# Lagrangian combinatorics of matroids

Federico Ardila  San Francisco State University and Universidad de Los Andes
<EMAIL_ADDRESS>,  Graham Denham  University of Western Ontario
<EMAIL_ADDRESS>and  June Huh  Princeton University<EMAIL_ADDRESS>
###### Abstract.

The _Lagrangian geometry of matroids_ was introduced in [ADH20] through the
construction of the _conormal fan_ of a matroid $\operatorname{\mathrm{M}}$.
We used the conormal fan to give a Lagrangian-geometric interpretation of the
$h$-vector of the broken circuit complex of $\operatorname{\mathrm{M}}$: its
entries are the degrees of the mixed intersections of certain convex piecewise
linear functions $\gamma$ and $\delta$ on the conormal fan of
$\operatorname{\mathrm{M}}$. By showing that the conormal fan satisfies the
Hodge-Riemann relations, we proved Brylawski’s conjecture that this $h$-vector
is a log-concave sequence.

This sequel explores the _Lagrangian combinatorics of matroids_ , further
developing the combinatorics of biflats and biflags of a matroid, and relating
them to the theory of basis activities developed by Tutte, Crapo, and Las
Vergnas. Our main result is a combinatorial strengthening of the $h$-vector
computation: we write the $k$-th mixed intersection of $\gamma$ and $\delta$
explicitly as a sum of biflags corresponding to the
$\operatorname{{\mathit{nbc}}}$-bases of internal activity $k+1$.

###### Contents

  1. 1 Introduction.
    1. 1.1 Enumerative combinatorics of matroids.
    2. 1.2 Lagrangian combinatorics of matroids.
  2. 2 Biflats and biflags of matroids.
    1. 2.1 Combinatorics of biflags.
    2. 2.2 Extended $\operatorname{{\mathit{nbc}}}$ biflags.
  3. 3 Lower bound for $\gamma^{k}\delta^{n-k-1}$.
    1. 3.1 The canonical expansion of $\delta^{m}$.
    2. 3.2 $h_{r-k}$-many $\operatorname{{\mathit{nbc}}}$ monomials in $\delta^{n-k-1}$.
    3. 3.3 Multiplying by a high power of $\gamma$ eradicates non-initial monomials.
    4. 3.4 $h_{r-k}$-many extended $\operatorname{{\mathit{nbc}}}$ monomials in $\gamma^{k}\delta^{n-k-1}$.
  4. 4 Upper bound for $\gamma^{k}\delta^{n-k-1}$.
    1. 4.1 The jump sets of a resistant term of $\delta^{n-k-1}$.
    2. 4.2 The resistant terms of $\delta^{n-k-1}$ are determined by the bottom row of their table.
    3. 4.3 The resistant terms of $\delta^{n-k-1}$ have no mixed biflats.
    4. 4.4 The resistant terms of $\delta^{n-k-1}$ are $\operatorname{{\mathit{nbc}}}$ monomials.

##  1\. Introduction.

Let $\operatorname{\mathrm{M}}$ be a matroid of rank $r+1$ on $n+1$ elements
with no loops and no coloops. The _Lagrangian geometry of matroids_ was
introduced in [ADH20] through the construction of the _conormal fan_ of
$\operatorname{\mathrm{M}}$. The conormal fan of $\operatorname{\mathrm{M}}$
is a Lagrangian analogue of the Bergman fan of $\operatorname{\mathrm{M}}$,
which in turn is a tropical geometric model of $\operatorname{\mathrm{M}}$. We
used the conormal fan to give a tropical geometric interpretation of the
$h$-vector of the broken circuit complex
$\operatorname{BC}(\operatorname{\mathrm{M}})$. Explicitly, we identified
convex piecewise linear functions $\gamma$ and $\delta$ such that

$\gamma^{k}\delta^{n-k-1}\cap
1_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}=h_{r-k}(\operatorname{BC}(\operatorname{\mathrm{M}}))\
\ \text{for all $k$,}$

where $1_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}$ is the
top-dimensional constant Minkowski weight $1$ on the conormal fan of
$\operatorname{\mathrm{M}}$ [ADH20, Theorem 1.2]. We also showed that the
conormal fan is _Lefschetz_ , and in particular, satisfies the Hodge–Riemann
relations [ADH20, Theorem 5.27]. Combining these results, we proved
Brylawski’s conjecture from [Bry82] that the $h$-vector of the broken circuit
complex forms a log-concave sequence, that is,

$h_{i}(\operatorname{BC}(\operatorname{\mathrm{M}}))^{2}\geq
h_{i-1}(\operatorname{BC}(\operatorname{\mathrm{M}}))h_{i+1}(\operatorname{BC}(\operatorname{\mathrm{M}}))\
\ \text{for all $i$.}$

In this followup paper, we explore the _Lagrangian combinatorics of matroids_
, which studies the combinatorial structure of the conormal fan. We further
develop the study of biflats and biflags of matroids, initiated in [ADH20],
unveiling a strong connection to the theory of basis activities developed by
Tutte [Tut67], Crapo [Cra69], and Las Vergnas [LV13]. In particular, we
directly relate the mixed intersections $\gamma^{k}\delta^{n-k-1}$ to the
reduced broken circuit complex using the _canonical expansion_ in Section 3,
obtaining the following bijective strengthening of [ADH20, Theorem 1.2] in the
_conormal Chow ring_ of Definition 1.7.

######  Theorem 1.1.

For $0\leq k\leq r$, in the conormal Chow ring of $\operatorname{\mathrm{M}}$,
we have

$\gamma^{k}\delta^{n-k-1}=\sum_{B}x_{{\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)},$

where the sum is over the $\operatorname{{\mathit{nbc}}}$-bases of
$\operatorname{\mathrm{M}}$ of internal activity $k+1$.

The symbol ${\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)$ stands for the extended
$\operatorname{{\mathit{nbc}}}$-biflag associated to $B$ in Section 2.2. By
construction, for every $B$,

$x_{{\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)}\cap
1_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}=1.$

Since $\operatorname{\mathrm{M}}$ has exactly
$h_{r-k}(\operatorname{BC}(\operatorname{\mathrm{M}}))$ bases with internal
activity $k+1$, Theorem 1.1 implies [ADH20, Theorem 1.2]. The original proof
of [ADH20, Theorem 1.2] relied on the special case $k=0$ of Theorem 1.1
[ADH20, Proposition 4.9] and on the theory of Chern–Schartz–MacPherson cycles
of matroids introduced by López de Medrano, Rincón, and Shaw [LdMRS20].

###  1.1. Enumerative combinatorics of matroids.

For the remainder of this paper, we fix a total ordering on the ground set $E$
of $\operatorname{\mathrm{M}}$ and identify $E$ with the set
$\\{0,1,\ldots,n\\}$. We refer to [Wel76] and [Oxl11] for any undefined
matroid terminology. We are interested in the following $r$-dimensional
simplicial complexes associated to $\operatorname{\mathrm{M}}$:

  1. $\bullet$

The _independence complex_ $\operatorname{{IN}}(\operatorname{\mathrm{M}})$,
the collection of subsets of $E$ which do not contain any circuit of
$\operatorname{\mathrm{M}}$.

  2. $\bullet$

The _broken circuit complex_ $\operatorname{BC}(\operatorname{\mathrm{M}})$,
the collection of subsets of $E$ which do not contain any broken circuit of
$\operatorname{\mathrm{M}}$.

A _broken circuit_ is a subset obtained from a circuit of
$\operatorname{\mathrm{M}}$ by deleting the least element in the fixed
ordering on $E$. The broken circuit complex
$\operatorname{BC}(\operatorname{\mathrm{M}})$ is the cone over the _reduced
broken circuit complex_
$\operatorname{\overline{BC}}(\operatorname{\mathrm{M}})$ with apex $0$.

For a simplicial complex $\Delta$ of dimension $r$, its _$f$ -vector_
$f(\Delta)=(f_{0},f_{1},\ldots,f_{r+1})$ is defined by

$f_{i}(\Delta)=\text{the number of faces in $\Delta$ with $i$ vertices}.$

The $f$-vector is often stored more compactly in the _$h$ -vector_
$h(\Delta)=(h_{0},h_{1},\ldots,h_{r+1})$, given by

$\sum_{i=0}^{r+1}f_{i}(\Delta)q^{r-i+1}=\sum_{i=0}^{r+1}h_{i}(\Delta)(q+1)^{r-i+1}.$

The $h$-vector of the broken circuit complex is given by

$h_{r-k}(\operatorname{BC}(\operatorname{\mathrm{M}}))=h_{r-k}(\operatorname{\overline{BC}}(\operatorname{\mathrm{M}}))=t_{k+1,0}(\operatorname{\mathrm{M}})\
\ \text{for all $k$},$

where $t_{i,j}(\operatorname{\mathrm{M}})$ is the coefficient of $x^{i}y^{j}$
in the Tutte polynomial $T_{\operatorname{\mathrm{M}}}(x,y)$ [Bjö92]. In
particular, $h_{r+1}(\operatorname{BC}(\operatorname{\mathrm{M}}))$ is zero,
and $h_{r}(\operatorname{BC}(\operatorname{\mathrm{M}}))$ is Crapo’s _beta
invariant_

$\beta_{\operatorname{\mathrm{M}}}=t_{1,0}(\operatorname{\mathrm{M}}).$

The authors of [AHK18] and [ADH20] proved the following results, conjectured
by Mason and Hoggar [Mas72, Hog74] and by Brylawski and Dawson [Bry82, Daw84],
respectively.

######  Theorem 1.2.

The following hold for any matroid $\operatorname{\mathrm{M}}$.

  1. (1)

The $f$-vectors of $\operatorname{{IN}}(\operatorname{\mathrm{M}})$ and
$\operatorname{BC}(\operatorname{\mathrm{M}})$ are log-concave [AHK18].

  2. (2)

The $h$-vectors of $\operatorname{{IN}}(\operatorname{\mathrm{M}})$ and
$\operatorname{BC}(\operatorname{\mathrm{M}})$ are log-concave [ADH20].

We note that the independence complex of any matroid is the reduced broken
circuit complex of another matroid [Bry77, Theorem 4.2], and, for any
simplicial complex, the log-concavity of its $h$-vector implies the log-
concavity of its $f$-vector [Bre94, Corollary 8.4]. Also, for Theorem 1.2, we
may suppose that $\operatorname{\mathrm{M}}$ has no loops and no coloops. We
thus focus on the $h$-vector of the broken circuit complex of a matroid with
no loops and no coloops. One of the main ingredients in the proof of Theorem
1.2 in this case is the above-mentioned formula for the $h$-vector [ADH20,
Theorem 1.2], which we strengthen in Theorem 1.1.

######  Remark 1.3.

For matroids representable over the field of complex numbers, the intersection
theoretic formula for the $h$-vector of the broken circuit complex was given
in [DGS12], and the connection to the Chern–Schwartz–MacPherson classes was
observed in [Huh13] and [Huh15]. Varchenko’s conjecture on the number of
critical points of products of linear forms [Var95], proved by Orlik and Terao
in [OT95], is equivalent to the central special case of the formula

$\delta^{n-1}\cap
1_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}=h_{r}(\operatorname{BC}(\operatorname{\mathrm{M}})).$

Recently, for any matroid, Berget, Eur, Spink, and Tseng proved a very general
and closely related formula

$\sum_{i+j+k+l=n}\Bigg{(}\int_{X_{E}}\alpha^{i}\beta^{j}c_{k}(\mathscr{S}^{\vee}_{\operatorname{\mathrm{M}}})c_{l}(\mathscr{Q}_{\operatorname{\mathrm{M}}})\Bigg{)}(x+y)x^{i}y^{j}z^{k}w^{l}=(y+z)^{r}(x+w)^{n-r+1}T_{\operatorname{\mathrm{M}}}\Big{(}\frac{x+y}{y+z},\frac{x+y}{x+w}\Big{)},$

where $c_{k}(\mathscr{S}^{\vee}_{\operatorname{\mathrm{M}}})$ and
$c_{l}(\mathscr{Q}_{\operatorname{\mathrm{M}}})$ are _tautological Chern
classes_ of $\operatorname{\mathrm{M}}$ [BEST21, Theorem A], and they used it
to give another proof of Theorem 1.2. To deduce Theorem 1.2 from their
formula, they used the fact [AFR10, DF10] that many functions of matroids
behave valuatively under matroid polytope subdivisions. This allowed them to
reduce key computations to the case of representable matroids and prove them
using an algebro-geometric argument, thus avoiding the combinatorics of
biflats and biflags. The final part of their proof also employs the fact,
proved in [ADH20], that the conormal fan of a matroid is a Lefschetz fan. We
refer to [BEST21, Remark 9.9] for a detailed comparison of the two proofs of
Theorem 1.2.

######  Example 1.4.

We will use two running examples throughout the paper. The first is the
graphical matroid of the graph $G$ of the pyramid, whose dual is also the
matroid of the pyramid $G^{\perp}$. The second is the graphical matroid of the
graph $H$ of the cube, whose dual is the graphical matroid of the graph
$H^{\perp}$ of the octahedron. These are shown in Figure 1 and Figure 2. The
$f$-vectors and $h$-vectors of their broken circuit complexes are shown in
Table 1.

|  $f$-vector of $\operatorname{\overline{BC}}(\operatorname{\mathrm{M}})$ |  $h$-vector of $\operatorname{\overline{BC}}(\operatorname{\mathrm{M}})$
---|---|---
pyramid $G$ |  $(1,7,17,14)$ |  $(1,4,6,3)$
cube $H$ |  $(1,11,55,159,282,290,133)$ |  $(1,5,15,29,40,32,11)$
Table 1. The $f$-vectors and $h$-vectors of the broken circuit complexes of
two graphs. 76543 210  01234 567  Figure 1. The graph $G$ of the pyramid and
its dual graph $G^{\perp}$. 3210$b$$a$984567 150$b$7$a$984326 Figure 2. The
graph $H$ of the cube and its dual graph $H^{\perp}$.

###  1.2. Lagrangian combinatorics of matroids.

This paper explores the algebraic-combinatorial structure of the conormal Chow
ring ${A_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}$
introduced in [ADH20, Section 3.5]. The conormal Chow ring is an extension of
the Chow ring $A_{\operatorname{\mathrm{M}}}$ studied in [FY04] and [AHK18].
We recall the central combinatorial notions from [ADH20].

  1. $\bullet$

A _biflat_ $F|G$ of $\operatorname{\mathrm{M}}$ consists of a flat $F$ of
$\operatorname{\mathrm{M}}$ and a flat $G$ of the dual matroid
$\operatorname{\mathrm{M}}^{\perp}$ such that they are nonempty, they are not
both equal to $E$, and their union is $E$.

  2. $\bullet$

Two biflats $F|G$ and $F^{\prime}|G^{\prime}$ of $\operatorname{\mathrm{M}}$
are _compatible_ if

$\text{($F\subseteq F^{\prime}$ and $G\supseteq G^{\prime}$)}\ \ \text{or}\ \
\text{($F\supseteq F^{\prime}$ and $G\subseteq G^{\prime}$)}.$

  3. $\bullet$

A _biflag_ of $\operatorname{\mathrm{M}}$ is a collection
${\mathscr{F}}|{\mathscr{G}}$ of pairwise compatible biflats of
$\operatorname{\mathrm{M}}$ satisfying

$\bigcup_{F|G\in{\mathscr{F}}|{\mathscr{G}}}F\cap G\neq E.$

The _length_ of ${\mathscr{F}}|{\mathscr{G}}$ is $k$, the number of biflats it
contains.

######  Remark 1.5.

Let ${\mathscr{F}}$ be an increasing sequence of $k$ nonempty flats of
$\operatorname{\mathrm{M}}$, say

${\mathscr{F}}=(\emptyset\subsetneq F_{1}\subseteq\cdots\subseteq
F_{k}\subseteq E),$

and let ${\mathscr{G}}$ be a decreasing sequence of $k$ nonempty flats of
$\operatorname{\mathrm{M}}^{\perp}$, say

${\mathscr{G}}=(E\supseteq G_{1}\supseteq\cdots\supseteq
G_{k}\supsetneq\emptyset).$

Then the collection ${\mathscr{F}}|{\mathscr{G}}$ consisting of the pairs
$F_{1}|G_{1},\ldots,F_{k}|G_{k}$ is a biflag of $\operatorname{\mathrm{M}}$ if
and only if

$F_{j}\cup G_{j}=E$ for all $1\leq j\leq k$ and $F_{j}\cup G_{j+1}\neq E$ for
some $0\leq j\leq k.$

See [ADH20, Proposition 2.15] for a straightforward verification.

######  Example 1.6.

We write ${\mathscr{F}}|{\mathscr{G}}$ as a table with rows ${\mathscr{F}}$
and ${\mathscr{G}}$, augmented with the columns $\emptyset|E$ and
$E|\emptyset$. For example, for the square pyramid graph $G$ of Figure 1,
$01256|1347$ is a biflat and

$\begin{array}[]{|c|c|c|}\hline\cr\emptyset&01256&E\\\ E&1347&\emptyset\\\
\hline\cr\end{array}\subset\begin{array}[]{|c|cccc|c|}\hline\cr\emptyset&5&56&01256&E&E\\\
E&E&E&1347&3&\emptyset\\\
\hline\cr\end{array}\subset\begin{array}[]{|c|cccccc|c|}\hline\cr\emptyset&5&56&01256&01256&E&E&E\\\
E&E&E&1347&347&347&3&\emptyset\\\ \hline\cr\end{array}$

are three biflags of biflats; the third one is maximal.

The biflats and biflags of a matroid encode the combinatorics of the conormal
fan, described in [ADH20, Section 3.4]. Our main algebraic object of interest
is the Chow ring of the conormal fan, which we now define independently. We
begin with a polynomial ring
${S_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}$ with real
coefficients and variables $x_{F|G}$ indexed by the biflats of
$\operatorname{\mathrm{M}}$. For any set of biflats
${\mathscr{F}}|{\mathscr{G}}$, we consider the monomial

$\displaystyle
x_{{\mathscr{F}}|{\mathscr{G}}}=\prod_{F|G\in{\mathscr{F}}|{\mathscr{G}}}x_{F|G}.$

We also define, for every element $i$ in the ground set $E$, the linear forms

$\gamma_{i}=\sum_{i\in F,\,F\neq
E}x_{F|G},\qquad\overline{\gamma}_{i}=\sum_{i\in G,\,G\neq
E}x_{F|G},\qquad\delta_{i}=\sum_{i\in F\cap G}x_{F|G}.$

These linear forms correspond to certain convex piecewise linear functions on
the conormal fan of $\operatorname{\mathrm{M}}$ [ADH20, Section 3.4]. We write

  1. $\bullet$

${I_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}$ for the
ideal generated by the monomials $x_{{\mathscr{F}}|{\mathscr{G}}}$, where
${\mathscr{F}}|{\mathscr{G}}$ is not a biflag, and

  2. $\bullet$

${J_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}$ for the
ideal generated by the linear forms $\gamma_{i}-\gamma_{j}$ and
$\overline{\gamma}_{i}-\overline{\gamma}_{j}$, for any $i$ and $j$ in $E$.

######  Definition 1.7.

The _conormal Chow ring_ of $\operatorname{\mathrm{M}}$ is the quotient

${A_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}={S_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}/({I_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}+{J_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}).$

The equivalence classes of $\gamma_{i}$ and $\overline{\gamma}_{i}$ in the
conormal Chow ring do not depend on $i$. We denote these classes by $\gamma$
and $\overline{\gamma}$, respectively.

We note that the equivalence class of $\delta_{i}$ in the conormal Chow ring
also do not depend on $i$: For every biflat $F|G$, the element $i$ must be in
$F$ or in $G$, and hence

$\delta_{i}=\gamma_{i}+\overline{\gamma}_{i}-\sum_{F\neq E,G\neq E}x_{F|G}.$

We write $\delta$ for the class of these elements in the conormal Chow ring.
In [ADH20, Section 3.5], we constructed the _degree map_

$\deg:A_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}^{n-1}\longrightarrow\mathbb{R},\qquad
x_{{\mathscr{F}}|{\mathscr{G}}}\longmapsto x_{{\mathscr{F}}|{\mathscr{G}}}\cap
1_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}=\begin{cases}1&\text{if
${\mathscr{F}}|{\mathscr{G}}$ is a biflag,}\\\ 0&\text{if
${\mathscr{F}}|{\mathscr{G}}$ is not a biflag.}\end{cases}$

In [ADH20, Theorem 1.2], we gave the following interpretation of
$h(\operatorname{BC}(\operatorname{\mathrm{M}}))$ in terms of the conormal
intersection theory of $\operatorname{\mathrm{M}}$:

$\deg(\gamma^{k}\delta^{n-k-1})=\gamma^{k}\delta^{n-k-1}\cap
1_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}=h_{r-k}(\operatorname{BC}(\operatorname{\mathrm{M}}))\
\ \text{for $0\leq k\leq r$.}$

Our goal is to give a bijective proof of this numerical identity.

Acknowledgments. The first author thanks the Mathematical Sciences Research
Institute, the Simons Institute for the Theory of Computing, the Sorbonne
Université, the Università di Bologna, and the Universidad de Los Andes for
providing wonderful settings to work on this project, and Felipe Rincón for
valuable conversations on this topic; his research is partially supported by
NSF grant DMS-1855610 and Simons Fellowship 613384. The second author’s
research is supported by NSERC of Canada. The third author is partially
supported by NSF Grant DMS-2053308 and the Simons Investigator Grant.

##  2\. Biflats and biflags of matroids.

Throughout the paper, we will fix a loopless and coloopless matroid
$\operatorname{\mathrm{M}}$ with ground set $E$ of size $n+1$ and rank $r+1$.
The dual matroid $\operatorname{\mathrm{M}}^{\perp}$ has rank
$r^{\perp}+1\coloneq(n+1)-(r+1)=n-r$.

###  2.1. Combinatorics of biflags.

The following lemma is the combinatorial manifestation of the fact that the
conormal fan is pure of dimension $n-1$. Its proof introduces several ideas
that will be useful in what follows.

######  Lemma 2.1.

Every maximal biflag of $\operatorname{\mathrm{M}}$ has length $n-1$.

###### Proof.

Let ${\mathscr{F}}|{\mathscr{G}}$ be a biflag consisting of biflats
$F_{1}|G_{1},\ldots,F_{k}|G_{k}$, say

${\mathscr{F}}=(\emptyset\subsetneq F_{1}\subseteq\cdots\subseteq
F_{k}\subseteq E)\ \ \text{and}\ \ {\mathscr{G}}=(E\supseteq
G_{1}\supseteq\cdots\supseteq G_{k}\supseteq\emptyset).$

For every $i$, we write $r_{i}$ for the rank of $F_{i}$ in
$\operatorname{\mathrm{M}}$ and $r_{i}^{\perp}$ for the rank of $G_{i}$ in
$\operatorname{\mathrm{M}}^{\perp}$. We have

$\sum_{i=1}^{k+1}\left((r_{i}-r_{i-1})+(r^{\perp}_{i-1}-r^{\perp}_{i})\right)=(r+1)+(r^{\perp}+1)=n+1.$

If $j$ is an index satisfying $F_{j}\cup G_{j+1}\neq E$, then we must have
$F_{j}\neq F_{j+1}$ and $G_{j}\neq G_{j+1}$, so

$(r_{j}-r_{j-1})+(r^{\perp}_{j-1}-r^{\perp}_{j})\geq 2.$

Since every summand of $n+1$ is positive, there can be at most $n$ summands,
so $k\leq n-1$.

We now show that ${\mathscr{F}}|{\mathscr{G}}$ is not maximal if $k<n-1$. We
consider two cases separately.

  1. (1)

We have $r_{i}-r_{i-1}\geq 2$ or $r^{\perp}_{i-1}-r^{\perp}_{i}\geq 2$ for
some $i$.

Suppose $r_{i}-r_{i-1}\geq 2$. Choose any flat $F$ with $F_{i-1}\subsetneq
F\subsetneq F_{i}$. If $F\cup G_{i}\neq E$, then $F|G_{i-1}$ is a biflat, and
hence
${\mathscr{F}}^{+}|{\mathscr{G}}^{+}\coloneq{\mathscr{F}}|{\mathscr{G}}\cup
F|G_{i-1}$ is a biflag. On the other hand, if $F\cup G_{i}=E$, then $F|G_{i}$
is a biflat, and
${\mathscr{F}}^{+}|{\mathscr{G}}^{+}\coloneq{\mathscr{F}}|{\mathscr{G}}\cup
F|G_{i}$ is a biflag.

  1. (2)

We have $r_{i}-r_{i-1}\leq 1$ and $r^{\perp}_{i-1}-r^{\perp}_{i}\leq 1$ for
all $i$.

In this case, $(r_{i}-r_{i-1})+(r^{\perp}_{i-1}-r^{\perp}_{i})\leq 2$ for all
$i$, so at least two summands of $n+1$ are equal to $2$. Therefore, at least
two values $i=j,j^{\prime}$ satisfy
$r_{i}-r_{i-1}=r^{\perp}_{i-1}-r^{\perp}_{i}=1$, and hence $F_{i-1}\subsetneq
F_{i}$ and $G_{i-1}\supsetneq G_{i}$. At least one of these, say $j^{\prime}$,
satisfies $F_{j^{\prime}-1}\cup G_{j}^{\prime}\neq E$. Consider the other one;
$F_{j}|G_{j-1}$ is a biflat, and
${\mathscr{F}}^{+}|{\mathscr{G}}^{+}\coloneq{\mathscr{F}}|{\mathscr{G}}\cup
F_{j}|G_{j-1}$ is a biflag, because it satisfies $F_{j^{\prime}-1}\cup
G_{j}^{\prime}\neq E$. ∎

We will often use the following basic result [ADH20, Lemma 3.15].

######  Lemma 2.2.

If $F$ and $G$ are nonempty flats of $\operatorname{\mathrm{M}}$ and
$\operatorname{\mathrm{M}}^{\perp}$ respectively, then $\left|F\cup
G\right|\neq n$.

We close this section with some definitions that will be useful throughout the
paper. Let ${\mathscr{F}}|{\mathscr{G}}$ be a biflag of
$\operatorname{\mathrm{M}}$. As above, we write ${\mathscr{F}}$ and
${\mathscr{G}}$ for the flags

${\mathscr{F}}=(\emptyset\subsetneq F_{1}\subseteq\cdots\subseteq
F_{k}\subseteq E)\ \ \text{and}\ \ {\mathscr{G}}=(E\supseteq
G_{1}\supseteq\cdots\supseteq G_{k}\supseteq\emptyset),$

where $k$ is the length of ${\mathscr{F}}|{\mathscr{G}}$, and
$F_{0}=G_{k+1}=\emptyset$ and $F_{k+1}=G_{0}=E$.

######  Definition 2.3.

The _gap sequence_ of ${\mathscr{F}}|{\mathscr{G}}$, denoted
$\operatorname{{\mathscr{D}}}({\mathscr{F}}|{\mathscr{G}})$, is the sequence
of _gaps_

$D_{0}|D_{1}|\cdots|D_{k},\ \ \text{where }\ \
D_{j}\coloneq(F_{j+1}-F_{j})\cap(G_{j}-G_{j+1})=E-(F_{j}\cup G_{j+1}).$

The notion of gaps appears naturally in the model of the conormal fan as a
configuration space [ADH20, Section 2]. Each gap must have size at least $2$,
as we noted above.

######  Definition 2.4.

The _jump sets_ of ${\mathscr{F}}|{\mathscr{G}}$ are

$\displaystyle\,\mathsf{J}({\mathscr{F}})$ $\displaystyle=\left\\{j\mid 0\leq
j\leq k\text{~{}and~{}}F_{j}\subsetneq F_{j+1}\right\\},$
$\displaystyle\,\mathsf{J}({\mathscr{G}})$ $\displaystyle=\left\\{j\mid 0\leq
j\leq k\text{~{}and~{}}G_{j}\supsetneq G_{j+1}\right\\}.$

The _double jump set_ is
$\,\mathsf{J}({\mathscr{F}})\cap\,\mathsf{J}({\mathscr{G}})$; the elements of
these sets are called _jumps_ of ${\mathscr{F}}$ and ${\mathscr{G}}$ and
_double jumps_ of ${\mathscr{F}}|{\mathscr{G}}$, respectively.

If the gap $D_{j}$ is nonempty, then $j$ must be a double jump. Thus, by
definition, every biflag has at least one double jump. The double jumps of a
biflag ${\mathscr{F}}|{\mathscr{G}}$ play an important role throughout the
paper, so we mark them
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}$
and
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}$
in the table for ${\mathscr{F}}|{\mathscr{G}}$.

######  Example 2.5.

Consider the biflag ${\mathscr{F}}|{\mathscr{G}}$ given by the table

$\begin{array}[]{|c|cccccc|c|}\hline\cr\emptyset&5&56&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&01256&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&E\\\
E&E&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&1347&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&3&\emptyset\\\
\hline\cr\end{array}\ .$

The biflag has jump sets $\,\mathsf{J}({\mathscr{F}})=\\{0,1,2,3\\}$ and
$\,\mathsf{J}({\mathscr{G}})=\\{2,3,4\\}$. The set of double jumps is
$\\{2,3\\}$, and the gap sequence is $\emptyset|\emptyset|02|47|\emptyset|$.

######  Lemma 2.6.

The gaps of ${\mathscr{F}}|{\mathscr{G}}$ are pairwise disjoint, and their
union is

$\bigsqcup_{j=0}^{k}D_{j}=E-\bigcup_{i=1}^{k}\left(F_{i}\cap G_{i}\right).$

###### Proof.

One easily verifies the first claim. For the second, first suppose $e\in
F_{i}\cap G_{i}$. Then $e\in F_{j}$ for all $j\geq i$, which means $e\not\in
D_{j}$ for $i\leq j\leq k$. Dually, $e\in G_{j}$ for all $j\leq i$, so
$e\not\in D_{j}$ for all $0\leq i\leq j-1$. Now suppose $e$ is not in any gap.
In this case, consider the index $1\leq i\leq k+1$ for which $e\in
F_{i}-F_{i-1}$. Since $e\in F_{i-1}\cup G_{i}$, we must have $e\in G_{i}$ and
hence $e\in F_{i}\cap G_{i}$. ∎

###  2.2. Extended $\operatorname{{\mathit{nbc}}}$ biflags.

In this section, we construct the biflag ${\mathscr{F}}(B)|{\mathscr{G}}(B)$
and the extended biflag ${\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)$ associated
to each $\operatorname{{\mathit{nbc}}}$ basis $B$ of
$\operatorname{\mathrm{M}}$: These are the biflags we need to give a
combinatorial formula for $\gamma^{k}\delta^{n-k-1}$ in Theorem 1.1.

We begin by recalling some basic facts about matroids on an ordered ground
set. For each basis $B\subseteq E$ of $\operatorname{\mathrm{M}}$, we write
$B^{\perp}\coloneq E-B$ for the corresponding dual basis of
$\operatorname{\mathrm{M}}^{\perp}$. Each $i\notin B$ has a unique dependence
on $B$, that is, there is a unique _fundamental circuit_ $C(B,i)$ contained in
$B\cup i$ and containing $i$. Dually, for each $i\in B$, there is a unique
_fundamental cocircuit_ in $B^{\perp}\cup{i}$ containing $i$, denoted
$C^{\perp}(B,i)$.

######  Definition 2.7.

The externally active set for $B$ in $\operatorname{\mathrm{M}}$ is defined to
be

$\displaystyle\operatorname{{EA}}(B)$ $\displaystyle=\left\\{i\notin B\mid
i=\min C(B,i)\right\\}$ $\displaystyle=\left\\{i\notin B\mid B\textrm{ is the
lexicographically largest basis contained in }B\cup i\right\\}.$

Dually, the internally active set for $B$ in $\operatorname{\mathrm{M}}$ is
defined to be

$\displaystyle\operatorname{{IA}}(B)$ $\displaystyle=\left\\{i\in B\mid i=\min
C^{\perp}(B,i)\right\\}$ $\displaystyle=\left\\{i\in B\mid B\textrm{ is the
lexicographically smallest basis containing }B-i\right\\}.$

The _internally passive_ and _externally passive sets_ of $B$ are
$\operatorname{{IP}}(B)=B-\operatorname{{IA}}(B)$ and
$\operatorname{{EP}}(B)=B^{\perp}-\operatorname{{EA}}(B)$, respectively.

Activites behave well with respect to matroid duality: The internally active
set of $B$ in $\operatorname{\mathrm{M}}$ equals the externally active set of
$B^{\perp}$, and the externally active set of $B$ in
$\operatorname{\mathrm{M}}$ equals the internally active set of $B^{\perp}$.
According to [Tut67, Cra69], the _Tutte polynomial_ of
$\operatorname{\mathrm{M}}$ equals

$T_{\operatorname{\mathrm{M}}}(x,y)=\sum_{B}x^{|\operatorname{{IA}}(B)|}y^{|\operatorname{{EA}}(B)|}.$

In particular, for any natural numbers $i$ and $e$, the number $t_{i,e}$ of
bases $B$ of $\operatorname{\mathrm{M}}$ with $|\operatorname{{IA}}(B)|=i$ and
$|\operatorname{{EA}}(B)|=e$ is independent of the ordering of the ground set
$E$.

######  Definition 2.8.

We say a basis $B$ of $\operatorname{\mathrm{M}}$ is a _no broken circuit
basis_ or _$\operatorname{{\mathit{nbc}}}$ basis_ if
$\operatorname{{EA}}(B)=\emptyset$. This is equivalent to demanding that $B$
does not contain any _broken circuit_ , that is, any set of the form $C-\min
C$ where $C$ is a circuit.

We now associate a biflag to each $\operatorname{{\mathit{nbc}}}$ basis of
$\operatorname{\mathrm{M}}$. We will later see how these biflags arise
naturally in the Lagrangian combinatorics of ordered matroids.

######  Proposition 2.9.

_( $\operatorname{{\mathit{nbc}}}$ biflags)_ Let $B$ be a
$\operatorname{{\mathit{nbc}}}$ basis with $|\operatorname{{IA}}(B)|=k+1$.
Define the sequence
$\mathcal{E}(B)=(e_{1},\ldots,e_{r-k},e_{r-k+1},\ldots,e_{n-k-1})$ by the
conditions

$B-\operatorname{{IA}}(B)=\\{e_{1}>\cdots>e_{r-k}\\}\ \ \text{and}\ \
B^{\perp}-\min B^{\perp}=\\{e_{r-k+1}<\cdots<e_{n-k-1}\\}.$

Let ${\mathscr{F}}(B)$ and ${\mathscr{G}}(B)$ be the flags of length $n-k-1$
given by

$F_{j}|G_{j}=\begin{cases}\operatorname{cl}\\{e_{1},e_{2},\ldots,e_{j}\\}|E&\text{for
$1\leq j\leq r-k$},\\\
E|\operatorname{cl}^{\perp}\\{e_{j},e_{j+1},\ldots,e_{n-k-1}\\}&\text{for
$r-k+1\leq j\leq n-k-1$,}\end{cases}$

where $\operatorname{cl}$ and $\operatorname{cl}^{\perp}$ stand for the
closure operators of $\operatorname{\mathrm{M}}$ and
$\operatorname{\mathrm{M}}^{\perp}$. Then ${\mathscr{F}}(B)|{\mathscr{G}}(B)$
is a biflag of $\operatorname{\mathrm{M}}$.

The biflag ${{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ is the
_$\operatorname{{\mathit{nbc}}}$ biflag_ associated to $B$. The table for
${{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ reads

$\begin{array}[]{|ccccccc|}\hline\cr\operatorname{cl}(e_{1})&\cdots&\operatorname{cl}(e_{1},\ldots,e_{r-k})&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&\cdots&E\\\
E&\cdots&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&\operatorname{cl}^{\perp}(e_{r-k+1},\ldots,e_{n-k-1})&\cdots&\operatorname{cl}^{\perp}(e_{n-k-1})\\\
\hline\cr\end{array}$

We call $x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ an
_$\operatorname{{\mathit{nbc}}}$ monomial_ of $\operatorname{\mathrm{M}}$.

###### Proof.

Let $x$ be the minimum element of $B^{\perp}$. We need to verify that

$\operatorname{cl}(B-\operatorname{{IA}}(B))\cup\operatorname{cl}^{\perp}(B^{\perp}-x)\neq
E.$

Assume this is not the case. Since
$x\notin\operatorname{cl}^{\perp}(B^{\perp}-x)$, we must have
$x\in\operatorname{cl}(B-\operatorname{{IA}}(B))$. Thus the fundamental
circuit $C(B,x)$ satisfies $C(B,x)\subseteq B-\operatorname{{IA}}(B)\cup x$.
Since $B$ is $\operatorname{{\mathit{nbc}}}$, the minimum element of this
circuit is some $y<x$. Also, since $y\in B-\operatorname{{IA}}(B)$, the
minimum element in the fundamental cocircuit $C^{\perp}(B,y)$ is some $z<y$,
and hence $z<x$ as well. However, $C^{\perp}(B,y)\subseteq B^{\perp}\cup y$
shows that $z\in B^{\perp}$, contradicting the minimality of $x$. ∎

######  Example 2.10.

We illustrate the construction above with the example of Figure 2. This
matroid has $n+1=12$ elements, rank $r+1=7$, and corank $r^{\perp}+1=5$. Let
us order the ground set $0<1<\cdots<9<a<b$. Consider the
$\operatorname{{\mathit{nbc}}}$ basis
$B={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}01}5{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}6}78b$,
whose internally active set
$\operatorname{{IA}}(B)={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}016}$
has $k+1=3$ elements, marked in green. We have
$\mathcal{E}(B)=(b,8,7,5;3,4,9,a)$, and ${{\mathscr{F}}(B)|{\mathscr{G}}(B)}$
is given by the table

$\begin{array}[]{|c|ccccccccc|c|}\hline\cr\emptyset&{\bf b}&{\bf 8}b&{\bf
7}8b&{\bf
5}78b&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&E&E&E&E\\\
E&E&E&E&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&0{\bf
3}469a&{\bf 4}69a&6{\bf 9}a&{\bf a}&\emptyset\\\ \hline\cr\end{array}\ .$

For each proper flat $F_{i}$ and proper coflat $G_{i}$, we have written in
bold the new element $e_{i}$ that is not present in $F_{i-1}$ and $G_{i+1}$,
respectively.

We now augment each $\operatorname{{\mathit{nbc}}}$ biflag to a maximal biflag
containing it. Let $B$ be a $\operatorname{{\mathit{nbc}}}$ basis with

$\operatorname{{IA}}(B)=\\{{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}c_{1}}>\cdots>{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}c_{k+1}}\\}.$

Define the sequence $\mathcal{E}(B)$ and the biflag
${\mathscr{F}}(B)|{\mathscr{G}}(B)$ as in Proposition 2.9, and set

$S=B-\operatorname{{IA}}(B)\ \ \text{and}\ \ T=B^{\perp}-\min B^{\perp}.$

######  Proposition 2.11.

_(Extended $\operatorname{{\mathit{nbc}}}$ biflags)_ Let $i$ be the largest
index such that

$c_{i}\notin\operatorname{cl}(S)\cup\operatorname{cl}^{\perp}(T),$

or equivalently, the smallest index such that

$\operatorname{cl}(S,c_{1},\ldots,c_{i})\cup\operatorname{cl}^{\perp}(T)=E.$

We define the flags ${\mathscr{F}}^{+}(B)$ and ${\mathscr{G}}^{+}(B)$ of
length $n-1$ by inserting the following $k$ columns at the double jump between
columns $r-k$ and $r-k+1$ of the table for
${{\mathscr{F}}(B)|{\mathscr{G}}(B)}$:

$\begin{array}[]{|ccccccc|}\hline\cr{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}(S,c_{1})}&\cdots&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}(S,c_{1},\ldots,c_{i-1})}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}(S,c_{1},\ldots,c_{i-1})}&\cdots&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}(S,c_{1},\ldots,c_{k})}\\\
{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}E}&\cdots&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}E}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}^{\perp}(T)}&\cdots&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}^{\perp}(T)}\\\
\hline\cr\end{array}$

Explicitly, we define ${\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)$ by setting

$\displaystyle F^{+}_{j}|G^{+}_{j}$
$\displaystyle=\begin{cases}\operatorname{cl}\\{e_{1},e_{2},\ldots,e_{j}\\}|E&\textrm{
for }1\leq j\leq r-k,\\\
\operatorname{cl}\\{S,c_{1},\ldots,c_{j-(r-k)}\\}|E&\textrm{ for }1\leq
j-(r-k)\leq i-1,\\\
\operatorname{cl}\\{S,c_{1},\ldots,c_{j-(r-k)}\\}|\operatorname{cl}^{\perp}(T)&\textrm{
for }i\leq j-(r-k)\leq k,\\\
E|\operatorname{cl}^{\perp}\\{e_{j-k},e_{j-k+1},\ldots,e_{n-k-1}\\}&\textrm{
for }1\leq j-r\leq n-r-1.\end{cases}$

Then ${\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)$ is a maximal biflag of
$\operatorname{\mathrm{M}}$.

We call ${{\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)}$ the _extended
$\operatorname{{\mathit{nbc}}}$ biflag_ of the $\operatorname{{\mathit{nbc}}}$
basis $B$.

###### Proof.

The only statement requiring proof is the equivalence between the two
different definitions of $i$. Let $i$ be the largest index for which
$c_{i}\notin\operatorname{cl}(S)\cup\operatorname{cl}^{\perp}(T)$. We verify
two statements:

  1. (1)

$\operatorname{cl}(S,c_{1},\ldots,c_{i-1})\cup\operatorname{cl}^{\perp}(T)\neq
E$.

To see this, notice that
$c_{i}\notin\operatorname{cl}(S,c_{1},\ldots,c_{i-1})$ because
$\\{S,c_{1},\ldots,c_{i-1},c_{i}\\}\subseteq B$ is independent.

  1. (2)

$\operatorname{cl}(S,c_{1},\ldots,c_{i})\cup\operatorname{cl}^{\perp}(T)=E$.

For this, let
$\operatorname{cl}(S,c_{1},\ldots,c_{i})\cup\operatorname{cl}^{\perp}(T)=U$,
and consider any element $e\neq\min B^{\perp}$.

  1. (2-1)

If $e\in B^{\perp}-\min B^{\perp}=T$, then $e\in U$.

  2. (2-2)

If $e\in\operatorname{{IP}}(B)=S$, then $e\in
S\subseteq\operatorname{cl}(S)\subseteq U$.

  3. (2-3)

If $e\in\operatorname{{IA}}(B)$, then $e=c_{j}$ for some index $j$. If $j>i$,
then by the maximality of $i$, we have
$c_{j}\in\operatorname{cl}(S)\cup\operatorname{cl}^{\perp}(T)\subseteq U$. If
$j\leq i$, then $c_{j}\in\\{S,c_{1},\ldots,c_{i})\subseteq U$. In either case,
$e\in U$.

We conclude that $U\supseteq E-\min B^{\perp}$. Since $\left|U\right|\neq n$
by Lemma 2.2, we must have $U=E$. ∎

######  Example 2.12.

We continue with Example 2.10, which described the
$\operatorname{{\mathit{nbc}}}$ biflag corresponding to the
$\operatorname{{\mathit{nbc}}}$ basis $B=015678b$ for Figure 2. We now augment
it to a maximal biflag. The set

$\operatorname{{IA}}(B)={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}016}=\\{c_{1}>c_{2}>c_{3}\\}$

determines the two top row entries to be added, namely,

$\operatorname{cl}(S,c_{1})=5678b\ \ \text{and}\ \
\operatorname{cl}(S,c_{1},c_{2})=1256789ab.$

Notice that $i=2$ is the largest index for which $c_{i}\in
E-\operatorname{cl}(S)\cup\operatorname{cl}^{\perp}(T)=12$ and also the
smallest index such that
$\operatorname{cl}(S,c_{1},\ldots,c_{i})\cup\operatorname{cl}^{\perp}(T)=E$.
Therefore the bottom row entries switch from $E$ to $03469a$ between positions
$i-1=1$ and $i=2$:

$\begin{array}[]{|c|ccccccccccc|c|}\hline\cr\emptyset&{\bf b}&{\bf 8}b&{\bf
7}8b&{\bf
5}78b&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}5{\bf
6}78b}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}{\bf
1}256789ab}&E&E&E&E&E\\\
E&E&E&E&E&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}E}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}03469a}&0{\bf
3}469a&{\bf 4}69a&6{\bf 9}a&{\bf a}&\emptyset\\\ \hline\cr\end{array}.$

The resulting table corresponds to the extended
$\operatorname{{\mathit{nbc}}}$ biflag
${{\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)}$.

##  3\. Lower bound for $\gamma^{k}\delta^{n-k-1}$.

We aim to compute the degree of $\gamma^{k}\delta^{n-k-1}$ by expressing it as
a sum of square-free monomials $x_{{\mathscr{F}}|{\mathscr{G}}}$. One
fundamental feature of the computation of $\gamma^{k}\delta^{n-k-1}$, which is
simultaneously an advantage and a difficulty, is that there are many different
ways to carry it out, since we have $n+1$ different definitions of $\gamma$
and $\delta$; namely $\gamma=\gamma_{i}$ and $\delta=\delta_{i}$ for every
$i\in E$. It is not clear from the outset how one should organize this
computation.

To have control over the computation, we require some structure amidst that
freedom. To achieve this, we introduce two key tools in this section:

  1. $\bullet$

(Definition 3.3) a canonical way of expanding powers of $\delta$, and

  2. $\bullet$

(Lemma 3.11) a criterion on a monomial $m$ that guarantees that
$m\cdot\gamma^{k}=0$.

As we will see in Section 4, the criterion in Lemma 3.11 shows that most
expressions of the form $m\cdot\gamma^{k}$ vanish in degree $n-1$. The
procedure in Definition 3.3 will provide the combinatorial structure necessary
to describe the terms that remain.

###  3.1. The canonical expansion of $\delta^{m}$.

Let ${\mathscr{F}}|{\mathscr{G}}$ be a biflag and
$x_{{\mathscr{F}}|{\mathscr{G}}}$ be the corresponding square-free monomial.
Recall from Lemma 2.6 that the union of the gaps of
${\mathscr{F}}|{\mathscr{G}}$ is nonempty and equals

$D_{0}\sqcup\cdots\sqcup D_{k}=E-\bigcup_{j=1}^{k}(F_{j}\cap G_{j}).$

######  Definition 3.1.

(_Canonical expansion of $x_{{\mathscr{F}}|{\mathscr{G}}}\,\delta$_) For a
monomial $x_{{\mathscr{F}}|{\mathscr{G}}}$, let

$e=e({\mathscr{F}}|{\mathscr{G}})\coloneq\max\big{(}E-\bigcup_{j=1}^{k}(F_{j}\cap
G_{j})\big{)},$

Define the _canonical expansion_ of $x_{{\mathscr{F}}|{\mathscr{G}}}\delta$ to
be the expression

$x_{{\mathscr{F}}|{\mathscr{G}}}\,\delta_{e}=x_{{\mathscr{F}}|{\mathscr{G}}}\sum_{e\in
F\cap G}x_{F|G}.$

We recursively obtain the _canonical expansion_ of
$x_{{\mathscr{F}}|{\mathscr{G}}}\delta^{m}$ by multiplying each monomial in
the canonical expansion of $x_{{\mathscr{F}}|{\mathscr{G}}}\delta^{m-1}$ by
$\delta$, again using the canonical expansion.

######  Lemma 3.2.

The canonical expansion of $x_{{\mathscr{F}}|{\mathscr{G}}}\,\delta$ is the
sum of the monomials $x_{({\mathscr{F}}\cup F)|({\mathscr{G}}\cup G)}$
corresponding to the biflags of the form ${({\mathscr{F}}\cup
F)|({\mathscr{G}}\cup G)}\supsetneq{{\mathscr{F}}|{\mathscr{G}}}$ such that
$e\in F\cap G$. If $j$ is the unique index for which $e\in F_{j+1}-F_{j}$,
then $e\in G_{j}-G_{j+1}$. Furthermore, the nonzero terms in the canonical
expansion correspond to the biflats ${F|G}$ with $F_{j}\subseteq F\subseteq
F_{j+1}$ and $G_{j}\supseteq G\supseteq G_{j+1}$.

###### Proof.

The first statement follows directly from definition. For the second one,
assume $e\in F_{j+1}-F_{j}$. Since $e\in F_{j+1}$ but $e\notin F_{j+1}\cap
G_{j+1}$, we have $e\notin G_{j+1}$. Since $e\notin F_{j}$ but $F_{j}\cup
G_{j}=E$, we have $e\in G_{j}$. Therefore $e\in G_{j}-G_{j+1}$ as desired.
Finally, if ${({\mathscr{F}}\cup F)|({\mathscr{G}}\cup G)}$ is a biflag with
$e\in F\cap G$, then $e\notin F_{j}$ and $e\notin G_{j+1}$ imply that the
biflat $F|G$ must be added in between indices $j$ and $j+1$ of
${{\mathscr{F}}|{\mathscr{G}}}$. Conversely, any such biflat arises in this
expansion. ∎

######  Definition 3.3.

(_Canonical expansion of $\delta^{m}$_) Given the canonical expansion
$\delta^{j}=x_{{\mathscr{F}}_{1}|{\mathscr{G}}_{1}}+\cdots+x_{{\mathscr{F}}_{t}|{\mathscr{G}}_{t}}$,
we compute the canonical expansion of
$\delta^{j+1}=x_{{\mathscr{F}}_{1}|{\mathscr{G}}_{1}}\,\delta+\cdots+x_{{\mathscr{F}}_{t}|{\mathscr{G}}_{t}}\,\delta$
by adding the canonical expansions of the individual terms, following
Definition 3.1.

We may think of the canonical expansion of $\delta^{m}$ as a recursive
procedure to produce a list of biflags of length $m$, where each biflag is
built up one biflat at a time according to the rules prescribed in Lemma 3.2.

######  Example 3.4.

For the graph $G$ of the square pyramid in Figure 1, the canonical expansion
of the highest nonzero power of $\delta$ in
$A_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}$, namely
$\delta^{n-1}=\delta^{6}$, is

$\displaystyle\delta^{6}$ $\displaystyle=$ $\displaystyle
x_{6|E}\,x_{56|E}\,x_{4567|E}\,x_{E|23467}\,x_{E|347}\,x_{E|7}$
$\displaystyle+x_{7|E}\,x_{57|E}\,x_{4567|E}\,x_{E|23467}\,x_{E|36}\,x_{E|6}$
$\displaystyle+x_{7|E}\,x_{67|E}\,x_{4567|E}\,x_{E|235}\,x_{E|35}\,x_{E|5}.$

This expression is deceivingly short. Carrying out this seemingly simple
computation by hand is very tedious; if one were to do it by brute force, one
would find that the number of terms of the canonical expansions of
$\delta^{0},\ldots,\delta^{6}$ are the following:

$\begin{array}[]{|l||l|l|l|l|l|l|l|}\hline\cr&\delta^{0}&\delta^{1}&\delta^{2}&\delta^{3}&\delta^{4}&\delta^{5}&\delta^{6}\\\
\hline\cr\hline\cr\text{\\# of monomials counted with
multiplicity}&1&29&352&658&383&69&3\\\ \hline\cr\text{\\# of distinct
monomials}&1&29&333&621&370&68&3\\\ \hline\cr\end{array}\ .$

This example shows typical behavior: for small $k$ the number of biflags in
the expansion of $\delta^{k}$ increases with $k$, but as $k$ approaches $n-1$,
increasingly many products $x_{{\mathscr{F}}|{\mathscr{G}}}\,\delta$ are zero,
and the canonical expansions become shorter.

Each monomial contribution $x_{{\mathscr{F}}|{\mathscr{G}}}$ to the canonical
expansion of $\delta^{m}$ is built up through a sequence of monomials

$1=x_{{\mathscr{F}}^{0}|{\mathscr{G}}^{0}},\ \
x_{{\mathscr{F}}^{1}|{\mathscr{G}}^{1}},\ \ \ldots,\ \
x_{{\mathscr{F}}^{m}|{\mathscr{G}}^{m}}=x_{{\mathscr{F}}|{\mathscr{G}}},$

where $x_{{\mathscr{F}}^{k+1}|{\mathscr{G}}^{k+1}}$ appears in the canonical
expansion of
$x_{{\mathscr{F}}^{k}|{\mathscr{G}}^{k}}\delta=x_{{\mathscr{F}}^{k}|{\mathscr{G}}^{k}}\delta_{e({\mathscr{F}}^{k}|{\mathscr{G}}^{k})}$.
We relabel the $e({\mathscr{F}}^{k}|{\mathscr{G}}^{k})$s, writing
$e_{i}=e({{\mathscr{F}}^{k}|{\mathscr{G}}^{k}})$ if the $i$-th biflat of the
final biflag ${\mathscr{F}}|{\mathscr{G}}$ is obtained in the canonical
expansion of $x_{{\mathscr{F}}^{k}|{\mathscr{G}}^{k}}\delta$. Thus both the
flat and the coflat of that $i$-th biflat must contain $e_{i}$. We call the
resulting sequence $\mathcal{E}=(e_{1},\ldots,e_{m})$ the _arrival sequence_
of this contribution. We summarize the definition of the canonical expansion
of $\delta^{m}$ and the arrival sequence of each term in the following
proposition.

######  Proposition 3.5.

The canonical expansion of $\delta^{m}$ consists of the monomials
$x_{{\mathscr{F}}|{\mathscr{G}}}$ indexed by the collection
${\mathscr{T}}^{m}_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}$
of all pairs $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ for which

  1. (1)

the biflag ${\mathscr{F}}|{\mathscr{G}}=\\{F_{1}|G_{1},\ldots,F_{m}|G_{m}\\}$
is a biflag of biflats of $\operatorname{\mathrm{M}}$, and

  2. (2)

the arrival sequence $\mathcal{E}=(e_{1},\ldots,e_{m})$ is a sequence of
distinct elements of $E$ such that

$e_{i}\in F_{i}\cap G_{i}\ \ \text{and}\ \ e_{i}=\max\big{(}E-\bigcup_{j\colon
e_{j}>e_{i}}(F_{j}\cap G_{j})\big{)}\ \ \text{for all $1\leq i\leq m$.}$

In symbols, the following identity holds in the conormal Chow ring of
$\operatorname{\mathrm{M}}$:

$\delta^{m}=\sum_{({\mathscr{F}}|{\mathscr{G}},\mathcal{E})\in{{\mathscr{T}}^{m}_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}}}x_{{\mathscr{F}}|{\mathscr{G}}}.$

We record each pair
$({\mathscr{F}}|{\mathscr{G}},\mathcal{E})\in{\mathscr{T}}^{m}_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}$
as a table, where we expand the table of biflag ${\mathscr{F}}|{\mathscr{G}}$
by placing element $e_{i}$ directly below $F_{i}$ and $G_{i}$. Each
contribution to the canonical expansion $\delta^{m}$ comes from such a table.

$\begin{array}[]{|cc|ccccccccccc|cc|}\hline\cr\emptyset&\subsetneq&F_{1}&\subseteq&\cdots&\subseteq&F_{d}&\subseteq&F_{d+1}&\subseteq&\cdots&\subseteq&F_{m}&\subseteq&E\\\
E&\supseteq&G_{1}&\supseteq&\cdots&\supseteq&G_{d}&\supseteq&G_{d+1}&\supseteq&\cdots&\supseteq&G_{m}&\supsetneq&\emptyset\\\
\hline\cr&&e_{1}&&\cdots&&e_{d}&&e_{d+1}&&\cdots&&e_{m}&&\\\
\hline\cr\end{array}\,$

The canonical expansion of $\delta^{m}$ may contain repeated terms
$x_{{\mathscr{F}}|{\mathscr{G}}}$ which lead to different tables
$({\mathscr{F}}|{\mathscr{G}};\mathcal{E})$ with the same biflat
${\mathscr{F}}|{\mathscr{G}}$ but different arrival sequences $\mathcal{E}$.

######  Example 3.6.

Let us revisit the canonical expansion of $\delta^{6}$, the highest power of
$\delta$, in Example 3.4. The first monomial arises from the following table.

$\begin{array}[]{|cc|ccccccccccc|cc|}\hline\cr\emptyset&\subset&6&\subsetneq&56&\subsetneq&4567&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&=&E&=&E&=&E\\\
{}E&=&E&=&E&=&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&23467&\supsetneq&347&\supsetneq&7&\supset&\emptyset\\\
\hline\cr&&e_{1}=6&&e_{2}=5&&e_{3}=4&&e_{4}=2&&e_{5}=3&&e_{6}=7&&\\\
\hline\cr\end{array}$

The terms $x_{F_{i}|G_{i}}$ arrive to the monomial
$x_{{\mathscr{F}}|{\mathscr{G}}}$ in the descending order of the $e_{i}$s,
namely,

$x_{E|7}x_{6|E}x_{56|E}x_{4567|E}x_{E|347}x_{E|23467}.$

This means that $x_{E|7}$ is in the canonical expansion of
$\delta=\delta_{7}$, $x_{E|7}x_{6|E}$ is in the canonical expansion of
$x_{E|7}\delta=x_{E|7}\delta_{6}$, $x_{E|7}x_{6|E}x_{56|E}$ is in the
canonical expansion of $x_{E|7}x_{6|E}\delta=x_{E|7}x_{6|E}\delta_{5}$, and so
on. The corresponding biflag is ${\mathscr{F}}(B)|{\mathscr{G}}(B)$ for the
$\operatorname{{\mathit{nbc}}}$ basis
$B={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}0}456$
with
$\operatorname{{IA}}(B)={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}0}$,
as introduced in Definition 2.9.

The two other monomials in our expansion of $\delta^{6}$ are
$x_{7|E}\,x_{57|E}\,x_{4567|E}\,x_{E|23467}\,x_{E|36}\,x_{E|6}$ and
$x_{7|E}\,x_{67|E}\,x_{4567|E}\,x_{E|235}\,x_{E|25}\,x_{E|5}$, which
correspond to the biflags of
${\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}0}457$
and
${\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}0}467$,
respectively. These are the other two $\operatorname{{\mathit{nbc}}}$ bases
whose only internally active element is $0$. Note that we must have
$0\in\operatorname{{IA}}(B)$ for any $\operatorname{{\mathit{nbc}}}$ basis
$B$.

This example illustrates a general phenomenon: The case $k=0$ of Theorem 1.1
says that the terms of the canonical expansion of $\delta^{n-1}$ in the
conormal Chow ring correspond to the $\operatorname{{\mathit{nbc}}}$ biflags
of the $\operatorname{{\mathit{nbc}}}$ bases $B$ of
$\operatorname{\mathrm{M}}$ with $|\operatorname{{IA}}(B)|=1$; these are also
known as the _$\beta$ -$\operatorname{{\mathit{nbc}}}$ bases_ of
$\operatorname{\mathrm{M}}$. They are enumerated by Crapo’s beta invariant
$\beta_{\operatorname{\mathrm{M}}}=h_{r}(\operatorname{BC}(\operatorname{\mathrm{M}}))=t_{1,0}(\operatorname{\mathrm{M}})$:
see [Zie92].

###  3.2. $h_{r-k}$-many $\operatorname{{\mathit{nbc}}}$ monomials in
$\delta^{n-k-1}$.

To each $\operatorname{{\mathit{nbc}}}$ basis $B$ of
$\operatorname{\mathrm{M}}$, we associated a sequence $\mathcal{E}(B)$ and a
biflag ${{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ in Definition 2.9. We will now
show that the table $({\mathscr{F}}(B)|{\mathscr{G}}(B),\mathcal{E}(B))$
satisfies the conditions of Proposition 3.5, and hence the
$\operatorname{{\mathit{nbc}}}$ monomial
$x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ appears in the canonical expansion of
$\delta^{n-k-1}$ with arrival sequence $\mathcal{E}(B)$, where
$k+1=|\operatorname{{IA}}(B)|$.

######  Example 3.7.

For the cube graph of Figure 2 and $k=2$, the $\operatorname{{\mathit{nbc}}}$
basis
${\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}01}5{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}6}78b$
with
$\operatorname{{IA}}(B)={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}016}$
gives rise to the $\operatorname{{\mathit{nbc}}}$ biflag of Example 2.10, and
the table

$\begin{array}[]{|c|ccccccccc|c|}\hline\cr\emptyset&b&8b&78b&578b&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&E&E&E&E\\\
E&E&E&E&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&03469a&469a&69a&9&\emptyset\\\
\hline\cr&b&8&7&5&&3&4&9&a&\\\ \hline\cr\end{array}$

gives rise to the following $\operatorname{{\mathit{nbc}}}$ monomial in the
canonical expansion of $\delta^{8}$:

$x_{b|E}\,x_{8b|E}\,x_{78b|E}\,x_{578b|E}\,x_{E|03469a}\,x_{E|469a}\,x_{E|69a}\,x_{E|9}.$

######  Proposition 3.8.

If $B$ is a $\operatorname{{\mathit{nbc}}}$ basis of
$\operatorname{\mathrm{M}}$ with $|I(B)|=k+1$, then the
$\operatorname{{\mathit{nbc}}}$ monomial
$x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ arises in the canonical expansion of
$\delta^{n-k-1}$.

###### Proof.

We verify that $({{\mathscr{F}}(B)|{\mathscr{G}}(B)},\mathcal{E}(B))$
satisfies the conditions of Proposition 3.5. We know that
${{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ is a biflag by Proposition 2.9, so it
remains to show that

$e=\max\big{(}E-\operatorname{cl}(\operatorname{{IP}}(B)_{>e})-\operatorname{cl}^{\perp}(B^{\perp}_{>e})\big{)}\
\ \text{for any $e\in\operatorname{{IP}}(B)$ or $e\in B^{\perp}-\min
B^{\perp}$.}$

First, we show that $e\in
E-\operatorname{cl}(\operatorname{{IP}}(B)_{>e})-\operatorname{cl}^{\perp}(B^{\perp}_{>e})$,
considering two cases:

  1. (1)

$e\in B^{\perp}-\min B^{\perp}$: In this case,
$e\notin\operatorname{cl}^{\perp}(B^{\perp}_{>e})$ since $B^{\perp}$ is
independent in $M^{\perp}$. Also, if we had
$e\in\operatorname{cl}(\operatorname{{IP}}(B)_{>e})$, then the fundamental
circuit $C(B,e)$ would be contained in $\operatorname{{IP}}(B)_{>e}\cup e$,
and hence its smallest element would be $e$. Thus $e$ would be externally
active in $B$, contradicting the assumption that $B$ is a
$\operatorname{{\mathit{nbc}}}$ basis.

  2. (2)

$e\in\operatorname{{IP}}(B)$: In this case,
$e\notin\operatorname{cl}(\operatorname{{IP}}(B)_{>e})$ since
$\operatorname{{IP}}(B)\subseteq B$ is independent in
$\operatorname{\mathrm{M}}$. Also, if we had
$e\in\operatorname{cl}^{\perp}(B^{\perp}_{>e})$, then the fundamental
cocircuit $C^{\perp}(B,e)$ would be contained in $B^{\perp}_{>e}\cup e$, and
hence its smallest element would be $e$, contradicting that $a$ is internally
passive in $B$.

Now, for the sake of contradiction, let us assume that
$\max\big{(}E-\operatorname{cl}(\operatorname{{IP}}(B)_{>e})-\operatorname{cl}^{\perp}(B^{\perp}_{>e})\big{)}=f>e$.
We consider three cases:

  1. (1)

$f\in B^{\perp}$: Then $f>e$ would imply $f\in
B^{\perp}_{>e}\subseteq\operatorname{cl}^{\perp}(B^{\perp}_{>e})$.

  2. (2)

$f\in\operatorname{{IP}}(B)$: Then $f>e$ would imply
$f\in\operatorname{{IP}}(B)_{>e}\subseteq\operatorname{cl}(\operatorname{{IP}}(B)_{>e})$.

  3. (3)

$f\in\operatorname{{IA}}(B)$: This means that $f=\min C^{\perp}(B,f)$, so this
fundamental cocircuit is contained in $B^{\perp}_{>f}\cup f$. Thus
$f\in\operatorname{cl}^{\perp}(B^{\perp}_{>f})$ while
$f\notin\operatorname{cl}^{\perp}(B^{\perp}_{>e})$, which is impossible since
$B^{\perp}_{>f}\subseteq B^{\perp}_{>e}$.

This completes the proof. ∎

###  3.3. Multiplying by a high power of $\gamma$ eradicates non-initial
monomials.

######  Lemma 3.9.

Consider a monomial $x_{{\mathscr{F}}|{\mathscr{G}}}$ and let $l$ be the
largest index for which $F_{l}\neq E$. If $c\notin F_{l}$, then
$x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma_{c}$ is the sum of the monomials
$x_{({\mathscr{F}}\cup F)|({\mathscr{G}}\cup G)}$ corresponding to the
variables $x_{F|G}$ with $F_{l}\cup c\subseteq F\subsetneq E$ and
$G_{l}\supseteq G\supseteq G_{l+1}$.

###### Proof.

We have
$x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma_{c}=x_{{\mathscr{F}}|{\mathscr{G}}}\sum_{c\in
F\neq E}x_{F|G}$. Now, if ${({\mathscr{F}}\cup F)|({\mathscr{G}}\cup G)}$ is a
biflag with $c\in F\neq E$, then we must have $F_{l}\subsetneq F\subsetneq
E=F_{l+1}$, so the biflat $F|G$ must be added in between indices $l$ and $l+1$
of ${{\mathscr{F}}|{\mathscr{G}}}$. Conversely, any such biflat arises in this
expansion. ∎

######  Definition 3.10.

Let us call a monomial $x_{{\mathscr{F}}|{\mathscr{G}}}$ _initial_ if the
distinct flats in ${\mathscr{F}}$ have ranks $1,2,\ldots,i,$ and $r+1$ for
some $i$.

The following technical lemma will play an important role. It shows that the
multiplication by a high power of $\gamma$ eradicates non-initial monomials.

######  Lemma 3.11.

Let ${\mathscr{F}}|{\mathscr{G}}$ be a biflag of $\operatorname{\mathrm{M}}$,
and let $s$ be the number of distinct proper flats in ${\mathscr{F}}$.

  1. (1)

If $s+k>r$, then $x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma^{k}=0$.

  2. (2)

If $s+k=r$ and ${\mathscr{F}}$ is not initial, then
$x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma^{k}=0$.

###### Proof.

The proof of the first part is nearly identical and simpler than the proof of
the second part. For the second part, as in Lemma 3.9, let $l$ be the largest
index for which $F_{l}\neq E$, and let $c\notin F_{l}$. We proceed by
descending induction on $s$.

The largest possible value of $s$ for a non-initial flag is $r-1$. Suppose
first that $s=r-1$ and $k=1$. If ${\mathscr{F}}$ is not initial, we must have
$\operatorname{{rk}}F_{l}=r$. Lemma 3.9 then implies that

$x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma=x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma_{c}=\sum_{\begin{subarray}{c}F_{l}\cup
c\subseteq F\neq E\\\ G_{l}\supseteq G\supseteq
G_{l+1}\end{subarray}}x_{({\mathscr{F}}\cup F)|({\mathscr{G}}\cup G)}=0,$

since the only flat containing $F_{l}$, which has corank 1, and $c$, which is
not in $F_{l}$, is $E$.

Now suppose that the result is true for some value of $s\leq r-1$, and
consider any biflag ${\mathscr{F}}|{\mathscr{G}}$, where ${\mathscr{F}}$ has
$s-1$ distinct proper flats. We have

$x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma^{r-(s-1)}=(x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma_{c})\gamma^{r-s}=\sum_{\begin{subarray}{c}F_{l}\cup
c\subseteq F\neq E\\\ G_{l}\supseteq G\supseteq
G_{l+1}\end{subarray}}x_{({\mathscr{F}}\cup F)|({\mathscr{G}}\cup
G)}\gamma^{r-s}=0,$

where each summand is $0$ by the inductive hypothesis because each flag
${\mathscr{F}}\cup F$ that arises is not initial and has $s$ distinct proper
flats. ∎

###  3.4. $h_{r-k}$-many extended $\operatorname{{\mathit{nbc}}}$ monomials in
$\gamma^{k}\delta^{n-k-1}$.

Lemma 3.11 shows that many monomials in $\delta^{n-k-1}$ are eliminated when
one multiplies them by $\gamma^{k}$. Let us now show that each one of the
$h_{r-k}(\operatorname{BC}(\operatorname{\mathrm{M}}))$
$\operatorname{{\mathit{nbc}}}$ monomials
$x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ of Proposition 3.8 resists
multiplication by $\gamma^{k}$, and gives rise to its corresponding extended
$\operatorname{{\mathit{nbc}}}$ monomial
$x_{{\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)}$ in $\gamma^{k}\delta^{n-k-1}$,
as introduced in Definition 2.11. We will later see that these are the only
monomials that resist multiplication by $\gamma^{k}$.

######  Proposition 3.12.

For every $\operatorname{{\mathit{nbc}}}$ basis $B$ with
$\left|\operatorname{{IA}}(B)\right|=k+1$, we have

$x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}\,\gamma^{k}=x_{{\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)}.$

###### Proof.

Recall from Proposition 2.9 that, if we write

$S=B-\operatorname{{IA}}(B)=\\{e_{1}>\cdots>e_{r-k}\\}\ \ \text{and}\ \
T=(E-B)-\min(E-B)=\\{e_{r-k+1}<\cdots<e_{n-k-1}\\},$

then the table of the $\operatorname{{\mathit{nbc}}}$ monomial
$x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ is

$\begin{array}[]{|cc|ccccccccccc|cc|}\hline\cr\emptyset&\subsetneq&F_{1}&\subsetneq&\cdots&\subsetneq&F_{r-k}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&=&\cdots&=&E&=&E\\\
E&=&E&=&\cdots&=&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&G_{r-k+1}&\supsetneq&\cdots&\supsetneq&G_{n-k-1}&\supsetneq&\emptyset\\\
\hline\cr&&e_{1}&>&\cdots&>&e_{r-k}&&e_{r-k+1}&<&\cdots&<&e_{n-k-1}&&\\\
\hline\cr\end{array}\ ,$

where $F_{j}=\operatorname{cl}(e_{1},\ldots,e_{j})$ for $j\leq r-k$ and
$G_{j}=\operatorname{cl}^{\perp}(e_{j},\ldots,e_{n-k-1})$ for $j>r-k$. In
particular,

$F_{r-k}=\operatorname{cl}(S)\ \ \text{and}\ \
G_{r-k+1}=\operatorname{cl}^{\perp}(T).$

We write
$\operatorname{{IA}}(B)=\\{{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}c_{1}}>{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}c_{2}}>\cdots>{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}c_{k+1}}\\}$,
and multiply $x_{{\mathscr{F}}|{\mathscr{G}}}$ by
$\gamma_{c_{1}},\ldots,\gamma_{c_{k}}$ to compute

$x_{{\mathscr{F}}|{\mathscr{G}}}\gamma^{k}=x_{{\mathscr{F}}|{\mathscr{G}}}\gamma_{c_{1}}\gamma_{c_{2}}\cdots\gamma_{c_{k}}.$

Since $c_{1}\notin F_{r-k}$, when we multiply
$x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma_{c_{1}}$, Lemma 3.9 tells us that
every resulting term $x_{({\mathscr{F}}\cup F)|({\mathscr{G}}\cup G)}$
corresponds to a biflat $F|G$ such that $F_{r-k}\cup c_{1}\subsetneq
F\subsetneq E$. Also, for such a term to resist further multiplication by
$\gamma^{k-1}$, ${\mathscr{F}}\cup F$ must be an initial flag by Lemma 3.11,
so $\operatorname{{rk}}(F)=r-k+1$. This implies that
$F=\operatorname{cl}(S,c_{1})$; we denote $F$ by $F_{r-k+1}$.

Similarly, since $c_{2}\notin F_{r-k+1}$, every term in
$x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma_{c_{1}}\gamma_{c_{2}}$ that resists
further multiplication by $\gamma^{k-2}$ must introduce
$F_{r-k+2}\coloneq\operatorname{cl}(S,c_{1},c_{2})$ as the new flat in the
flag. Continuing with this line of reasoning, we see that the nonzero terms in
$x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma_{c_{1}}\gamma_{c_{2}}\cdots\gamma_{c_{k}}$
are obtained from the table of $x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ by
adding new columns between columns $r-k$ and $r-k+1$ as

$\begin{array}[]{|ccccccccccc|}\hline\cr\cdots&\operatorname{cl}(S)&\subsetneq&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}(S,c_{1})}&\subsetneq&\cdots&\subsetneq&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}(S,c_{1},\ldots,c_{k})}&\subsetneq&E&\cdots\\\
\cdots&E&\supseteq&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}G^{\prime}_{r-k+1}}&\supseteq&\cdots&\supseteq&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}G^{\prime}_{r}}&\supseteq&\operatorname{cl}^{\perp}(T)&\cdots\\\
\hline\cr&e_{r-k}&&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}c_{1}}&>&\cdots&>&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}c_{k}}&&e_{r-k+1}&\\\
\hline\cr\end{array}\ ,$

for any choice of coflats $G^{\prime}_{r-k+1}\supseteq\cdots\supseteq
G^{\prime}_{r}$, each of which is either $E$ or
$\operatorname{cl}^{\perp}(T)$, since $r^{\perp}(T)=r^{\perp}-1$.

The only freedom we appear to have left is the choice of the unique value of
$1\leq i\leq k+1$ for which $G^{\prime}_{r-k+i-1}=E$ and
$G^{\prime}_{r-k+i}=\operatorname{cl}^{\perp}(T)$:

$\begin{array}[]{|ccccc|}\hline\cr\cdots&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}(S,c_{1},\ldots,c_{i-1})}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}(S,c_{1},\ldots,c_{i})}&\cdots\\\
\cdots&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}E}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}\operatorname{cl}^{\perp}(T)}&\cdots\\\
\hline\cr\end{array}$

However, for this to be a valid flag, we must have

$\operatorname{cl}(S,c_{1},\ldots,c_{i-1})\cup\operatorname{cl}^{\perp}(T)\neq
E\ \ \text{and}\ \
\operatorname{cl}(S,c_{1},\ldots,c_{i})\cup\operatorname{cl}^{\perp}(T)=E.$

Therefore, we do not have the freedom to choose $i$; it must be the smallest
index for which
$\operatorname{cl}(S,c_{1},\ldots,c_{i})\cup\operatorname{cl}^{\perp}(T)=E$.

This precisely matches the description of the extended
$\operatorname{{\mathit{nbc}}}$ biflag of $B$ in Definition 2.11. Thus the
unique surviving term in
$x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}\gamma_{c_{1}}\gamma_{c_{2}}\cdots\gamma_{c_{k}}$
is precisely $x_{{\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)}$, as we wished to
show. ∎

######  Example 3.13.

Let us return to Example 3.7 and multiply the $\operatorname{{\mathit{nbc}}}$
monomial of
$B={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}01}5{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}6}78b$
with
$\operatorname{{IA}}(B)={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}016}$
by $\gamma^{2}$ by computing

$(x_{b|E}\,x_{8b|E}\,x_{78b|E}\,x_{578b|E}\,x_{E|03469a}\,x_{E|469a}\,x_{E|69a}\,x_{E|9})\cdot\gamma_{6}\cdot\gamma_{1}.$

This adds two new columns to the middle of the table in Example 3.7, shown in
green. The new entries in the top row must be
$\operatorname{cl}(578b{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}6})=5678b$
and
$\operatorname{cl}(578b{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}61})=1256789ab$.
The new entries in the bottom row can equal $E$ or $03469a$. Since $5678b\cup
03469a\neq E$, and we must have $F_{i}\cup G_{i}\neq E$ for all $i$, the first
one must be $E$. Since $1256789ab\cup 03469a=E$, and we must have $F_{i}\cup
G_{i+1}\neq E$ for some $i$, the second one must be $03469a$. Thus the only
possible table is

$\begin{array}[]{|c|cccccccccc|c|}\hline\cr\emptyset&b&8b&78b&578b&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}5678b}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}1256789ab}&E&E&E&E&E\\\
E&E&E&E&E&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}E}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}03469a}&03469a&469a&69a&9&\emptyset\\\
\hline\cr&b&8&7&5&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}6}&{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}1}&3&4&9&a&\\\
\hline\cr\end{array}\ ,$

and the resulting monomial is the extended $\operatorname{{\mathit{nbc}}}$
monomial of $B$

$x_{b|E}\,x_{8b|E}\,x_{78b|E}\,x_{578b|E}\,{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}x_{5678b,E}\,x_{1256789ab,03469a}}\,x_{E|03469a}\,x_{E|469a}\,x_{E|69a}\,x_{E|9},$

corresponding to the extended $\operatorname{{\mathit{nbc}}}$ biflag of
Example 2.12.

We have now made some progress towards our proof of Theorem 1.1, which says
that

$\gamma^{k}\delta^{n-k-1}=\sum_{\begin{subarray}{c}B\operatorname{{\mathit{nbc}}}\textrm{basis}\\\
|\operatorname{{IA}}(B)|=k+1\end{subarray}}x_{{\mathscr{F}}^{+}(B)|{\mathscr{G}}^{+}(B)}.$

Propositions 3.8 and 3.12 show that all terms in the right hand side of this
expression do arise in $\gamma^{k}\delta^{n-k-1}$. Proving that no other terms
appear requires significantly more work; this is the content of Section 4.

##  4\. Upper bound for $\gamma^{k}\delta^{n-k-1}$.

Proposition 3.5 shows that the terms $x_{{\mathscr{F}}|{\mathscr{G}}}$ in the
canonical expansion of $\delta^{n-k-1}$ are given by the set
${\mathscr{T}}_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}$
of combinatorially determined tables
$({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$. Lemma 3.11 shows that
multiplication by high powers of $\gamma$ eradicates many of these monomials.
The main goal of this section will be to characterize those monomials in the
canonical expansion of $\delta^{n-k-1}$ that resist multiplication by
$\gamma^{k}$.

######  Definition 4.1.

A monomial $x_{{\mathscr{F}}|{\mathscr{G}}}$ of degree $n-k-1$ and the
corresponding to table
$({\mathscr{F}}|{\mathscr{G}},\mathcal{E})\in{\mathscr{T}}_{\operatorname{\mathrm{M}},\operatorname{\mathrm{M}}^{\perp}}$
are said to be _resistant_ if $x_{{\mathscr{F}}|{\mathscr{G}}}\,\gamma^{k}\neq
0$ in the conormal Chow ring of $\operatorname{\mathrm{M}}$.

We saw in Proposition 3.12 that any $\operatorname{{\mathit{nbc}}}$ monomial
does resist multiplication by $\gamma^{k}$, and gives rise to its
corresponding extended $\operatorname{{\mathit{nbc}}}$ monomial. We will
eventually show in Proposition 4.15 that these are in fact the only resistant
monomials, and hence that
$\deg(\gamma^{k}\delta^{n-k-1})=h_{r-k}(\operatorname{BC}(\operatorname{\mathrm{M}}))$.
This proof will require several steps, which we carry out in the following
subsections.

###  4.1. The jump sets of a resistant term of $\delta^{n-k-1}$.

Recall the notion of jump sets of ${\mathscr{F}}|{\mathscr{G}}$ from
Definition 2.4. We write $\operatorname{{Tr}}_{k}\operatorname{\mathrm{M}}$
for the $k$-th truncation of $\operatorname{\mathrm{M}}$.

######  Lemma 4.2.

If the monomial $x_{{\mathscr{F}}|{\mathscr{G}}}$ arises in the canonical
expansion of $\delta^{n-k-1}$ and resists multiplication by $\gamma^{k}$, then

  1. (1)

${\mathscr{F}}|{\mathscr{G}}$ has a unique double jump, and

  2. (2)

${\mathscr{F}}$ and ${\mathscr{G}}$ are complete flags of nonempty flats in
$\operatorname{{Tr}}_{k}\operatorname{\mathrm{M}}$ and
$\operatorname{\mathrm{M}}^{\perp}$, with possible repetitions.

###### Proof.

Since $x_{{\mathscr{F}}|{\mathscr{G}}}\gamma^{k}$ is nonzero, the first part
of Lemma 3.11 tells us that ${\mathscr{F}}$ contains $s\leq r-k$ distinct
proper flats, so $\left|\,\mathsf{J}({\mathscr{F}})\right|=s+1\leq r-k+1$.
Also, since $\operatorname{\mathrm{M}}^{\perp}$ has rank $r^{\perp}+1=n-r$, we
have $\left|\,\mathsf{J}({\mathscr{G}})\right|\leq n-r$. On the other hand,
since a square-free monomial cannot contain repeated terms,
$\,\mathsf{J}({\mathscr{F}})\cup\,\mathsf{J}({\mathscr{G}})=\left\\{0,1,\ldots,n-k-1\right\\}$.
Therefore,

$n-k\leq\left|\,\mathsf{J}({\mathscr{F}})\cup\,\mathsf{J}(G)\right|+\left|\,\mathsf{J}({\mathscr{F}})\cap\,\mathsf{J}({\mathscr{G}})\right|=\left|\,\mathsf{J}({\mathscr{F}})\right|+\left|\,\mathsf{J}({\mathscr{G}})\right|\leq
n-k+1,$

and the number of double jumps is
$\left|\,\mathsf{J}({\mathscr{F}})\cap\,\mathsf{J}({\mathscr{G}})\right|\leq
1$. But ${\mathscr{F}}|{\mathscr{G}}$ has at least one nonempty gap $D_{d}$,
which guarantees a double jump $d$.

The above analysis also implies that ${\mathscr{F}}$ contains $s=r-k$ distinct
proper flats – which must have ranks $1,2,\ldots,r-k$ by the second part of
Lemma 3.11 – and that ${\mathscr{G}}$ has flats in every rank of
$\operatorname{\mathrm{M}}^{\perp}$. ∎

It follows that any table $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ arising
in the canonical expansion of $\delta^{n-k-1}$ that resists multiplication by
$\gamma^{k}$ has the form

$\begin{array}[]{|cc|ccccccccccc|cc|}\hline\cr\emptyset&\subsetneq&F_{1}&\subseteq&\cdots&\subseteq&F_{d}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&F_{d+1}&\subseteq&\cdots&\subseteq&F_{n-k-1}&\subseteq&E\\\
E&\supseteq&G_{1}&\supseteq&\cdots&\supseteq&G_{d}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&G_{d+1}&\supseteq&\cdots&\supseteq&G_{n-k-1}&\supsetneq&\emptyset\\\
\hline\cr&&e_{1}&&\cdots&&e_{d}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{D}&e_{d+1}&&\cdots&&e_{n-k-1}&&\\\
\hline\cr\end{array}\,$

where $d$ is the unique double jump of ${\mathscr{F}}|{\mathscr{G}}$. We write

$D\coloneq D_{d}=E-(F_{d}\cup G_{d+1})=(F_{d+1}-F_{d})\cap(G_{d}-G_{d+1})$

for the unique nonempty gap. In every column other than the $d$-th, one
inclusion is strict and the other one is an equality. From now on, we will
also record the nonempty gap $D$ in the bottom row of the table
$({\mathscr{F}}|{\mathscr{G}};\mathcal{E})$. This is redundant information,
but it will be useful visually in the proofs that follow.

######  Remark 4.3.

For each index $i$, the flat $F_{i}$ contains the bottom row entries below it
and to its left; namely, $e_{1},\ldots,e_{i}$, and $D$ if $i\geq d+1$.
Similarly, the coflat $G_{i}$ contains the bottom row entries below it and to
its right; $e_{i},\ldots,e_{n-k-1}$, and $D$ if $i\leq d$.

We continue with two more easy but important properties of the canonical
expansion. The first one tells us that the arrival sequence $\mathcal{E}$ and
the index $d$ of the double jump completely determine which inclusions are
strict in the table of $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$. We define
the _descent set_ and _ascent set_ of $\mathcal{E}$ by

$\operatorname{{Des}}(\mathcal{E})=\\{i\,:\,e_{i}>e_{i+1}\\}\ \ \text{and}\ \
\operatorname{{Asc}}(\mathcal{E})=\\{i\,:\,e_{i}<e_{i+1}\\}.$

######  Lemma 4.4.

If $i\in\,\mathsf{J}({\mathscr{F}})-\,\mathsf{J}({\mathscr{G}})$, then
$e_{i}>e_{i+1}$. If
$i\in\,\mathsf{J}({\mathscr{G}})-\,\mathsf{J}({\mathscr{F}})$, then
$e_{i}<e_{i+1}$. Therefore,

$\,\mathsf{J}({\mathscr{F}})=\operatorname{{Des}}(\mathcal{E})\cup\\{0,d\\}\ \
\text{and}\ \
\,\mathsf{J}({\mathscr{G}})=\operatorname{{Asc}}(\mathcal{E})\cup\\{d,n-k-1\\}.$

###### Proof.

It is clear from the definitions that $0\in\,\mathsf{J}({\mathscr{F}})$ and
$n-k-1\in\,\mathsf{J}({\mathscr{G}})$. By symmetry, it is enough to prove the
first assertion. Assume for the sake of contradiction that
$i\in\,\mathsf{J}({\mathscr{F}})-\,\mathsf{J}({\mathscr{G}})$ and
$e_{i}<e_{i+1}$, so the table of $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$
contains

$\begin{array}[]{|ccccc|}\hline\cr\cdots&F_{i}&\subsetneq&F_{i+1}&\cdots\\\
\cdots&G_{i}&=&G_{i+1}&\cdots\\\ \hline\cr\cdots&e_{i}&<&e_{i+1}&\cdots\\\
\hline\cr\end{array}\,.$

Then the biflat $F_{i}|G_{i}$ arrives to the monomial
$x_{{\mathscr{F}}|{\mathscr{G}}}$ after the biflat $F_{i+1}|G_{i+1}$ does, so
$e_{i}\notin F_{i+1}\cap G_{i+1}$. This contradicts the fact that $e_{i}\in
F_{i}\cap G_{i}\subseteq F_{i+1}\cap G_{i+1}$. ∎

######  Lemma 4.5.

If $i<j$ and $e_{i}<e_{j}$, then $e_{i}\not\in G_{j}$. If $i<j$ and
$e_{i}>e_{j}$, then $e_{j}\not\in F_{i}$.

###### Proof.

It suffices to prove the first assertion. A part of the table of
$({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ reads

$\begin{array}[]{|ccccc|}\hline\cr F_{i}&\subseteq&\cdots&\subseteq&F_{j}\\\
G_{i}&\supseteq&\cdots&\supseteq&G_{j}\\\ \hline\cr e_{i}&&<&&e_{j}\\\
\hline\cr\end{array}\ ,$

and hence $F_{i}|G_{i}$ appears in the term $x_{{\mathscr{F}}|{\mathscr{G}}}$
after $F_{j}|G_{j}$ and $e_{i}\notin F_{j}\cap G_{j}$. Since $e_{i}\in
F_{i}\subseteq F_{j}$, we must have $e_{i}\notin G_{j}$. ∎

###  4.2. The resistant terms of $\delta^{n-k-1}$ are determined by the bottom
row of their table.

We now show that any resistant table
$({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ is completely determined by the
bottom row of its table, that is, by $\mathcal{E}$, $d$, and $D$.

######  Proposition 4.6.

Let $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$, $D$, and $d$ be as above.

  1. (1)

For any $x\in D$, we have

$\displaystyle B(x)$ $\displaystyle\coloneq\text{$\\{e_{i+1}\mid
i\in\,\mathsf{J}({\mathscr{F}})-d\\}\cup x$ is a basis of
$\operatorname{{Tr}}_{k}\operatorname{\mathrm{M}}$, and}$ $\displaystyle
B^{\perp}(x)$ $\displaystyle\coloneq\text{$\\{e_{i}\mid
i\in\,\mathsf{J}({\mathscr{G}})-d\\}\cup x$ is a basis of
$\operatorname{\mathrm{M}}^{\perp}$}.$

  2. (2)

For any $x\in D$, the flags ${\mathscr{F}}$ and ${\mathscr{G}}$ are given by

$\displaystyle F_{j}$
$\displaystyle=\begin{cases}\operatorname{cl}(\\{e_{i+1}\mid
i\in\,\mathsf{J}({\mathscr{F}})-d,\,i+1\leq j\\})&\textrm{ if }j\leq d,\\\
\operatorname{cl}(x\cup\\{e_{i+1}\mid
i\in\,\mathsf{J}({\mathscr{F}})-d,\,i+1\leq j\\})&\textrm{ if
}j>d,\end{cases}$ $\displaystyle G_{j}$
$\displaystyle=\begin{cases}\operatorname{cl}^{\perp}(x\cup\\{e_{i}\mid
i\in\,\mathsf{J}({\mathscr{G}})-d,\,i\geq j\\})&\textrm{ if }j\leq d,\\\
\operatorname{cl}^{\perp}(\\{e_{i}\mid
i\in\,\mathsf{J}({\mathscr{G}})-d,\,i\geq j\\})&\textrm{ if }j>d.\end{cases}$

  3. (3)

The ranks of the flags ${\mathscr{F}}$ and ${\mathscr{G}}$ are given by

$\operatorname{{rk}}(F_{j})=\left|\,\mathsf{J}({\mathscr{F}})_{<j}\right|\ \
\text{and}\ \
\operatorname{{rk}}^{\perp}(G_{j})=\left|\,\mathsf{J}({\mathscr{G}})_{\geq
j}\right|.$

The jump sets $\,\mathsf{J}({\mathscr{F}})$ and $\,\mathsf{J}({\mathscr{G}})$
are given by Lemma 4.4, and hence ${\mathscr{F}}$ and ${\mathscr{G}}$ are
determined by $\mathcal{E}$, $d$, and $D$. Thus, the biflag
${\mathscr{F}}|{\mathscr{G}}$ is determined by the arrival sequence
$\mathcal{E}$, the double jump $d$, and the nonempty gap $D$.

###### Proof.

Let us treat $B(x)$ as an ordered set, ordered from left to right. If
$i\in\,\mathsf{J}({\mathscr{F}})-\\{d\\}$ then $F_{i}\subsetneq F_{i+1}$ and
$\operatorname{{rk}}(F_{i+1})=\operatorname{{rk}}(F_{i})+1$ by the first part
of Lemma 3.11. Also, Lemmas 4.4 and 4.5 tell us that $e_{i}>e_{i+1}$ and
$e_{i+1}\notin F_{i}$. This implies that $F_{i+1}=\operatorname{cl}(F_{i}\cup
e_{i+1})$, and that $e_{i+1}$ is independent from the earlier terms in $B(x)$.
For $i=d$, since $x\in F_{d+1}-F_{d}$ by definition, we have that
$F_{d+1}=\operatorname{cl}(F_{d}\cup x)$ and $x$ is independent from the
earlier terms in $B(x)$. The same argument shows the analogous claims for
$B^{\perp}(x)$. This proves the first and the second parts of the proposition.
The third part follows from

$\left|\,\mathsf{J}({\mathscr{F}})_{<j}\right|=\begin{cases}\left|(\,\mathsf{J}({\mathscr{F}})-d)_{<j}\right|&\textrm{
if }j\leq d,\\\ \left|(\,\mathsf{J}({\mathscr{F}})-d)_{<j}\right|+1&\textrm{
if }j>d,\\\ \end{cases}\qquad\left|\,\mathsf{J}({\mathscr{G}})_{\geq
j}\right|=\begin{cases}\left|(\,\mathsf{J}({\mathscr{F}})-d)_{\geq
j}\right|+1&\textrm{ if }j\leq d,\\\
\left|(\,\mathsf{J}({\mathscr{F}})-d)_{\geq j}\right|&\textrm{ if }j>d.\\\
\end{cases}.$

This completes the proof. ∎

###  4.3. The resistant terms of $\delta^{n-k-1}$ have no mixed biflats.

Proposition 4.6 tells us that, in order to describe the tables
$({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ arising in the canonical expansion
of $\delta^{n-k-1}$ that resist multiplication by $\gamma^{k}$, we may focus
on the bottom row of their tables, that is, on $\mathcal{E}$, $d$, and $D$. We
now pursue this analysis further. Call a biflat $F|G$ _mixed_ if both $F$ and
$G$ are proper flats of $\operatorname{\mathrm{M}}$ and
$\operatorname{\mathrm{M}}^{\perp}$, respectively.

######  Proposition 4.7.

If the table $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ arises in the
canonical expansion of $\delta^{n-k-1}$ and the monomial
$x_{{\mathscr{F}}|{\mathscr{G}}}$ resists multiplication by $\gamma^{k}$, then

  1. (1)

its unique double jump is at $d=r-k$, and

  2. (2)

the resulting monomial $x_{{\mathscr{F}}|{\mathscr{G}}}$ has no mixed biflats,
and its table is of the form

$\begin{array}[]{|cc|ccccccccccc|cc|}\hline\cr\emptyset&\subsetneq&F_{1}&\subsetneq&\cdots&\subsetneq&F_{r-k}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&=&\cdots&=&E&=&E\\\
E&=&E&=&\cdots&=&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&G_{r-k+1}&\supsetneq&\cdots&\supsetneq&G_{n-k-1}&\supsetneq&\emptyset\\\
\hline\cr&&e_{1}&>&\cdots&>&e_{r-k}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{D}&e_{r-k+1}&<&\cdots&<&e_{n-k-1}&&\\\
\hline\cr\end{array}\,.$

For the remainder of this subsection, we write $Y$ for the set
$E-\\{e_{1},\ldots,e_{n-k-1}\\}-D$ consisting of indices that does not appear
in the bottom row of the table of $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$
when augmented with the entry
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{D}$.

######  Example 4.8.

Before proving Proposition 4.7, let us illustrate it using Example 2.10. The
graphical matroid of the cube shown in Figure 2 has $n+1=12$ elements, rank
$r+1=7$, and corank $r^{\perp}+1=5$. We saw in Examples 2.10 and 2.12 that,
for $k=2$, one of the resistant tables
$({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ in the canonical expansion of
$\delta^{n-k-1}=\delta^{8}$ is the $\operatorname{{\mathit{nbc}}}$ monomial of
the basis
$B={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}01}5{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}6}78b$
with
$\operatorname{{IA}}(B)={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}016}$,
given by the table

$\begin{array}[]{|c|ccccccccc|c|}\hline\cr\emptyset&b&8b&78b&578b&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&E&E&E&E\\\
E&E&E&E&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&03469a&469a&69a&a&\emptyset\\\
\hline\cr&b&8&7&5&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{12}&3&4&9&a&\\\
\hline\cr\end{array}\,.$

The double jump occurs at $d=4$ and we have $D=\\{1,2\\}$ and $Y=\\{0,6\\}$.
For either $x=1$ or $x=2$ the flats in ${\mathscr{F}}$ are the closures of the
independent sets $b,8b,78b,578b,x578b$ of $\operatorname{\mathrm{M}}$, and the
coflats in ${\mathscr{G}}$ are the coclosures of the independent sets
$a,9a,49a,349a,x349a$ of $\operatorname{\mathrm{M}}^{\perp}$.

###### Proof of Proposition 4.7.

We first show that $\min\,\mathsf{J}({\mathscr{G}})=d$. Suppose otherwise that
${\mathscr{G}}$ has jumps before $d$, and let $j-1<d$ be the position at which
the first one occurs. Then
$\left\\{0,1,\ldots,j-2\right\\}\in\,\mathsf{J}({\mathscr{F}})$, and the table
of $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ reads

$\small\begin{array}[]{|ccccccccccccccccc|}\hline\cr
F_{1}&\subsetneq&\cdots&\subsetneq&F_{j-1}&=&F_{j}&\subseteq&\cdots&\subseteq&F_{d}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&F_{d+1}&\subseteq&\cdots&\subseteq&F_{n-k-1}\\\
E&=&\cdots&=&E&\supsetneq&G_{j}&\supseteq&\cdots&\supseteq&G_{d}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&G_{d+1}&\supseteq&\cdots&\supsetneq&G_{n-k-1}\\\
\hline\cr
e_{1}&&\cdots&&e_{j-1}&&e_{j}&&\cdots&&e_{d}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{D}&e_{d+1}&&\cdots&&e_{n-k-1}\\\
\hline\cr\end{array}\,.$

Proposition 4.6 guarantees that $\left\\{e_{1},\ldots,e_{j-1}\right\\}$ is
independent and spans the flat $F_{j-1}=F_{j}$ of $\operatorname{\mathrm{M}}$.
Notice that $F_{j}\neq E$, since $j<d$. Now, Lemma 4.2 tells us that
${\mathscr{G}}$ is a complete flag of $\operatorname{\mathrm{M}}^{\perp}$ with
possible repetitions, so the coflat $G_{j}$ must be a hyperplane in
$\operatorname{\mathrm{M}}^{\perp}$, and hence $C_{j}=E-G_{j}$ is a circuit of
$\operatorname{\mathrm{M}}$. But we have that
$G_{j}\supseteq\\{e_{j},e_{j+1},\ldots,e_{n-k-1}\\}\cup D$, which implies that
$C_{j}=E-G_{j}\subseteq\left\\{e_{1},\ldots,e_{j-1}\right\\}\cup Y$. But
$\left\\{e_{1},\ldots,e_{j-1}\right\\}$ is independent in
$\operatorname{\mathrm{M}}$, so there must be an element $y\in Y$ such that
$y\in C_{j}$. Then $y\notin G_{j}$, so $y\in F_{j}$.

Recall from Proposition 4.6 that $B^{\perp}(x)\coloneq\left\\{e_{i}\mid
i\in\,\mathsf{J}({\mathscr{G}})-d\right\\}\cup x$ is a basis of
$\operatorname{\mathrm{M}}^{\perp}$, so its complement
${\mathscr{B}}^{\perp}(x)^{\prime}\coloneq\left\\{e_{i}\mid
i\in\,\mathsf{J}({\mathscr{F}})-0\right\\}\cup(D-x)\cup Y$ is a basis of
$\operatorname{\mathrm{M}}$, and hence

$I\coloneq\left\\{e_{i}\mid i\in\,\mathsf{J}({\mathscr{F}})_{\leq
d}-0\right\\}\cup y$ is independent in $\operatorname{\mathrm{M}}$.

Notice that $I$ is a subset of $F_{d}$ and
$\left|I\right|=\left|\,\mathsf{J}({\mathscr{F}})_{\leq
d}-0\right|+1=\left|\,\mathsf{J}({\mathscr{F}})_{<d}\right|+1$; this
contradicts the third part of Proposition 4.6, which says that
$\operatorname{{rk}}(F_{d})=\left|\,\mathsf{J}({\mathscr{F}})_{<d}\right|$. We
conclude that ${\mathscr{G}}$ has no jumps before $d$, that is,
$G_{1}=\cdots=G_{d}=E$.

We next show that $\max\,\mathsf{J}({\mathscr{F}})=d$. Suppose ${\mathscr{F}}$
has jumps after $d$, and let $j>d$ be the position at which the first such
jump occurs. Since
$\,\mathsf{J}({\mathscr{F}})\cup\,\mathsf{J}({\mathscr{G}})=\left\\{0,1,\ldots,n-k-1\right\\}$
and we previously showed that
$0,1,\ldots,d-1\notin\,\mathsf{J}({\mathscr{G}})$, we must have
$\,\mathsf{J}({\mathscr{F}})\supseteq\left\\{0,1,\ldots,d-1\right\\}$.
Therefore, the table of $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ reads

$\small\begin{array}[]{|ccccccccccccccccc|}\hline\cr
F_{1}&\subsetneq&\cdots&\subsetneq&F_{d}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&F_{d+1}&=&\cdots&=&F_{j}&\subsetneq&F_{j+1}&\subseteq&\cdots&\subsetneq&F_{n-k-1}\\\
E&=&\cdots&=&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&G_{d+1}&\supsetneq&\cdots&\supsetneq&G_{j}&=&G_{j+1}&\supseteq&\cdots&\supsetneq&G_{n-k-1}\\\
\hline\cr
e_{1}&&\cdots&&e_{d}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{D}&e_{d+1}&&\cdots&&e_{i}&&e_{i+1}&&\cdots&&e_{n-k-1}\\\
\hline\cr\end{array}\,.$

This implies that the basis $B^{\perp}(x)^{\prime}$ of
$\operatorname{\mathrm{M}}$ contains
$J\coloneq\left\\{e_{1},\ldots,e_{d},e_{j}\right\\}\cup(D-x)$, which must then
be independent. But $J\subseteq F_{j}=F_{d+1}$, and
$\operatorname{{rk}}(F_{d+1})=d+1$ because ${\mathscr{F}}$ is a complete flag
of $\operatorname{{Tr}}_{k}(\operatorname{\mathrm{M}})$, with possible
repetitions, by Lemma 4.2, so

$d+1\geq\operatorname{{rk}}(\left|J\right|)=\left|J\right|=d+1+\left|D-x\right|.$

which implies $D=\left\\{x\right\\}$. It follows that $F_{d}\cup G_{d+1}=E-x$,
contradicting Lemma 2.2. It follows that ${\mathscr{F}}$ has no jumps after
$d$, that is, $F_{d+1}=\cdots=F_{n-k-1}=E$. We conclude that
$\,\mathsf{J}({\mathscr{F}})=\\{0,1,\ldots,d\\}$ and
$\,\mathsf{J}({\mathscr{G}})=\\{d,d+1,\ldots,n-k\\}$. The first part of
Proposition 4.6 then implies that $d=r-k$.

The above discussion shows that $x_{{\mathscr{F}}|{\mathscr{G}}}$ has no mixed
biflats. Furthermore, Lemma 4.4 tells us that $e_{1}>\cdots>e_{d}$ and
$e_{d+1}<\cdots<e_{n-k-1}$. This completes the proof. ∎

We now strengthen Proposition 4.6 by showing that a resistant table
$({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ in the canonical expansion of
$\delta^{n-k-1}$ is completely determined by the arrival sequence
$\mathcal{E}$.

######  Corollary 4.9.

If $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ be a table arising in the
canonical expansion of $\delta^{n-k-1}$ such that
$x_{{\mathscr{F}}|{\mathscr{G}}}$ resists multiplication by $\gamma^{k}$, then

$F_{j}|G_{j}=\begin{cases}\operatorname{cl}\\{e_{1},\ldots,e_{j}\\}|E&\textrm{
if }j\leq r-k,\\\
E|\operatorname{cl}^{\perp}\\{e_{j},\ldots,e_{n-k+1}\\}&\textrm{ if
}j>r-k.\end{cases}$

In particular, the biflag ${\mathscr{F}}|{\mathscr{G}}$ is determined uniquely
by the arrival sequence $\mathcal{E}$.

###### Proof.

This is a direct consequence of the second part of Proposition 4.6, since
$\,\mathsf{J}({\mathscr{F}})=\\{0,1,\ldots,r-k\\}$ and
$\,\mathsf{J}({\mathscr{G}})=\\{r-k,r-k+1,\ldots,n-k\\}$. Note that
$\mathcal{E}$ also determines the nonempty gap $D=E-(F_{r-k}\cup G_{r-k+1})$.
∎

Although the resistant monomials in $\delta^{n-k-1}$ do not contain mixed
biflats, note however that the multiplication by $\gamma^{k}$ may introduce
mixed biflats in the canonical expansion of $\gamma^{k}\delta^{n-k-1}$, as we
saw in Section 3.4.

###  4.4. The resistant terms of $\delta^{n-k-1}$ are
$\operatorname{{\mathit{nbc}}}$ monomials.

In order to identify the resistant terms in the canonical expansion of
$\delta^{n-k-1}$, we need to recall a few fundamental facts from the theory of
basis activities of a matroid $\operatorname{\mathrm{M}}$ on a linearly
ordered ground set, as developed by Tutte [Tut67] and Crapo [Cra69].

######  Definition 4.10.

For a subset $S\subseteq E$ of the ground set of $\operatorname{\mathrm{M}}$,
we set

$\displaystyle\operatorname{{IA}}(S)$ $\displaystyle\coloneq\left\\{e\in
S\mid\text{there exists a cocircuit $C^{\perp}\subseteq(E-S)\cup e$ for which
$e=\min C^{\perp}$}\right\\},$ $\displaystyle\operatorname{{EA}}(S)$
$\displaystyle\coloneq\left\\{e\in E-S\mid\text{there exists a circuit
$C\subseteq S\cup e$ for which $e=\min C$}\right\\}.$

The following set will play a very important role.

######  Proposition/Definition 4.11.

For an independent set $S\subseteq E$ of $\operatorname{\mathrm{M}}$, let

$\displaystyle{\mathscr{P}}(S)$ $\displaystyle\coloneq\left\\{e\in
E-S\mid\text{there exists a cocircuit $C^{\perp}\subseteq E-S$ for which
$e=\min C^{\perp}$}\right\\}$ $\displaystyle=\left\\{e\in E-S\mid\text{there
exists a cocircuit $C^{\perp}\subseteq E-\operatorname{cl}(S)$ for which
$e=\min C^{\perp}$}\right\\}$ $\displaystyle=\text{lexicographically smallest
set such that $S\sqcup{\mathscr{P}}(S)$ is a basis}.$

###### Proof.

The equivalence of the first and third definitions is shown in [Cra69]. To
show the equivalence of the first two, suppose $C^{\perp}\subseteq E-S$ is a
cocircuit with $e=\min C^{\perp}$. If we had an element $f\in
C^{\perp}\cap\operatorname{cl}(S)$, there would be a circuit $C\subseteq S\cup
f$ containing $f$. But then we would have $C\cap
C^{\perp}=\left\\{f\right\\}$, which is impossible by Lemma 2.2, so
$C^{\perp}\subseteq E-\operatorname{cl}(S)$. This proves one inclusion, and
the reverse inclusion is trivial. ∎

The next proposition shows that there is a close relationship between $S$ and
the lexicographically smallest basis $B$ containing it [LV13, Section 2].

######  Proposition 4.12.

The following holds for any independent subset $S\subseteq E$ of
$\operatorname{\mathrm{M}}$.

  1. (1)

$B\coloneq S\cup{\mathscr{P}}(S)$ is the lexicographically smallest basis of
$\operatorname{\mathrm{M}}$ containing $S$.

  2. (2)

$\operatorname{{IA}}(B)=\operatorname{{IA}}(S)\cup{\mathscr{P}}(S)$.

  3. (3)

$\operatorname{{EA}}(B)=\operatorname{{EA}}(S)$.

In the upcoming arguments, the reader may find it useful to consult Figure 3,
which summarizes Proposition 4.12.

$E$$\operatorname{{IA}}(B)$$\operatorname{{EA}}(B)$$B$${\mathscr{P}}(S)$$S$$\operatorname{{IA}}(S)$$\operatorname{{EA}}(S)$
Figure 3. Activities for an independent $S$ and its greedy completion to a
basis $B$.

######  Example 4.13.

For the graphical matroid of the pyramid in Figure 1 and $S=15$, we have
${\mathscr{P}}(S)=23$ and $B=01234$. Then
$\operatorname{{IA}}(B)=23,\operatorname{{IA}}(S)=\emptyset,$ and
$\operatorname{{EA}}(B)=\operatorname{{EA}}(S)=0.$

We will make use of the following fact.

######  Lemma 4.14.

Let $S$ be an independent set of $\operatorname{\mathrm{M}}$, and let
$F=\operatorname{cl}(S)$. If $G$ is a hyperplane of
$\operatorname{\mathrm{M}}^{\perp}$ satisfying $F\cup G\neq E$, then $F\cup
G\cup{\mathscr{P}}(S)\neq E$ as well.

###### Proof.

Proposition 4.12 tells us that $B\coloneq S\cup{\mathscr{P}}(S)$ is a basis of
$\operatorname{\mathrm{M}}$ and
${\mathscr{P}}(S)\subseteq\operatorname{{IA}}(B)$. Thus, for each element
$p\in{\mathscr{P}}(S)$, we can find a unique cocircuit
$C^{\perp}(p)\subseteq(E-B)\cup p$ of $\operatorname{\mathrm{M}}$ for which
$p=\min C^{\perp}_{p}$. Notice that
$C^{\perp}(p)\cap{\mathscr{P}}(S)=\left\\{p\right\\}$. Set $H(p)\coloneq
E-C^{\perp}(p)$, a hyperplane of $\operatorname{\mathrm{M}}$.

We claim $F=\bigcap_{p\in{\mathscr{P}}(S)}H(p)$. Since each $C^{\perp}(p)$ is
contained in $E-S$ we have $S\subseteq\bigcap_{p\in{\mathscr{P}}(S)}H(p)$.
Since the intersection of flats is a flat,
$F=\operatorname{cl}(S)\subseteq\bigcap_{p\in{\mathscr{P}}(S)}H(p)$ as well.
On the other hand, each $H(p)$ is a hyperplane, so the submodular inequality
gives

$\operatorname{{rk}}\big{(}\bigcap_{p\in{\mathscr{P}}(S)}H(p)\big{)}\leq\operatorname{{rk}}(\operatorname{\mathrm{M}})-\left|{\mathscr{P}}(S)\right|=\left|S\right|=\operatorname{{rk}}(F)$

which proves the claim.

Now, the complement $C\coloneq E-G$ is a circuit of
$\operatorname{\mathrm{M}}$. Let $D=E-(F\cup G)\neq\emptyset$. Then

$D=(E-F)\cap(E-G)=\big{(}\bigcup_{p\in{\mathscr{P}}(S)}C^{\perp}(p)\big{)}\cap
C=\bigcup_{p\in{\mathscr{P}}(S)}(C^{\perp}(p)\cap C).$

Since $D$ is nonempty, $C^{\perp}(p)\cap C\neq\emptyset$ for some $p$. Since
$C^{\perp}(p)$ is a cocircuit and $C$ is a circuit, $C^{\perp}(p)\cap C$ – and
hence $D$ – contains at least one element $q\neq p$. Since
${C^{\perp}(p)\cap{\mathscr{P}}(S)}=\left\\{p\right\\}$, it follows that
$q\notin{\mathscr{P}}(S)$ so $q\notin F\cup G\cup{\mathscr{P}}(S)$, as
desired. ∎

In Proposition 4.7, we showed that the terms of the canonical expansion of
$\delta^{n-k-1}$ that resist multiplication by $\gamma^{k}$ are given by
tables $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ of the form

$\begin{array}[]{|cc|ccccccccccc|cc|}\hline\cr\emptyset&\subsetneq&F_{1}&\subsetneq&\cdots&\subsetneq&F_{r-k}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\subsetneq}&E&=&\cdots&=&E&=&E\\\
E&=&E&=&\cdots&=&E&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\supsetneq}&G_{r-k+1}&\supsetneq&\cdots&\supsetneq&G_{n-k-1}&\supsetneq&\emptyset\\\
\hline\cr&&e_{1}&>&\cdots&>&e_{r-k}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{D}&e_{r-k+1}&<&\cdots&<&e_{n-k-1}&&\\\
\hline\cr\end{array}\,.$

The following proposition describes precisely which tables arise.

######  Proposition 4.15.

Let $S$ be the independent set $\left\\{e_{1}>\cdots>e_{r-k}\right\\}$ of
$\operatorname{\mathrm{M}}$, and let $B$ be the basis
$B=S\sqcup{\mathscr{P}}(S)$ of $\operatorname{\mathrm{M}}$ in Proposition
4.12.

  1. (1)

$\operatorname{{IA}}(S)=\operatorname{{EA}}(S)=\emptyset$.

  2. (2)

$\operatorname{{EA}}(B)=\emptyset$, so $B$ is a
$\operatorname{{\mathit{nbc}}}$ basis of $\operatorname{\mathrm{M}}$.

  3. (3)

$\operatorname{{IA}}(B)={\mathscr{P}}(S)$ and $|\operatorname{{IA}}(B)|=k+1$.

  4. (4)

$e_{i}=\min F_{i}$ for $1\leq i\leq r-k$.

  5. (5)

$e_{i}=\min(G_{i}-{\mathscr{P}}(S))$ for $r-k+1\leq i\leq n-k-1$.

  6. (6)

${\mathscr{F}}|{\mathscr{G}}={{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ and
$x_{{\mathscr{F}}|{\mathscr{G}}}$ is the $\operatorname{{\mathit{nbc}}}$
monomial of the $\operatorname{{\mathit{nbc}}}$ basis $B$, as in Proposition
2.9.

######  Example 4.16.

For the cube graph of Figure 2 and $k=2$, let us revisit the resistant
monomial

$x_{b|E}\,x_{8b|E}\,x_{78b|E}\,x_{578b|E}\,x_{E|03469a}\,x_{E|469a}\,x_{E|69a}\,x_{E|9}.$

The arrival sequence $\mathcal{E}=(b,8,7,5;3,4,9,a)$ was computed in Example
4.8. We have $S=b875$, and its lexicographically smallest completion to a
basis is given by ${\mathscr{P}}(S)=016$. Then
$B={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}01}5{\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}6}78b$
is indeed a $\operatorname{{\mathit{nbc}}}$ basis with
$\operatorname{{IA}}(B)={\color[rgb]{0,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,0}016}$
. Conditions (3) and (4) are easily checked directly, and
${{\mathscr{F}}|{\mathscr{G}}}={{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ as
described in Example 2.10.

We prepare the proof of Proposition 4.15 with some technical lemmas.

######  Lemma 4.17.

Let $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ be as in Proposition 4.15.

  1. (1)

For each $1\leq i\leq r-k$, there is an index $r-k+1\leq j\leq n-k$ such that

$e_{i}=\max\big{(}E-(F_{i-1}\cup G_{j})\big{)}.$

It is the smallest index $j\geq r-k+1$ such that $e_{j}>e_{i}$, or $j=n-k$ if
there is no such index.

  2. (2)

For each $r-k+1\leq j\leq n-k-1$, there is an index $0\leq i\leq r-k$ such
that

$e_{j}=\max\big{(}E-(F_{i}\cup G_{j+1})\big{)}.$

It is the largest index $i\leq r-k$ such that $e_{i}>e_{j}$, or $i=0$ if there
is no such index.

###### Proof.

This is a straightforward restatement of Proposition 3.5, since we have
determined the table for $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ in
Proposition 4.7. ∎

######  Lemma 4.18.

Every element of the nonempty gap $D$ is smaller than every $e_{i}$.

###### Proof.

The smallest $e_{i}$ is either $e_{r-k}$ or $e_{r-k+1}$. If
$e_{r-k}<e_{r-k+1}$, then Lemma 4.17 implies that

$e_{r-k}=\max\big{(}E-(F_{r-k-1}\cup G_{r-k+1})\big{)}\geq\max D,$

where we used $F_{r-k-1}\cup G_{r-k+1}\subseteq F_{r-k}\cup G_{r-k+1}=E-D$.
Since $D\cap\left\\{e_{1},\ldots,e_{n-k-1}\right\\}$ is empty, the inequality
is strict. If $e_{r-k}>e_{r-k+1}$, a similar argument shows $e_{r-k+1}>\max
D$. ∎

The first part of Proposition 4.6 tells us that $S$ is independent in
$\operatorname{\mathrm{M}}$, and the first part of Proposition 4.12 tells us
that the set $B\coloneq S\sqcup{\mathscr{P}}(S)$ is a basis. In particular,
$\left|{\mathscr{P}}(S)\right|=k+1$ and ${\mathscr{Q}}(S)=\emptyset$. Our next
result relates ${\mathscr{P}}(S)$ with the partition
$E=\left\\{e_{1},\ldots,e_{n-k-1}\right\\}\sqcup D\sqcup Y$. We illustrate
this in Figure 4, which is a refinement of Figure 3 in the case
$\operatorname{{IA}}(S)=\operatorname{{EA}}(S)=\emptyset$.

$E$$D$$Y$$\operatorname{{IA}}(B)$$B$$x$${\mathscr{P}}(S)$$S=\left\\{e_{1},\ldots,e_{r-k}\right\\}$$B^{\perp}-x=\left\\{e_{r-k+1},\ldots,e_{n-k-1}\right\\}$$\operatorname{{IA}}(S)=\emptyset$$\operatorname{{EA}}(S)=\emptyset$
Figure 4. Activities and the partition of $E$.

######  Lemma 4.19.

For $x=\min(E-B)\in D$, we have

$E-\left\\{e_{1},\ldots,e_{n-k-1}\right\\}=D\sqcup
Y={\mathscr{P}}(S)\sqcup\left\\{x\right\\}.$

###### Proof.

We first show that ${\mathscr{P}}(S)\subseteq D\sqcup Y$. By way of a
contradiction, suppose $e_{i}\in{\mathscr{P}}(S)$ for some $i$. Since
$e_{1},\ldots,e_{r-k}\in S$, we must have $i\geq r-k+1$. By Definition 4.11,
there is a cocircuit $C^{\perp}$ for which $e_{i}=\min C^{\perp}$ and
$C^{\perp}\subseteq E-\operatorname{cl}(S)=E-F_{r-k}$. Also, Lemma 4.17 tells
us that

$e_{i}=\max(E-F_{j})\cap(E-G_{i+1})\ \ \text{for some $j\leq r-k$}.$

However, since $e_{i}\notin G_{i+1}$ by Lemma 4.5, we have

$(E-F_{j})\cap(E-G_{i+1})\supseteq(E-F_{r-k})\cap(E-G_{i+1})\supseteq
C^{\perp}\cap(E-G_{i+1})\ni e_{i}.$

Therefore, we must also have $e_{i}=\max\big{(}(E-G_{i+1})\cap
C^{\perp}\big{)}$. Since $e_{i}=\min C^{\perp}$ we must have $(E-G_{i+1})\cap
C^{\perp}=\left\\{e_{i}\right\\}$, and hence
$\left|G_{i+1}\cup(E-C^{\perp})\right|=n$; this contradicts Lemma 2.2.

Next we show that there must be an element $x\in D$ that is not in
${\mathscr{P}}(S)$. To do that, we invoke Lemma 4.14. Since
$\operatorname{cl}(S)\cup G_{r-k+1}=F_{r-k}\cup G_{r-k+1}\neq E$, we must have
some element $x\notin F_{r-k}\cup G_{r-k+1}\cup{\mathscr{P}}(S)$. However,
$x\notin F_{r-k}\cup G_{r-k+1}$ means that $x\in D$.

We observe that $\left|D\sqcup
Y\right|=\left|E\right|-\left|\left\\{e_{1},\ldots,e_{n-k-1}\right\\}\right|=(n+1)-(n-k-1)=k+2$,
while
$\left|{\mathscr{P}}(S)\right|=\left|B\right|-\left|S\right|=(r+1)-(r-k)=k+1$.
Since our $x\not\in{\mathscr{P}}(S)$, the inclusion
${\mathscr{P}}(S)\sqcup\left\\{x\right\\}\subseteq D\sqcup Y$ must in fact be
an equality.

Finally, Lemma 4.18 and $E-B=\\{x,e_{r-k+1},\ldots,e_{n-k}\\}$ give
$x=\min(E-B)$. ∎

######  Lemma 4.20.

We have the inclusion $Y\subseteq G_{r-k+1}$.

###### Proof.

We have that $Y\subseteq F_{r-k}\cup G_{r-k+1}=E-D$. If the claim were not
true, there would be an element $y\in Y$ such that $y\in
F_{r-k}=\operatorname{cl}(S)$, so $S\cup y$ would be dependent. But Lemma 4.19
would then imply that $y\in Y\subseteq{\mathscr{P}}(S)$, contradicting the
fact that $S\cup P(S)$ is a basis. ∎

We now prove our description of the resistant terms in the canonical expansion
of $\delta^{n-k-1}$.

###### Proof of Proposition 4.15.

We prove parts (4), (5), (1), (2), (3), (6), in that order.

  1. (4)

Let $1\leq i\leq r-k$. Lemma 4.17 provides a $j\geq r-k+1$ for which
$e_{i}=\max\big{(}E-(F_{i-1}\cup G_{j})\big{)}$. We claim that

$F_{i}\cup
G_{j}\subseteq\left\\{e_{1},\ldots,e_{i}\right\\}\sqcup\left\\{e_{j},e_{j+1},\ldots,e_{n-k-1}\right\\}\sqcup
D\sqcup Y.$

In view of the decomposition $E=\left\\{e_{1},\ldots,e_{n-k-1}\right\\}\sqcup
D\sqcup Y$, this amounts to checking that $e_{h}\not\in F_{i}\cup G_{j}$ for
$i<h<j$, which follows from Lemma 4.5.

Now assume, for the sake of contradiction, that $\min F_{i}=a<e_{i}$. Since
$e_{1}>\cdots>e_{i}$ and $e_{n-k-1}>\cdots>e_{j}>e_{i}$, we must have $a\in
D\sqcup Y={\mathscr{P}}(S)\sqcup x$, by Lemma 4.19. However, $D\cap
F_{i}=\emptyset$ and $a\in F_{i}$, so $a$ cannot be $x$. Additionally,
${\mathscr{P}}(S)$ is independent from $S$ and $a\in
F_{i}\subseteq\operatorname{cl}(S)$, so $a$ cannot be in ${\mathscr{P}}(S)$
either. We conclude $e_{i}=\min F_{i}$ for $i\leq r-k$.

  2. (5)

Let $r-k+1\leq j\leq n-k-1$. As before, Lemma 4.17 provides a $0\leq i\leq
r-k$ for which

$F_{i}\cup
G_{j}\subseteq\left\\{e_{1},\ldots,e_{i}\right\\}\sqcup\left\\{e_{j},e_{j+1},\ldots,e_{n-k-1}\right\\}\sqcup
D\sqcup Y.$

Assume, for the sake of contradiction, that
$\min\big{(}G_{j}-{\mathscr{P}}(S)\big{)}=a<e_{j}$. Since
$e_{1}>\cdots>e_{i}>e_{j}$ and $e_{n-k-1}>\cdots>e_{j}$, we must have $a\in
D\cup Y$. But $a\notin{\mathscr{P}}(S)$ by definition, and $D\cap
G_{j}=\emptyset$ so $a\neq x$. The desired result follows.

  3. (1)

If we had $a\in\operatorname{{EA}}(S)$, then $S\cup a$ would contain a circuit
$C$ with $a=\min C$. For the largest $i$ with $e_{i}\in C$, we would then have
$C-a\subseteq F_{i}$. Since $C$ is a circuit and $F_{i}$ is a flat, this would
imply that $a\in F_{i}$, contradicting that $e_{i}=\min F_{i}$ as shown in 4.
Thus $\operatorname{{EA}}(S)=\emptyset$.

Suppose we had $e_{i}\in\operatorname{{IA}}(S)$ for some $1\leq i\leq r-k$.
Then $e_{i}\in\operatorname{{IA}}(B)$, so there is a cocircuit
$C^{\perp}\subseteq(E-B)\cup e_{i}=\\{e_{r-k+1},\ldots,e_{n-k},x,e_{i}\\}$
with $e_{i}=\min C^{\perp}$. By Lemma 4.18, this means that $x\notin
C^{\perp}$. Therefore $C^{\perp}-e_{i}\subseteq G_{r-k+1}$. But then, since
$C^{\perp}$ is a circuit and $G_{r-k+1}$ is a flat in
$\operatorname{\mathrm{M}}^{\perp}$, we must have $e_{i}\in G_{r-k+1}$ as
well; this contradicts Lemma 4.4 since $i<r-k+1$ and $e_{i}<e_{r-k+1}$.

  4. (2)

Proposition 4.12 and 1. tell us that $B$ is a basis and
$\operatorname{{EA}}(B)=\operatorname{{EA}}(S)=\emptyset$.

  5. (3)

Proposition 4.12 and 2 tell us that $\operatorname{{IA}}(B)={\mathscr{P}}(S)$.

  6. (6)

By (2) and (3), we have

$B-\operatorname{{IA}}(B)=S=\\{e_{1}>\cdots>e_{r-k}\\}$

and Lemma 4.19 tells us that

$E-B-\min(E-B)=\\{e_{r-k+1}<\cdots<e_{n-k-1}\\}$

Therefore, by Corollary 4.9, the flags ${\mathscr{F}}$ and ${\mathscr{G}}$ are
precisely the flags ${\mathscr{F}}(B)$ and ${\mathscr{G}}(B)$ of the
$\operatorname{{\mathit{nbc}}}$ biflag of $B$, as described in Proposition
2.9. ∎

######  Corollary 4.21.

Let $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ be a table arising in the
canonical expansion of $\delta^{n-k-1}$ such that
$x_{{\mathscr{F}}|{\mathscr{G}}}$ resists multiplication by $\gamma^{k}$. Then
the arrival sequence $\mathcal{E}$ is determined uniquely by the biflag
${\mathscr{F}}|{\mathscr{G}}$.

###### Proof.

The arrival sequence $\mathcal{E}$ is determined by (3) and (4) of Proposition
4.15. ∎

We are finally ready to prove our description of the canonical expansion of
$\gamma^{k}\delta^{n-k-1}$ into monomials.

###### Proof of Theorem 1.1.

Every resistant table $({\mathscr{F}}|{\mathscr{G}},\mathcal{E})$ in the
canonical expansion of $\delta^{n-k-1}$ gives a
$\operatorname{{\mathit{nbc}}}$ monomial
$x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}$ by the last part of Proposition 4.15.
Every such monomial does appear in this expansion by Proposition 3.8.
Furthermore, it appears only once by Corollary 4.21. Therefore, we have

$\gamma^{k}\delta^{n-k-1}=\sum_{\begin{subarray}{c}B\operatorname{{\mathit{nbc}}}\textrm{basis}\\\
|\operatorname{{IA}}(B)|=k+1\end{subarray}}\gamma^{k}x_{{\mathscr{F}}(B)|{\mathscr{G}}(B)}.$

The desired formula for $\gamma^{k}\delta^{n-k-1}$ then follows by Proposition
3.12. ∎

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