number density is conserved, i.e., $\sum_{\alpha}n_{\alpha}=1$,
while allowing for fluctuations in $n_{\alpha}$.

We numerically investigate the system by utilizing worm-algorithm Monte Carlo
Prokof’ev  _et al._ (1998); Capogrosso-Sansone  _et al._ (2007); Blomquist
(2021); Sellin (2018); Lingua  _et al._ (2018)—a quantum Monte-Carlo method
which samples path-integral configurations of the partition function in real
space and imaginary time. To extract the numerical values of $\rho_{\text{k}}$
and $\rho_{\text{d}}$ appearing in the free-energy density, Eq. (1), we
generalize Pollock and Ceperley’s formula Pollock and Ceperley (1987); Sellin
and Babaev (2018); Syrwid  _et al._ (2021):
$\rho_{\text{k}}=T\langle\mathbf{w}_{\alpha}^{2}\rangle/2$, and
$\rho_{\text{d}}=T\langle\mathbf{w}_{\alpha}\cdot\mathbf{w}_{\beta}\rangle/2$
where $T$ is the temperature ($k_{\text{B}}=1$) and $\alpha\neq\beta$. The
winding numbers $\mathbf{w}_{\alpha}$ encode the net number of times, and in
which direction, $\alpha$-type particles cross the periodic boundaries. The
notation $\langle\cdot\rangle$ refers to the standard statistical Monte Carlo
average.

Figure 2:  The superfluid density $\rho_{\mathrm{k}}$ (triangles), the
Andreev–Bashkin drag-coefficient $\rho_{\mathrm{d}}$ (diamonds), and the ratio
$\rho_{\mathrm{d}}/\rho_{\mathrm{k}}$ (inset) as a function of $t/U$. For
$t/U\in[0.02,0.05]$ the ratio $\rho_{\mathrm{d}}/\rho_{\mathrm{k}}$ saturates
to its lower bound, $-1/2$, which indicates the presence of the Borromean
supercounterfluid phase. The computations were performed using $U=1$,
$U^{\prime}=0.9$, $T=t/L$, $L=12,18$, and a fixed particle number density
$n_{\alpha}=1/3$.

The calculated coefficients $\rho_{\text{k}}$, $\rho_{\text{d}}$, and their
ratios $\rho_{\text{d}}/\rho_{\text{k}}$ are presented in Fig. 2 as functions
of $t/U$ for the interactions strengths $U=1$ and $U^{\prime}=0.9$. For small
$t/U$ we observe $\rho_{\text{k}}=\rho_{\text{d}}=0$, which indicates a Mott
insulating phase. However, at $t/U\gtrsim 0.02$ the system enters the
Borromean supercounterfluid phase, where $\rho_{\text{d}}/\rho_{\text{k}}$
rapidly saturates at the value $-1/2$ for which the coefficient $\uprho_{N=3}$
in Eq. (1) vanishes. This result demonstrates that the corresponding effective
free-energy density is given by gradients of phase differences between all the
three components. When further increasing $t/U\gtrsim 0.05$, the system
undergoes a second transition to the three-component superfluid phase where
the $U(1)\times U(1)\times U(1)$ symmetry is broken. Deep in this regime
$\rho_{\text{d}}/\rho_{\text{k}}$ becomes very small.

Figure 3:  In panel (a) the particle number-density imbalance $\Delta n$, Eq.
(3), is plotted versus the system size $L$ for various parameter values $t/U$
with the inverse temperature given by $\beta=L/t$. The total particle number-
density $n_{\text{r}}+n_{\text{g}}+n_{\text{b}}=1$ is fixed while the particle
number of each of the three symmetric components is allowed to fluctuate. The
plot reveals a decay of $\Delta n$ with increasing $L$. The latter is more
apparent in the inset which presents $\Delta n$ for the largest $L$s on
logarithmic scales. This shows that $\Delta n\rightarrow 0$ as
$L\rightarrow\infty$. The bottom panels present two representative real-space
particle-number distributions, i.e., imaginary-time slices of the world-line
configuration, with $L=60$ for $t/U=0.02$ (b) and $t/U=0.05$ (c). The three
different components are represented by the colors red, green, and blue,
respectively. The darker the color, the fewer the particles occupying the
corresponding site such that an empty site becomes black. These particle-
numbers distributions clearly display a miscible phase which is corroborated
by $\Delta n$ of panel (a). In all cases the values $U=1$ and $U^{\prime}=0.9$
were used.

Let us now characterize the absence of phase separation in the system. Namely,
suppose the system spontaneously forms bound pairs between two components
only, and one relaxes the restriction on individually fixed particle numbers
$n_{\alpha}$. In that case, a disproportion between the components’ particle
densities is expected. To that end, we investigate the density imbalance
$\Delta n$ defined through

$\langle\hat{n}_{\text{r}}\hat{n}_{\text{g}}\hat{n}_{\text{b}}\rangle=(1/3-2\Delta
n)(1/3+\Delta n)^{2},$ (3)

while constraining the total particle-number density to
$n_{\text{r}}+n_{\text{g}}+n_{\text{b}}=1$, and leaving $n_{\alpha}$
unrestricted such that only on average $\langle\hat{n}_{\alpha}\rangle=1/3$.
Here $\hat{n}_{\alpha}=\sum_{i}\hat{n}_{i\alpha}/L^{2}$. If the particles of
one component are completely exchanged in favor of particles belonging to the
other two components, then
$\langle\hat{n}_{\text{r}}\hat{n}_{\text{g}}\hat{n}_{\text{b}}\rangle=0$
leading to $\Delta n=1/6$. Similarly, if two components are completely
expelled then $\Delta n=-1/3$. In contrast, if there is no density imbalance
and all components are equally represented one should have
$\langle\hat{n}_{\text{r}}\hat{n}_{\text{g}}\hat{n}_{\text{b}}\rangle=(1/3)^{3}$
such that $\Delta n=0$. In Fig. 3 (a) we demonstrate a decay of $\Delta n$
with increased system size $L$ and inverse temperature $\beta$ in the regime
of the anticipated Borromean supercounterfluid phase. As expected, we find a
nonzero $\Delta n$ at finite temperatures due to thermal density fluctuations,
however, the magnitude of $\Delta n$ is small in comparison to $1/3$. In
addition, $\Delta n$ show a clear decay with increased $L$ indicating $\Delta
n\rightarrow 0$ for $L\rightarrow\infty$. This is further corroborated in Fig.
3 (b,c) by real-space particle-number distributions obtained from two
representative imaginary-time slices of the world-line configurations. This
demonstrates that there is no density imbalance nor phase separation in the
ground state.

The Borromean supercounterfluid phase can be further studied by inspecting a
typical partition function world-line configurations sampled with the help of
the worm-algorithm Monte Carlo method. A representative configuration of the
Borromean superfluid phase is illustrated in Fig. 4, which indeed reveals
three mixed components. The net flow of particles in imaginary time further
exhibits the counterflowlike behavior, resulting in the winding numbers
$\langle\mathbf{w}_{\alpha}^{2}\rangle\neq 0$ and
$\langle(\sum_{\alpha}\mathbf{w}_{\alpha})^{2}\rangle=0$, or alternatively
$\rho_{\text{d}}/\rho_{\text{k}}=-1/2$.

Figure 4:  Spatial projection of a typical partition function world-line
configuration of the Borromean supercounterfluid phase. Here the three
different components are represented by red, green, and blue, respectively.
The directional loops indicate the net particle propagation in imaginary time
between adjacent sites, and the site’s color indicates its average particle
number population. The sites reveal three miscible components, and the loops
display on average counter-propagating world-line trajectories. These
properties are indicative of the Borromean supercounterfluid phase. By
counting the flux of loops across the boundaries we find the winding numbers
$\mathbf{w}_{\text{r}}=\mathbf{e}_{y}$, $\mathbf{w}_{\text{g}}=\mathbf{0}$,
$\mathbf{w}_{\text{b}}=-\mathbf{e}_{y}$ which satisfy
$\mathbf{w}_{\text{r}}+\mathbf{w}_{\text{g}}+\mathbf{w}_{\text{b}}=\mathbf{0}$.
The results were obtained using $t=0.04$, $U=1$, $U^{\prime}=0.9$, $L=12$,
$T=t/L$, and a fixed particle number density $n_{\alpha}=1/3$.

In conclusion, we have demonstrated microscopically that a strongly correlated
three-component bosonic mixture—realizable in optical lattice setups—has a
phase with “super” transport phenomenon different from conventional
superfluidity. In this phase, the simultaneous co-flow of all three bosonic
components is arrested, while the system retains dissipationless counterflows
between any pair of components. These three counterflows are not independent
but rather described by two superfluid degrees of freedom. At the microscopic
level, the types of counterpropagating partners can vary.

Possible realization of these states in optical lattices could be obtained by
trapping mixtures of bosonic isotopes of Na and K. Possible ways to detect the
Borromean supercounterfluid state experimentally is through tilting the
lattice and detecting the ratio between transport of different individual
components. However, the more striking signature can be obtained by observing
a dramatic change in the system’s rotational response. Namely, a rotating
conventional superfluid can be described by introducing a fictitious vector
potential $\bm{\Theta}$, leading to a vortex lattice formation (see, e.g.,
chapter 1 in Svistunov  _et al._ (2015)). In the Borromean supercounterfluid
state the vector potential $\bm{\Theta}$ couples to the phase gradients
through mass differences according to
$f\approx\sum_{\alpha,\beta\neq\alpha}[\nabla\theta_{\alpha}-\nabla\theta_{\beta}-(m_{\alpha}-m_{\beta})\bm{\Theta}]^{2}$.
If all the components have equal particle masses, i.e.,
$m_{\alpha}=m_{\beta}$, the system is unaffected by rotation. Indeed, in that
case, the counterflow involves no mass transfer, and vortices carry no angular
momentum. On the other hand, if $m_{\alpha}\neq m_{\beta}$ the system forms a
vortex lattice to compensate for the superextensive increase of the free
energy due to the rotation. In the latter case, the critical velocities and
number of rotation-induced vortices will be proportional to the mass
difference $m_{\alpha}-m_{\beta}$ rather than to masses of individual
components.

## Acknowledgements

We would like to thank Martin Zwierlein and Immanuel Bloch for discussions on
experimental realizations of bosonic mixtures. E. Bl. E. Bl. and E. Ba. were
supported by the Swedish Research Council Grants No. 2016-06122, 2018-03659,
and Göran Gustafsson Foundation for Research in Natural Sciences. A. S. and E.
Ba. acknowledge the support from Olle Engkvists stiftelse. The computations
were enabled by resources provided by the Swedish National Infrastructure for
Computing (SNIC) at the National Supercomputer Centre (NSC) partially funded
by the Swedish Research Council through grant agreement no. 2018-05973.

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##  Appendix A Iterations of FEM

In Fig. 13, we show ICL scores reported at each of up to $25$ iterations by
various combinations of submodel and $k$ for our GSWLC-2 sample. These
‘iteration profiles’ are mostly quite flat; hence, $25$ iterations are more
than sufficient for allowing FEM to stabilise to an outcome. In addition, the
bulk of the clustering structure appears to be determined during the k-means
initialisation step, which spreads the cluster centres out ahead of the first
iteration. The ICL criterion rewards separated clusters, so k-means
initialisations are particularly well suited to yielding useful clustering
outcomes. Trials of the use of uniform random initialisations resulted in more
combinations of submodels and $k$ failing to converge.

Variations in the ICL values reported by individual combinations of submodel
and $k$ over successive iterations arise due to the Fisher step of FEM, in
which the subspace within which the clusters are to be modelled is found.
Hence, the updating of the model parameters during the Maximisation step is
indirectly related to the probabilities calculated in the Expectation step.
For traditional EM algorithms, these steps are directly related and thereby
guarantee convergence. The large changes between successive iterations
exhibited by some combinations (e.g. $\Sigma$, $\delta_{k}$, $k=9$) are most
often due to the emptying of clusters; a reduction in the number of clusters
used by FEM leads, in these cases, to a sudden increase in ICL.

Figure 13: ICL scores reported at iterations $1$ through $25$ by various
combinations of submodel and $k$ for our GSWLC-2 sample. For each submodel, we
show the value of $k$ which yields the highest ICL score. These iteration
profiles are generally quite flat, indicating that FEM quickly converges to a
stable outcome. The large changes exhibited by $\Sigma$, $\delta_{k}$, $k=9$
are due to the emptying of clusters as it iterates.

##  Appendix B Smoothing of feature data for our GSWLC-2 sample

Preliminary tests revealed that a truncated, bimodal substructure among
passive galaxies within the nine-dimensional colour space representing our
GSWLC-2 sample (see the left-hand plot of Fig. 14; also visible in Fig. 3) led
to an inability of FEM to converge for the majority of submodels and values of
$k$. This truncated bimodal substructure is due to the lack of input NIR
photometry to the CIGALE SED estimation of GSWLC-2 galaxies, such that their
NIR SEDs must be inferred from UV and optical photometry. This, in turn, leads
to poorly constrained, discretised metallicities: galaxies at $r-K_{s}\lesssim
0.67$ peak strongly at $\log_{10}(Z)\sim-2.4$, and those at $r-K_{s}\gtrsim
0.67$ at $\log_{10}(Z)\sim-2.1$. The NIR SEDs of VIPERS galaxies, on the other
hand, are constrained by $K_{s}$-band photometry and hence have slightly more
freedom to vary. This smooths their colour and metallicity distributions.

Figure 14: The effect of our smoothing on the distribution of GSWLC-2 galaxies
in the passive region of the $NUV-r-K_{s}$ colour-colour plane. Substructures
in the distribution of galaxies within this region are preserved post-
smoothing.

We hence opt to apply a small level of Gaussian smoothing to the GSWLC-2
distributions of the rest-frame absolute magnitudes reported by CIGALE. The
smoothing scale for the rest-frame absolute magnitude of a given galaxy is
given by its Bayesian error. These errors are winsorised at the mean value of
the logarithmic distribution of errors (i.e. errors larger than the mean value
are set to the mean value). This winsorisation ensures that the smoothing
scale is kept small enough to avoid the potential loss of astrophysically
meaningful substructures, while still enabling FEM to converge more readily.
The absolute rest-frame magnitude most affected by this smoothing is $FUV$,
whose errors are winsorised at a maximum value of $0.25$ (all other magnitudes
have a maximum error $<0.1$ after winsorisation). The right-hand plot of Fig.
14 demonstrates the effect of our smoothing, showing that the bimodality in
the colours of passive galaxies is retained post-smoothing. While this
bimodality is likely to be an artefact, trends in the astrophysical features
of galaxies between its peaks are still likely to be genuine (see also Section
4.4.2).

##  Appendix C Behaviour of the various submodels of FEM for our samples

Our model selection approach considers ICL scores for $72$ different
combinations of submodel and $k$ for each of our samples. The comparison of
these $72$ combinations is simplified greatly by the realisation that several
submodels exhibit consistent patterns of behaviour across all values of $k$.

FEM is unable to converge to an outcome for several combinations of submodel
and $k$. The most common diagnosis made by FEM in the case of non-convergence
is that a cluster has become empty (i.e. that it no longer contains galaxies).
Table 1 shows that several submodels are unable to converge beyond a maximum
value of $k$, suggesting a limit to their ability to properly partition the
samples. Alternatively, submodels that converge at $k$, but fail to converge
at $k-1$ and $k+1$ appear to be striking a ‘sweet spot’ in terms of this
ability. Different combinations are generally very consistent with respect to
convergence, converging for either all or none of our $100$ initialisations.

Given their flexibility and their high levels of parametrisation, the
$\Sigma_{k}$, $\delta_{k}$ and $\Sigma_{k}$, $\delta$ submodels offer the
greatest promise among all of the FEM submodels for yielding detailed and
astrophysically meaningful partitions of our samples. The outcomes they
produce are similar; they exhibit near-identical trends in their ICL scores
for $k=2$ through $k=5$ for our GSWLC-2 sample in Table 1. They differ only in
their treatment of the noise terms, which appears to be a minor detail in
comparison with their shared use of full, unique covariance matrices. Outcomes
at higher values of $k$ generally consist of splits of clusters present in
outcomes at lower values of $k$.

Submodels featuring non-unique covariance matrices for the Gaussian density
functions representing the clusters (i.e. submodels with $\Sigma$ and
$\alpha$, such that they all have the same shape) consistently produce
clusters with highly disparate sizes. Some clusters are large, containing $30$
to $60$ per cent of the galaxies in our samples each (and each often spanning
both blue and red galaxies); others are empty or nearly empty, containing
$\lesssim 1$ per cent of the galaxies in our samples each. Nearly-empty
clusters appear to capture small, undesirable artefacts in the structure of
our samples within their input feature spaces. While it is unclear why FEM
registers a valid ICL score for these outcomes when they include empty
clusters (often cited as a cause for the failure of FEM; see above), it is
clear that these submodels are too crude to return more than a very broad
partition of our samples, and that their outcomes are limited in their
capacity for astrophysical interpretation. All of this is also true for the
$\Sigma$, $\delta$ clustering outcome at $k=9$ for our GSWLC-2 sample, which
achieved the highest ICL score in our model selection search despite including
empty and nearly-empty clusters. For these reasons, we reject this outcome for
analysis.

A general property of clustering outcomes reported by submodels which assume
diagonal covariance matrices ($\alpha_{k,j}$, $\alpha_{k}$) for the Gaussian
density functions within the discriminative latent subspace is that they
segment our samples principally along a single dimension. Several
representative examples of their clustering structures are shown in Fig. 15,
revealing that this single dimension is most strongly associated with the UV
colours among our nine input features, with little-to-no distinction made
between galaxies based on their NIR colours. We note that these submodels
scored highest when we tested clustering of our samples using $i$-band
magnitudes of galaxies as a reference point for defining colours (as in Siudek
et al. 2018b; see also Section 3.4), producing the same striping pattern
within the $NUV-r-K_{s}$ plane. While this simple segmentation does correspond
broadly with incremental changes in the star formation activity of galaxies
within our samples, other submodels (with $\Sigma_{k}$) return more detailed
partitions and achieve higher ICL scores anyway.

The large spread in the ICL scores reported in Table 1 arises directly from a
large spread in the log-likelihood values of the fits. This large spread in
the log-likelihood values arises, in turn, primarily from a $1/\delta_{k}$
coefficient in the log-likelihood function of DLM model (which may be seen in
full in appendix 2 of Bouveyron & Brunet 2012). Submodels which yield very
large but negative log-likelihood (and hence, ICL) values tend to have very
small $\delta_{k}$ values for most (if not all) of their clusters; usually
$0.001$, which is the floor that FEM imposes upon the value of $\delta_{k}$.
Very small values of $\delta_{k}$ produce very large, positive values of
$1/\delta_{k}$, and (via a $-1/2$ coefficient of the log-likelihood function)
very large, negative values of the log-likelihood and, thus, of the ICL
criterion. The addition of this especially low-variance noise to subspace
Gaussians leads to highly peaked full space Gaussians which are unlikely to
reflect the more continuous distributions of both samples (see Fig. 3).

Figure 15: Examples of the clustering structures determined by $\alpha_{k,j}$
and $\alpha_{k}$ submodels for our GSWLC-2 sample, shown in the $NUV-r-K_{s}$
colour-colour plane. The combination of submodel and $k$ for each outcome is
shown to the lower-right of each plot. Individual galaxies are coloured in
accordance with the cluster to which they belong. The choice of colours in
this figure is not intended to imply any trends within or between plots. The
horizontal striping pattern exhibited by these examples in these plots, which
is a general property of $\alpha_{k,j}$\- and $\alpha_{k}$-based outcomes,
indicates segmentation mainly along a single axis.