Rook placements and Jordan forms of upper-triangular nilpotent matrices

TL;DR

This paper provides a combinatorial formula for counting upper-triangular nilpotent matrices over finite fields by their Jordan forms, linking rook placements and Young tableaux to refine enumeration methods.

Contribution

It introduces a new combinatorial formula for F_____(q) using standard Young tableaux and explores rook placements to refine this enumeration.

Findings

Derived a weighted sum formula for F_____(q)

Established a connection between matrices and non-attacking rook placements

Provided a combinatorial refinement of the counting formula

Abstract

The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions lambda of n. We present a combinatorial formula for computing the number F_\lambda(q) of matrices of Jordan type lambda as a weighted sum over standard Young tableaux. We also study a connection between these matrices and non-attacking rook placements, which leads to a refinement of the formula for F_\lambda(q).

Equations

$$J_{\lambda }=J_{{\lambda }_1}\oplus J_{{\lambda }_2}\oplus \cdots \oplus J_{{\lambda }_r},where{J}_i={\left[\begin{array}{cccccc}0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ 0& 0& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 1\\ 0& 0& 0& \cdots & 0& 0\end{array}\right]}_{i\times i}$$

$$F_{\lambda }(q)=\left|{\mathfrak{g}}_{n,\lambda }({\mathbb{F}}_q)\right|.$$

$$R_{B,k}(q)=\sum _{c\in \mathcal{C}(B,k)}{q}^{\mathrm{inv}(c)},$$

$$S_{n,k}(q)=q^{k-1}{S}_{n-1,k-1}(q)+[k{]}_q{S}_{n-1,k}(q)for0\le k\le n,$$

$$P_{B_n,k}(q)=(q-1{)}^k{q}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)-k}{R}_{B_n,k}(q^{-1}).$$

$$P_{B_n,k}(q)=\sum _{\lambda ⊢n:\ell (\lambda )=n-k}{F}_{\lambda }(q).$$

$$\lambda =(4,2,2,1)⊢9has\; diagram\text{yng}(4,2,2,1)$$

$$\mathrm{rank}\left(J_{\lambda }^k\right)=\{\begin{array}{ll}{\lambda }_{k+1}^{\prime }+\cdots +{\lambda }_c^{\prime },& if\; 0\le k<c,\\ 0,& ifk\ge c.\end{array}$$

$$\mathrm{rank}\left(J_{\lambda }^k\right)=\sum _{i=1}^r\mathrm{rank}\left(J_{{\lambda }_i}^k\right)=\sum _{i:{\lambda }_i\ge k}\mathrm{rank}\left(J_{{\lambda }_i}^k\right)=\sum _{j=k+1}^c{\lambda }_j^{\prime },$$

$$X=\left[\begin{array}{cc}{J}_{\mu }& \mathbf{v}\\ 0& 0\end{array}\right],$$

$$X^k=\left[\begin{array}{cc}{J}_{\mu }^k& {\mathbf{v}}^k\\ 0& 0\end{array}\right].$$

$$v_i^k=\{\begin{array}{ll}{v}_{i+k-1},& ifi,i+k-1\le {\alpha }_{b}for\; allb\ge 1,\\ 0,& otherwise.\end{array}$$

$$X=\left[\begin{array}{ccccccccc}0& 1& 0& 0& & & & & v_1\\ 0& 0& 1& 0& & & & & v_2\\ 0& 0& 0& 1& & & & & v_3\\ 0& 0& 0& 0& & & & & v_4\\ & & & & & & & & \\ & & & & 0& 1& & & v_5\\ & & & & 0& 0& & & v_6\\ & & & & & & & & \\ & & & & & & 0& & v_7\\ & & & & & & & & \\ & & & & & & & 0& v_8\\ & & & & & & & & \\ & & & & & & & & 0\end{array}\right]so\; that{X}^2=\left[\begin{array}{ccccccccc}0& 0& 1& 0& & & & & v_2\\ 0& 0& 0& 1& & & & & v_3\\ 0& 0& 0& 0& & & & & v_4\\ 0& 0& 0& 0& & & & & 0\\ & & & & & & & & \\ & & & & 0& 0& & & v_6\\ & & & & 0& 0& & & 0\\ & & & & & & & & \\ & & & & & & 0& & 0\\ & & & & & & & & \\ & & & & & & & 0& 0\\ & & & & & & & & \\ & & & & & & & & 0\end{array}\right].$$

$$\mathbf{v}=\text{young}(\text{raise}1.5pt{v}_4\text{raise}1.5pt{v}_3\text{raise}1.5pt{v}_2\text{raise}1.5pt{v}_1,\text{raise}1.5pt{v}_6\text{raise}1.5pt{v}_5,\text{raise}1.5pt{v}_7,\text{raise}1.5pt{v}_8)and{\mathbf{v}}^2=\text{young}(0\text{raise}1.5pt{v}_4\text{raise}1.5pt{v}_3\text{raise}1.5pt{v}_2,0\text{raise}1.5pt{v}_6,0,0).$$

$$\mathrm{rank}\left(X^k\right)-\mathrm{rank}\left(J_{\mu }^k\right)=\{\begin{array}{ll}0,& if{v}_{{\alpha }_i}=0\; for\; allisuch\; that{\mu }_i\ge k,\\ 1,& otherwise.\end{array}$$

$$\mathrm{rank}(X^k)-\mathrm{rank}(J_{\mu }^k)=\{\begin{array}{ll}0,& ifk\ge c,\\ 1,& ifk<c.\end{array}$$

$${\lambda }_k^{\prime }-{\mu }_k^{\prime }=\left(\mathrm{rank}(X^{k-1})-\mathrm{rank}(X^k)\right)-\left(\mathrm{rank}(J_{\mu }^{k-1})-\mathrm{rank}(J_{\mu }^k)\right)=\{\begin{array}{ll}1,& ifk=c,\\ 0,& ifk\ne c.\end{array}$$

$$c_{\mu,\lambda}(q)=\left\{\begin{array}[]{ll}q^{|\mu|-\mu_{j}^{\prime}},&\hbox{if $j=1$},\\ q^{|\mu|-\mu_{j-1}^{\prime}}\left(q^{\mu_{j-1}^{\prime}-\mu_{j}^{\prime}}-1\right),&\hbox{if $j\geq 2$}.\end{array}\right.$$

$$X=\left[\begin{array}{cc}{J}_{\mu }& \mathbf{v}\\ 0& 0\end{array}\right]$$

$$q^{n-1-{\mu }_{j-1}^{\prime }}\left(q^{{\mu }_{j-1}^{\prime }-{\mu }_j^{\prime }}-1\right)$$

$$F_{\lambda }(q)=\sum _{\mu :\mu ⋖\lambda }{c}_{\mu ,\lambda }(q)F_{\mu }(q),$$

$$\text{yng}(1,1,1,1)$$

$$F_{\lambda }(q)=\sum _{P\in P_{\mathcal{Y}}(\lambda )}{F}_P(q),$$

$$T^{(i)}(q)=\{\begin{array}{ll}{q}^{i-\ell ({\lambda }^{(i)})},& if\; theith\; box\; is\; in\; the\; first\; column,\\ q^{i-{{\lambda }_j^{(i)}}^{\prime }}-q^{i-1-{{\lambda }_{j-1}^{(i)}}^{\prime }},& if\; theith\; box\; is\; in\; thejth\; column,j\ge 2.\end{array}$$

$$F_{\lambda }(q)=\sum _{T\in \mathrm{SYT}(\lambda )}{F}_T(q),$$

$$n_{\lambda }=\sum _{i\ge 1}(i-1){\lambda }_i=\sum _{b\in \lambda }\mathrm{coleg}(b),$$

$$\mathrm{deg}{F}_{\lambda }(q)=\left(\genfrac{}{}{0ex}{}{n}{2}\right)-n_{\lambda }.$$

$$\mathrm{deg}{F}_P(q)=\sum _{k=1}^n\mathrm{deg}{c}_{{\pi }^{(k-1)},{\pi }^{(k)}}(q)=\sum _{k=1}^{n}k-\sum _{k\ge 1}k{\lambda }_k=\left(\genfrac{}{}{0ex}{}{n}{2}\right)-n_{\lambda }.$$

$$F_{(3,1)}(q)$$

$$=(q-1{)}^2\left(3q^3+q^2\right),$$

$$F_{(2,2)}(q)$$

Full text

1. Introduction

In the beautiful paper Variations on the Triangular Theme [Kir95], Kirillov studied various structures on the set of triangular matrices. Let $G=G_{n}(\mathbb{F}_{q})$ denote the group of $n$ by $n$ invertible upper-triangular matrices over the field $\mathbb{F}_{q}$ having $q$ elements, and let $\mathfrak{g}=\mathfrak{g}_{n}(\mathbb{F}_{q})=\mathrm{Lie}(G_{n}(\mathbb{F}_{q}))$ denote the corresponding Lie algebra of $n$ by $n$ upper-triangular nilpotent matrices over $\mathbb{F}_{q}$. The problem of determining the set $\mathcal{O}_{n}(\mathbb{F}_{q})$ of adjoint $G$-orbits in $\mathfrak{g}$ remains challenging, and a more tractable task is to study a decomposition of $\mathcal{O}_{n}(\mathbb{F}_{q})$ via the Jordan canonical form. Let $\lambda\vdash n$ be a partition of $n$ with $r$ positive parts $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r}>0$, and let

[TABLE]

is the $i$ by $i$ elementary Jordan matrix with all eigenvalues equal to zero. If $X\in\mathfrak{g}_{n}(\mathbb{F}_{q})$ is similar to $J_{\lambda}$ under $GL_{n}(\mathbb{F}_{q})$, then $X$ is said to have Jordan type $\lambda$. Each conjugacy class contains a unique Jordan matrix $J_{\lambda}$, so these classes are indexed by the partitions of $n$. Evidently, the Jordan type of $X$ depends only on its adjoint $G$-orbit.

Let $\mathfrak{g}_{n,\lambda}(\mathbb{F}_{q})\subseteq\mathfrak{g}_{n}(\mathbb{F}_{q})$ be the set of upper-triangular nilpotent matrices of fixed Jordan type $\lambda$, and let

[TABLE]

Springer showed that $\mathfrak{g}_{n,\lambda}(\mathbb{F}_{q})$ is an algebraic manifold with $f^{\lambda}$ irreducible components, where $f^{\lambda}$ is the number of standard Young tableaux of shape $\lambda$, and each of which has dimension ${n\choose 2}-n_{\lambda}$, where $n_{\lambda}$ is an integer defined in Equation 10. These quantities appear in the study of $F_{\lambda}(q)$.

In Section 2, we show that the numbers $F_{\lambda}(q)$ satisfy a simple recurrence equation, and that they are polynomials in $q$ with integer coefficients. As a consequence of the recurrence equation in Theorem 2.8, it follows that the coefficient of the highest degree term in $F_{\lambda}(q)$ is $f^{\lambda}$, and $\deg F_{\lambda}(q)={n\choose 2}-n_{\lambda}$. Equation (9) is a combinatorial formula for $F_{\lambda}(q)$ as a sum over standard Young tableaux of shape $\lambda$ that can be derived from the recurrence equation.

The cases $F_{(1^{n})}(q)=1$ and $F_{(n)}(q)=(q-1)^{n-1}q^{n-1\choose 2}$ are readily computed, since the matrix in $\mathfrak{g}_{n}(\mathbb{F}_{q})$ of Jordan type $(1^{n})$ is the matrix of zero rank, and the matrices in $\mathfrak{g}_{n}(\mathbb{F}_{q})$ of Jordan type $(n)$ are the matrices of rank equal to $n-1$. Section 2 concludes with explicit formulas for $F_{\lambda}(q)$ in several other special cases of $\lambda$, including hook shapes, two-rowed partitions and two-columned partitions.

In Section 3, we explore a connection of $F_{\lambda}(q)$ with rook placements. In their study of a formula of Frobenius, Garsia and Remmel [GR86] introduced the $q$-rook polynomial

[TABLE]

which is a sum over the set $\mathcal{C}(B,k)$ of non-attacking placements of $k$ rooks on a Ferrers board $B$, and $\mathrm{inv}(c)$, defined in Equation (13), is the number of inversions of $c$. In the case when $B=B_{n}$ is the staircase-shaped board, Garsia and Remmel showed that $R_{B_{n},k}(q)=S_{n,n-k}(q)$ is a $q$-Stirling number of the second kind. These numbers are defined by the recurrence equation

[TABLE]

with initial conditions $S_{0,0}(q)=1$, and $S_{n,k}(q)=0$ for $k<0$ or $k>n$.

It was shown by Solomon [Sol90] that non-attacking placements of $k$ rooks on rectangular $m\times n$ boards are naturally associated to $m$ by $n$ matrices with rank $k$ over $\mathbb{F}_{q}$. By identifying a Ferrers board $B$ inside an $n$ by $n$ grid with the entries of an $n$ by $n$ matrix, Haglund [Hag98] generalized Solomon’s result to the case of non-attacking placements of $k$ rooks on Ferrers boards, and obtained a formula for the number of $n$ by $n$ matrices with rank $k$ whose support is contained in the Ferrers board region. A special case of Haglund’s formula shows that the number of $n$ by $n$ nilpotent upper-triangular matrices of rank $k$ is

[TABLE]

Now, a matrix in $\mathfrak{g}_{n,\lambda}(\mathbb{F}_{q})$ has rank $n-\ell(\lambda)$, where $\ell(\lambda)$ is the number of parts of $\lambda$, so the number of matrices in $\mathfrak{g}_{n}(\mathbb{F}_{q})$ with rank $k$ is

[TABLE]

Given Equations 2 and 3, it would be natural to ask whether it is possible to partition the placements $\mathcal{C}(B_{n},k)$ into disjoint subsets so that the sum over each subset of placements gives $F_{\lambda}(q)$. A central goal of this paper is to study the connection between upper-triangular nilpotent matrices over $\mathbb{F}_{q}$ and non-attacking rook placements on the staircase-shaped board $B_{n}$. Theorem 3.7 shows that there is a weight-preserving bijection $\Phi$ between rook placements on $B_{n}$ and paths in a graph $\mathcal{Z}$ (see Figure 5), which is a multi-edged version of Young’s lattice. As a result, we obtain Corollary 3.9, which gives a formula for $F_{\lambda}(q)$ as a sum over certain rook placements that can be viewed as a generalization of Haglund’s formula in Equation (2).

There is a classically known bijection between rook placements in $\mathcal{C}(B_{n},k)$ and set partitions of $[n]$ with $n-k$ parts, so it is logical to next study the connection between $F_{\lambda}(q)$ and set partitions. We do this in Section 4. Theorem 4.3 describes the construction of a new (weight-preserving) bijection $\Psi$ between rook placements and set partitions. These bijections allow us to refine Equation (9) to a sum over set partitions (or rook placements). We also discuss the significance of the polynomials $F_{C}(q)$ indexed by rook placements in a special case.

Acknowledgements. I would like to thank Jim Haglund and Alexandre Kirillov for their invaluable guidance, and Yue Cai and Alejandro Morales for enlightening conversations.

2. Formulas for $F_{\lambda}(q)$

The recurrence equation for $F_{\lambda}(q)$ in Theorem 2.8 can be found in [Bor95, Division Theorem], where Borodin considers the matrices as particles of a certain mass and studies the asymptotic behaviour of the formula. A preliminary version of the idea first appeared in [Kir95]. In this section, we give an elementary proof of the formula, and investigate some of the combinatorial properties of $F_{\lambda}(q)$.

2.1. The recurrence equation for $F_{\lambda}(q)$

A partition $\lambda$ of a nonnegative integer $n$, denoted by $\lambda\vdash n$, is a non-increasing sequence of nonnegative integers $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0$ with $|\lambda|=\sum_{i=1}^{n}\lambda_{i}=n$. If $\lambda$ has $r$ positive parts, write $\ell(\lambda)=r$. A partition $\lambda$ can be represented by its Ferrers diagram in the English notation, which is an array of $\lambda_{i}$ boxes in the $i$th row, with the boxes justified upwards and to the left. Let $\lambda_{j}^{\prime}$ denote the size of the $j$th column of $\lambda$.

Young’s lattice $\mathcal{Y}$ is the lattice of partitions ordered by the inclusion of their Ferrers diagrams; that is, $\mu\leq\lambda$ if and only if $\mu_{i}\leq\lambda_{i}$ for every $i$. In particular, $\mu$ is covered by $\lambda$ in the Hasse diagram of $\mathcal{Y}$ and we write $\mu\lessdot\lambda$ if the Ferrers diagram of $\lambda$ can be obtained by adding a box to the Ferrers diagram of $\mu$. See Figure 1.

Example 2.1**.**

\Yboxdim

10pt The partition

[TABLE]

and columns $\lambda_{1}^{\prime}=4,\lambda_{2}^{\prime}=3,\lambda_{3}^{\prime}=1,\lambda_{4}^{\prime}=1$.

Lemma 2.2**.**

Let $\lambda\vdash n$ be a partition whose Ferrers diagram has $r$ rows and $c$ columns. The Jordan matrix $J_{\lambda}$ satisfies

[TABLE]

Proof.

The $i$ by $i$ elementary Jordan matrix $J_{i}$ has $\mathrm{rank}\left(J_{i}^{k}\right)=i-k$ if $0\leq k\leq i$, and its rank is zero otherwise, so the Jordan matrix $J_{\lambda}=J_{\lambda_{1}}\oplus\cdots\oplus J_{\lambda_{r}}$ has

[TABLE]

for $0\leq k<c$, which is the number of boxes in the last $c-k$ columns of $\lambda$. ∎

Remark 2.3**.**

Matrices which are similar have the same rank, so if $X\in\mathfrak{g}_{n,\lambda}(\mathbb{F}_{q})$, then $\mathrm{rank}\left(X^{k}\right)=\mathrm{rank}\left(J_{\lambda}^{k}\right)$ for all $k\geq 0$. Conversely, let $\lambda,\nu\vdash n$. It follows from Lemma 2.2 that $\mathrm{rank}\left(J_{\lambda}^{k}\right)=\mathrm{rank}\left(J_{\nu}^{k}\right)$ for all $k\geq 0$ if and only if $\lambda=\nu$. Thus if $X\in\mathfrak{g}_{n}(\mathbb{F}_{q})$ is a matrix such that $\mathrm{rank}\left(X^{k}\right)=\mathrm{rank}\left(J_{\lambda}^{k}\right)$ for all $k\geq 0$, then $X$ is similar to $J_{\lambda}$.

Example 2.4**.**

If a matrix $X\in\mathfrak{g}_{n}(\mathbb{F}_{q})$ has Jordan type $\lambda=(4,2,2,1)$, then $\mathrm{rank}(X)=5$, $\mathrm{rank}(X^{2})=2$, $\mathrm{rank}(X^{3})=1$, and $\mathrm{rank}(X^{4})=0$.

If $X\in\mathfrak{g}_{n}(\mathbb{F}_{q})$ is a matrix of the form

[TABLE]

where $\mu\vdash n-1$, and $\mathbf{v}=[v_{1},\ldots,v_{n-1}]^{T}\in\mathbb{F}_{q}^{n-1}$, then the first order leading principal submatrix of $X^{k}$ is $J_{\mu}^{k}$, and for $1\leq k\leq n$, we define column vectors $\mathbf{v}^{k}=[v_{1}^{k},\ldots,v_{n-1}^{k}]^{T}\in\mathbb{F}_{q}^{n-1}$ by

[TABLE]

For $i\geq 1$, let $\alpha_{i}=\mu_{1}+\cdots+\mu_{i}$ be the sum of the first $i$ parts of $\mu$. The $(i,j)$th entry of $J_{\mu}^{k}$ is nonzero if and only if $j=i+k$, and $i,i+k\leq\alpha_{b}$ for all $b\geq 1$. It follows from this that

[TABLE]

There is a simple way to visualize the vectors $\mathbf{v}^{k}$, which we illustrate with an example.

Example 2.5**.**

Let $\mu=(4,2,1,1)$, so that $\alpha_{1}=4,\alpha_{2}=6,\alpha_{3}=7$, and $\alpha_{4}=8$. Let

[TABLE]

We may visualize the vectors $\mathbf{v}$ and $\mathbf{v}^{2}$ as fillings of the Ferrers diagram for $\mu$: \Yboxdim16pt

[TABLE]

This way, a basis of $\ker X^{k}$ is the set of vectors filling the first $k$ columns of the diagram.

Lemma 2.6**.**

If $X\in\mathfrak{g}_{n,\lambda}(\mathbb{F}_{q})$ and its first order leading principal submatrix $Y\in\mathfrak{g}_{n-1,\mu}(\mathbb{F}_{q})$, then $\lambda\gtrdot\mu$.

Proof.

We first consider the case $Y=J_{\mu}$. If $\mu$ has $s$ parts, let $\alpha_{i}=\mu_{1}+\cdots+\mu_{i}$ for $1\leq i\leq s$. Then

[TABLE]

Let $c\leq n$ be the smallest positive integer for which $\mathrm{rank}(X^{c})-\mathrm{rank}(J_{\mu}^{c})=0$. Then Equation (5) implies that

[TABLE]

Together with Lemma 2.2, we deduce that

[TABLE]

Therefore, $\lambda\gtrdot\mu$ in the case $Y=J_{\mu}$.

In the general case where $Y$ is any matrix of Jordan type $\mu$, then $\mathrm{rank}(Y^{k})=\mathrm{rank}(J_{\mu}^{k})$ for all $k\geq 0$, so the argument is the same. ∎

Let $\lambda$ be the partition whose diagram is obtained by adding a box to the $i$th row and $j$th column of the diagram of the partition $\mu$. Define the coefficient

[TABLE]

Note that in the case $j\geq 2$, we have $\mu_{j-1}^{\prime}-\mu_{j}^{\prime}\geq 1$.

Lemma 2.7**.**

Let $Y$ be an upper-triangular nilpotent matrix of Jordan type $\mu\vdash n-1$. If $\mu\lessdot\lambda$, then there are $c_{\mu,\lambda}(q)$ upper-triangular nilpotent matrices $X$ of Jordan type $\lambda$ whose first order leading principal submatrix is $Y$.

Proof.

By similarity, it suffices to consider the case $Y=J_{\mu}=J_{\mu_{1}}\oplus\cdots\oplus J_{\mu_{m}}$, where $\ell(\mu)=m$. Suppose $X$ is a matrix of the form

[TABLE]

of Jordan type $\lambda$ such that $\lambda$ is obtained by adding a box to $\mu$ in the $i$th row and $j$th column.

First consider the case $j\geq 2$. Following the proof of Lemma 2.6, we know that $j$ is the unique integer where $\mathrm{rank}(X^{j-1})=\mathrm{rank}(J_{\mu}^{j-1})+1,$ and $\mathrm{rank}(X^{j})=\mathrm{rank}(J_{\mu}^{j})$. In order to satisfy the first condition, the entries in the vector $\mathbf{v}^{j-1}$ corresponding to the boxes in the $(j-1)$th column and rows $\geq i$ must not simultaneously be zero (refer to Equation (4) and Example 2.5), while in order to satisfy the second condition, the entries in the vector $\mathbf{v}^{j}$ corresponding to the boxes in the $j$th column of $\mu$ must all be zero. The remaining $n-1-\mu_{j-1}^{\prime}$ entries of the vector $\mathbf{v}$ are free to be any element in $\mathbb{F}_{q}$, so there are

[TABLE]

possible matrices $X$ whose leading principal submatrix is $J_{\mu}$.

The case $j=1$ is simpler. The necessary and sufficient condition that $X$ and $J_{\mu}$ must satisfy is that $\mathrm{rank}(X^{k})=\mathrm{rank}(J_{\mu}^{k})$ for all $k\geq 1$, so the entries in the vector $\mathbf{v}$ corresponding to the boxes in the first column of the diagram for $\mathbf{v}^{1}$ must all be zero, while the remaining $n-1-\mu_{1}^{\prime}$ entries are free to be any element in $\mathbb{F}_{q}$, so there are $q^{n-1-\mu_{1}^{\prime}}$ matrices $X$ whose leading principal submatrix is $J_{\mu}$ in this case. ∎

Theorem 2.8**.**

The number of $n$ by $n$ upper-triangular nilpotent matrices over $\mathbb{F}_{q}$ of Jordan type $\lambda\vdash n$ is

[TABLE]

with $F_{\emptyset}(q)=1$.

Proof.

Proceed by induction on $n$. For $n=1$, the zero matrix is the only upper-triangular nilpotent matrix, and it has Jordan type $(1)$, agreeing with the formula $c_{\emptyset,(1)}(q)=1$.

Suppose $\lambda\vdash n$. By Lemma 2.6, any matrix of Jordan type $\lambda$ has a leading principal submatrix of type $\mu\vdash n-1$ for some $\mu\lessdot\lambda$. Furthermore, by Lemma 2.7, for each matrix $Y\in\mathfrak{g}_{n-1,\mu}(\mathbb{F}_{q})$, there are $c_{\mu\lambda}(q)$ matrices $X\in\mathfrak{g}_{n,\lambda}(\mathbb{F}_{q})$ having $Y$ as its leading principal submatrix. Summing over all $\mu\lessdot\lambda$ gives the desired formula. ∎

Remark 2.9** (Formulation in terms of standard Young tableaux).**

The formula for $F_{\lambda}(q)$ in Theorem 2.8 can be re-phrased as a sum over the set $\mathcal{P}_{\mathcal{Y}}(\lambda)$ of paths in Young’s lattice $\mathcal{Y}$ from the empty partition $\emptyset$ to $\lambda$. If $\mu\lessdot\lambda$ in $\mathcal{Y}$, we assign the weight $c_{\mu,\lambda}(q)$ to the corresponding edge in $\mathcal{Y}$. Figure 1 shows Young’s lattice with weighted edges for partitions with up to four boxes. Let $P=(\emptyset=\pi^{(0)},\pi^{(1)},\ldots,\pi^{(n)}=\lambda)$ denote a path in $\mathcal{Y}$ from $\emptyset$ to $\lambda$, where $\pi^{(i)}$ is a partition of $i$. To simplify notation, let $\epsilon_{i}(q)=c_{\pi^{(i-1)},\pi^{(i)}}(q)$. Theorem 2.8 is equivalently re-phrased as

[TABLE]

where the weight of the path $P$ is $F_{P}(q)=\prod_{i=1}^{n}\epsilon_{i}(q)$.

The set of paths $\mathcal{P}_{\mathcal{Y}}(\lambda)$ is in bijection with the set $\mathrm{SYT}(\lambda)$ of standard Young tableaux of shape $\lambda$, so we can also give an equation for $F_{\lambda}(q)$ as a sum over standard Young tableaux.

A standard Young tableau $T$ of shape $\lambda$ is a filling of the Ferrers diagram of $\lambda\vdash n$ with the integers $1,\ldots,n$ such that the integers increase weakly along each row and strictly along each column. For $1\leq i\leq n$, Let $T^{(i)}$ denote the Young tableau of shape $\lambda^{(i)}$ consisting of the boxes containing $1,\ldots,i$, and define weights

[TABLE]

Then

[TABLE]

where the weight of the standard Young tableau $T$ is $F_{T}(q)=\prod_{i=1}^{n}T^{(i)}(q)$.

2.2. Properties of $F_{\lambda}(q)$

Several properties of $F_{\lambda}(q)$ follow readily from Theorem 2.8. For $\lambda\vdash n$, let

[TABLE]

where if a box $b\in\lambda$ lies in the $i$th row of $\lambda$, then $\mathrm{coleg}(b)=i-1$.

Corollary 2.10**.**

Let $\lambda\vdash n$. As a polynomial in $q$,

[TABLE]

Moreover, the coefficient of the highest degree term in $F_{\lambda}(q)$ is $f^{\lambda}$, the number of standard Young tableaux of shape $\lambda$.

Proof.

Suppose $P=(\emptyset=\pi^{(0)},\pi^{(1)},\ldots,\pi^{(n)}=\lambda)$ is a path in $\mathcal{Y}$ such that $\pi^{(k)}$ is obtained by adding a box to the $i$th row and $j$th column of $\pi^{(k-1)}$. Then $\deg c_{\pi^{(k-1)},\pi^{(k)}}(q)=k-i$, and therefore

[TABLE]

In particular, every polynomial $F_{P}(q)$ arising from a path $P\in\mathcal{P}_{\mathcal{Y}}(\lambda)$ has the same degree, so $\deg F_{\lambda}(q)={n\choose 2}-n_{\lambda}$. Moreover, each $F_{P}(q)$ is monic, so the coefficient of the highest degree term in $F_{\lambda}(q)$ is the number of paths in $\mathcal{Y}$ from $\emptyset$ to $\lambda$, which is $f^{\lambda}$. ∎

Corollary 2.11**.**

Let $\lambda\vdash n$. The multiplicity of the factor $q-1$ in $F_{\lambda}(q)$ is $n-\ell(\lambda)$.

Proof.

The weight $c_{\pi^{(k-1)},\pi^{(k)}}(q)$ corresponding to the $k$th step in the path $P$ contributes a single factor of $q-1$ to $F_{P}(q)$ if and only if the $k$th box added is not in the first column of $\lambda$. Therefore, the multiplicity of $q-1$ in $F_{P}(q)$ is $n-\ell(\lambda)$, and it follows that the multiplicity of $q-1$ in $F_{\lambda}(q)$ is $n-\ell(\lambda)$. ∎

Example 2.12**.**

There are two partitions of $4$ with two parts, namely $(3,1)$ and $(2,2)$.

There are three paths from $\emptyset$ to $(3,1)$ in $\mathcal{Y}$, giving

[TABLE]

and there are two paths from $\emptyset$ to $(2,2)$ in $\mathcal{Y}$, giving

[TABLE]

Summing these gives a shift of the $q$-Stirling polynomial $(q-1)^{2}q^{4}S_{4,2}(q^{-1})=(q-1)^{2}(3q^{3}+3q^{2}+q)$.

2.3. Explicit formulas

In this section, we derive non-recursive formulas for some special cases of $\lambda$. Previously, we have noted the simple cases $F_{(1^{n})}=1$ and $F_{(n)}=q^{n-1\choose 2}(q-1)^{n-1}$.

Proposition 2.13** (Hook shapes).**

Let $n>k\geq 2$, and let $\lambda=(n-k+1,1^{k-1})$ be a hook-shaped partition of $n$ with $\ell(\lambda)=k$ parts. Then

[TABLE]

Proof.

We make use of Equation (7). We enumerate paths from $\emptyset$ to $\lambda$ according to the first time a box is added to the second column, so for $0\leq r\leq k-1$, let $\mathrm{S}_{r}$ be the set of paths in the sublattice $[\emptyset,\lambda]$ which contains the edge $((1,1^{r}),(2,1^{r}))$. Such paths are formed by the concatenation of the unique path between $\emptyset$ and $\nu=(2,1^{r})$, which has weight $q^{r+1}-1$, with any path in the sublattice $[\nu,\lambda]$. The sublattice $[\nu,\lambda]$ is the Cartesian product of a $(n-k)$-chain and a $(k-r-1)$-chain, so it forms a rectangular grid, and therefore $|\mathrm{S}_{r}|={n-r-1\choose k-r-1}$. Notice that in any sublattice of the form

[TABLE]

the product of the edge weights is $(q-1)q^{2a+b-1}$ no matter which path is taken from $(a,1^{b})$ to $(a+1,1^{b+1})$, so it follows that every path from $\nu$ to $\lambda$ has the same weight. By considering the path $(\nu,21^{r+1},\ldots,21^{k-1},31^{k-1},\ldots,\lambda)$, this weight is easily seen to be $(q-1)^{n-k-1}q^{\alpha-r},$ for $\alpha={n-1\choose 2}-{k-1\choose 2}$. Altogether,

[TABLE]

For $0\leq i\leq k-1$, the coefficient of $q^{\alpha-i}$ in $F_{\lambda}(q)/(q-1)^{n-k}$ is

[TABLE]

since $\sum_{u=0}^{M}{N+u\choose u}={N+M+1\choose M}$. Therefore,

[TABLE]

as claimed. ∎

We next consider the case when $\lambda$ is a partition with two parts. For $n\geq k\geq 1$, let

[TABLE]

and let $C_{n,0}=1$ for all $n\geq 0$. These generalized Catalan numbers $C_{n,k}$ enumerate lattice paths from $(0,0)$ to $(n,k)$, using the steps $(1,0)$ and $(0,1)$, which do not rise above the line $y=x$. In the remainder of this section, we shall refer to these as Dyck paths.

The generalized Catalan numbers satisfy the simple recursive formula $C_{n,k}=C_{n,k-1}+C_{n-1,k}.$ Also, these are the usual Catalan numbers $C_{n}=\frac{1}{n+1}{2n\choose n}=C_{n,n}=C_{n,n-1}$ when $k=n$ or $n-1$. These facts will be used in the computations which follow.

Proposition 2.14** (Partitions with two parts).**

If $\lambda=(r,s)\vdash n$ such that $r>s\geq 1$, then

[TABLE]

If $r=s$, then

[TABLE]

Proof.

Proceed by induction on $r+s$. The base cases are $F_{(r)}(q)=(q-1)^{r-1}q^{r-1\choose 2}$ for $r\geq 1$ and $F_{(1,1)}(q)=1$.

We first handle the case $s=1$ separately. For $r\geq 2$,

[TABLE]

Next, consider the case $s=r$. For $r\geq 2$,

[TABLE]

The case $s=r-1$ is obtained as follows. For $r\geq 3$,

[TABLE]

Since $C_{n,n-1}=C_{n,n}$, then $C_{r,r-2}+C_{r-1,r-2}=C_{r,r-1}$. Similarly, we obtain $C_{2r-1-i,i-1}+C_{2r-2-i,i-1}+C_{2r-3-i,i}=C_{2r-1-i,i}$ by applying the recurrence equation for the generalized Catalan numbers. Lastly, $C_{n,0}=1$ for all $n\geq 0$, thus

[TABLE]

which agrees with the formula for the case $s=r-1$.

The last case to consider is the general case $r-s\geq 2$ where $s\geq 2$.

[TABLE]

∎

The next equation is a formula for $F_{(r,r)}(q)$ with a different factorization.

Proposition 2.15** (Two equal parts).**

Let $\lambda=(r,r)\vdash n$, and $r\geq 1$. Then

[TABLE]

Proof.

The set of paths in the sublattice $[\emptyset,\lambda]$ are in bijection with the set of lattice paths from $(0,0)$ to $(r,r)$. In any sublattice of the form

[TABLE]

where $b\leq a-2$, the product of the edge weights is $(q-1)^{2}q^{2a+2b-2}$ no matter which path is taken from $(a,b)$ to $(a+1,b+1)$. As for sublattices of the form

[TABLE]

the product of the edge weights is $(q-1)^{2}q^{4a-4}$ via the lower horizontal edge, versus $(q-1)^{2}q^{4a-5}(q+1)$ via the upper horizontal edge. It follows that if a path $P$ from $\emptyset$ to $\lambda$ contains $i$ partitions of the form $(a,a)$, then it has the weight

[TABLE]

Dyck paths may be enumerated according to the points at which they touch the diagonal line $y=x$, and the set of touch points are indexed by compositions $\alpha=(\alpha_{1},\ldots,\alpha_{i+1})\vDash r$ where $\alpha_{j}\geq 1$. The number of Dyck paths from $(0,0)$ to $(r,r)$ which touch the diagonal exactly $i$ times, not including the initial and the end points, is

[TABLE]

On the other hand, the number $C_{r-1,r-1-i}$ of Dyck paths from $(0,0)$ to $(r-1,r-1-i)$ satisfies the same recurrence equation

[TABLE]

but the sum is over the set of weak compositions so that $\beta_{j}\geq 0$. Under the appropriate shift in indices, it follows that the number of Dyck paths from $(0,0)$ to $(r,r)$ which touch the diagonal exactly $i$ times is $C_{r-1,r-1-i}$. The result follows from this.∎

Corollary 2.16**.**

For $k\geq m\geq 0$,

[TABLE]

Proof.

The two formulas for $F_{(k+1,k+1)}(q)$ yields the identity

[TABLE]

Extracting the coefficient of $q^{2k^{2}-m}$ in the above expressions yields the result.∎

Remark 2.17**.**

The formula for $F_{(r,r)}(q)$ provided in Proposition 2.15 can be viewed as a sum over Dyck paths, where each Dyck path $\pi$ contributes a term of the form $q^{s_{1}(\pi)}(q+1)^{s_{2}(\pi)}$ for some statistics $s_{1}$ and $s_{2}$ on the Dyck paths. This particular factorization for $F_{(r,r)}(q)$ is related to the work of Cai and Readdy on the $q$-Stirling numbers of the second kind, since the polynomials $F_{\lambda}(q)$ can be viewed as a refinement of $S_{n,k}(q)$, as explained in Section 3.

Cai and Readdy obtained a formula [CR15, Theorem 3.2] for $\widetilde{S}_{n,k}(q)$ (they use a different recursive formula to define the $q$-Stirling numbers, and the two are related by $S_{n,k}(q)=q^{k\choose 2}\widetilde{S}_{n,k}(q)$) as a sum over allowable restricted-growth words, where each allowable word $w$ gives rise to a term of the form $q^{a(w)}(q+1)^{b(w)}$ for some statistics $a(w)$ and $b(w)$. They also showed that this enumerative result has an interesting extension to the study of the Stirling poset of the second kind, providing a decomposition of that poset into Boolean sublattices.

For example, if we let $(q-1)^{n-\ell(\lambda)}G_{\lambda}(q)=F_{\lambda}(q)$, then $G_{(3,1)}(q)+G_{(2,2)}(q)=q^{3}\widetilde{S}_{4,2}(q^{-1}).$ The formula of Cai and Readdy yields $q^{3}S_{4,2}(q^{-1})=q(q+1)^{2}+q^{2}(q+1)+q^{3}$, while our factorization yields $G_{(3,1)}(q)+G_{(2,2)}(q)=\left(q^{3}+q^{3}+q^{2}(q+1)\right)+\left(q^{2}+q(q+1)\right)$. So, the result of Proposition 2.15 gives a different expression for $S_{n,n-2}(q)$ as a sum with terms of the form $q^{s_{1}(\pi)}(q+1)^{s_{2}(\pi)}$, and it may be interesting to further investigate such factorizations of $F_{\lambda}(q)$.

Example 2.18**.**

The first few $F_{(k,k)}$ are

[TABLE]

We end this section with one more closed formula for $F_{\lambda}(q)$ where $\lambda$ is a rectangular shape with two columns. Let $\mathcal{D}(n,k)$ denote the set of Dyck paths from $(0,0)$ to $(n,k)$. The coarea of a Dyck path $\pi$ is the number of whole unit squares lying between the path and the $x$-axis. For $i=1,\ldots,n$, let $\rho_{i}(\pi)$ be one plus the number of unit squares lying between the path and the line $y=x+1$ in the $i$th row. For example, the following Dyck path $\pi$ has $\mathrm{coarea}(\pi)=12$, and $(\rho_{1}(\pi),\rho_{2}(\pi),\rho_{3}(\pi),\rho_{4}(\pi))=(2,2,1,1)$.

[TABLE]

For $n\geq 1$, let $[n]_{q}=1+q+\cdots+q^{n-1}$.

Proposition 2.19**.**

(Partitions with two columns) Let $\lambda=(2^{r},1^{s})\vdash n$ such that $r,s\geq 0$. Then

[TABLE]

Proof.

By Corollary 2.11, we know the multiplicity of the factor $q-1$ in $F_{(2^{r},1^{s})}(q)$ is $n-\ell(\lambda)=\lambda_{2}^{\prime}=r$, so we focus on computing $F_{(2^{r},1^{s})}(q)/(q-1)^{r}$. The paths in Young’s lattice from $\emptyset$ to $(2^{r},1^{s})$ are in bijection with the Dyck paths $\mathcal{D}(r+s,r)$, so we identify these paths; adding a box in the first column of a partition corresponds to a $(1,0)$ step in the Dyck path, and adding a box in the second column of a partition corresponds to a $(0,1)$ step in the Dyck path. As seen in Figure 4, a vertical step $(i,j)$ to $(i,j+1)$ has weight $q^{j}[i]_{q}$, while a horizontal step $(i,j)$ to $(i+1,j)$ has weight $q^{j}$. Thus the product of the edge weights of the $r$ vertical steps of a given Dyck path $\pi$ is $q^{r\choose 2}\prod_{i=1}^{r}[\rho_{i}(\pi)]_{q}$, while the product of the edge weights of the $r+s$ horizontal steps of a given Dyck path is $q^{\mathrm{coarea}(\pi)}$. The result follows. ∎

Example 2.20**.**

The first few $F_{(2^{n})}$ are

[TABLE]

Remark 2.21**.**

Kirillov and Melnkov [KM95] considered the number $A_{n}(q)$ of $n$ by $n$ upper-triangular matrices over $\mathbb{F}_{q}$ satisfying $X^{2}=0$. In their first characterization of these polynomials, they considered the number $A_{n}^{r}(q)$ of matrices of a given rank $r$, so that $A_{n}(q)=\sum_{r\geq 0}A_{n}^{r}(q)$, and observed that $A_{n}^{r}(q)$ satisfies the recurrence equation

[TABLE]

We may think of $A_{n}(q)$ as the sum of $F_{\lambda}(q)$ over $\lambda\vdash n$ with at most two columns, so Theorem 2.8 is a generalization of this recurrence equation.

It was also conjectured in [KM95] that the same sequence of polynomials arise in a number of different ways. Ekhad and Zeilberger [EZ96] proved that one of the conjectured alternate definitions of $A_{n}(q)$, namely

[TABLE]

is a sum over all $s\in[-n-1,n+1]$ which satisfy $s\equiv n+1\ \mathrm{mod}\ 2$ and $s\equiv(-1)^{n}\ \mathrm{mod}\ 3$, and $c_{n+1,s}$ are entries in the signed Catalan triangle, is indeed the same as $A_{n}(q)$. It would be interesting to see what other combinatorics may arise from considering the sum of $F_{\lambda}(q)$ over $\lambda\vdash n$ with at most $k$ columns for a fixed $k$.

3. Jordan canonical forms and $q$-rook placements

In light of Corollary 2.11, we define polynomials $G_{\lambda}(q)\in\mathbb{Z}[q]$ by

[TABLE]

In fact, we can deduce from Corollary 2.11 that $G_{\lambda}(q)\in\mathbb{N}[q]$. In this section, we explore the connection between the nonnegative coefficients of $G_{\lambda}(q)$ and rook placements.

3.1. Background on rook polynomials

A board $B$ is a subset of an $n$ by $n$ grid of squares. In this paper, we follow Haglund [Hag98] and Solomon [Sol90], and index the squares using the convention for the entries of a matrix. A Ferrers board is a board $B$ where if a square $s\in B$, then every square lying north and/or east of $s$ is also in $B$. Our Ferrers boards have squares justified upwards and to the right. Let $B_{n}$ denote the staircase-shaped board with $n$ columns of sizes $0,1,\ldots,n-1$. Let $\mathrm{area}(B)$ be the number of squares in $B$, so that in particular, $\mathrm{area}(B_{n})={n\choose 2}$.

A placement of $k$ rooks on a board $B$ is non-attacking if there is at most one rook in each row and each column of $B$. Let $\mathcal{C}(B,k)$ denote the set of non-attacking placements of $k$ rooks on $B$. All rook placements considered in this article are non-attacking, so from this point forward, we drop the qualifier. For a placement $C\in\mathcal{C}(B,k)$, let $\mathrm{ne}(C)$ be the number of squares in $B$ lying directly north or directly east of a rook. The inversion of the placement is the number

[TABLE]

As noted in [GR86], the statistic $\mathrm{inv}(C)$ is a generalization of the number of inversions of a permutation, since permutations can be identified with rook placements on a square-shaped board.

For $i=1,\ldots,n$, the weight of the $i$th column $C_{i}$ of $C$ is

[TABLE]

and the weight of $C$ is defined by $F_{C}(q)=\prod_{i=1}^{n}C_{i}(q)$. Alternatively, if $C\in\mathcal{C}(B,k)$, then $F_{C}(q)=(q-1)^{k}q^{\mathrm{ne}(C)}$.

Example 3.1**.**

We use $\times$ to mark a rook and use $\,\bullet\,$ to mark squares lying directly north or directly east of a rook (these squares shall be referred to as the north-east squares of the placement). The following illustration is a placement of four rooks on the staircase-shaped board $B_{7}$.

[TABLE]

This rook placement has $\mathrm{ne}(C)=11$, $\mathrm{inv}(C)=6$, and weight $F_{C}(q)=(q-1)^{4}q^{11}$.

For $k\geq 0$, the $q$-rook polynomial of a Ferrers board $B$ is defined by Garsia and Remmel [GR86, I.4] as

[TABLE]

The following result explains the role of rook polynomials in the enumeration of matrices of given rank. The support of a matrix $X$ is $\{(i,j)\mid x_{ij}\neq 0\}$. Given a Ferrers board $B$ with $n$ columns, we may identify the squares in $B$ with the entries in an $n$ by $n$ matrix.

Theorem 3.2** (Haglund).**

If $B$ is a Ferrers board, then the number $P_{B,k}(q)$ of $n$ by $n$ matrices of rank $k$ with support contained in $B$ is

[TABLE]

Looking ahead, it will be convenient to consider Theorem 3.2 in the following equivalent form:

[TABLE]

Example 3.3**.**

We list the seven rook placements on $B_{4}$ with two rooks, along with their weights.

[TABLE]

[TABLE]

Thus $P_{B_{4},2}(q)=(q-1)^{2}(3q^{3}+3q^{2}+q)$.

3.2. Rook placements and Jordan forms

The purpose of this section is to generalize Haglund’s formula (16) to a formula for $F_{\lambda}(q)$ (Corollary 3.9) as a sum over a set of rook placements. We achieve this by defining a multigraph $\mathcal{Z}$ that is related to $\mathcal{Y}$, and show that paths in $\mathcal{Z}$ are equivalent to rook placements.

The multigraph $\mathcal{Z}$ is constructed from $\mathcal{Y}$ by replacing each edge of $\mathcal{Y}$ by one or more edges as follows. If there is an edge from $\mu$ to $\lambda$ in $\mathcal{Y}$ of weight $q^{|\mu|-\mu_{j-1}^{\prime}}\left(q^{\mu_{j-1}^{\prime}-\mu_{j}^{\prime}}-1\right)$, then this edge is replaced by $\mu_{j-1}^{\prime}-\mu_{j}^{\prime}$ edges from $\mu$ to $\lambda$ with weights

[TABLE]

in $\mathcal{Z}$. All other edges remain as before. See Figure 5.

Let $\mathcal{P}_{\mathcal{Z}}(\lambda)$ denote the set of paths in the graph $\mathcal{Z}$ from the empty partition $\emptyset$ to $\lambda$. For a path $P=(\emptyset=\pi^{(0)},\pi^{(1)},\ldots,\pi^{(n)}=\lambda)$ in $\mathcal{P}_{\mathcal{Z}}(\lambda)$, let $\epsilon_{i}(q)$ denote the weight of the $i$th edge, for $i=1,\ldots,n$. Naturally, we define the weight of the path by $F_{P}(q)=\prod_{i=1}^{n}\epsilon_{i}(q)$, so that

[TABLE]

Lemma 3.4**.**

Let $\mu\vdash n-1$ be a partition with $\ell(\mu)=\ell$ parts. Then there are $\ell+1$ edges leaving $\mu$ in the graph $\mathcal{Z}$, with weights

[TABLE]

Proof.

If a partition $\lambda\vdash n$ is obtained by adding a box to the first column of $\mu$, then there is a unique edge from $\mu$ to $\lambda$ in $\mathcal{Z}$ with weight $q^{|\mu|-\ell}$. Otherwise, if we consider the set of all partitions which can be obtained from $\mu$ by adding a box anywhere except in the first column, then there are a total of

[TABLE]

edges from $\mu$ to some partition of $n$. Moreover, by Equation (19), these $\ell$ weights are $(q-1)q^{|\mu|-i}$ for $i=1,\ldots,\ell$. ∎

A sequence of nonnegative integers is $\mathcal{P}_{\mathcal{Z}}$-admissible if it is the degree sequence of a path $P=(\emptyset,\pi^{(1)},\ldots,\pi^{(n)})$ in $\mathcal{Z}$. That is, $(d_{1},\ldots,d_{n})=(\deg\epsilon_{1}(q),\ldots,\deg\epsilon_{n}(q))$.

Corollary 3.5**.**

*A $\mathcal{P}_{\mathcal{Z}}$-admissible sequence determines a unique path in $\mathcal{Z}$. *

Proof.

Induct on $n$. When $n=1$, the only path is the from $\emptyset$ to $(1)$, and it has degree sequence $(0)$.

Given a $\mathcal{P}_{\mathcal{Z}}$-admissible sequence $(d_{1},\ldots,d_{n})$, the subsequence $(d_{1},\ldots,d_{n-1})$ determines a unique path $P^{\prime}=(\emptyset,\pi^{(1)},\ldots,\pi^{(n-1)})$. Suppose $\mu=\pi^{(n-1)}$ has $\ell$ parts. Then $|\mu|-\ell+1\leq d_{n}\leq|\mu|$, and by Lemma 3.4, there is a unique edge leaving $\mu$ with degree $d_{n}$. ∎

3.3. The construction of $\Phi$

Let $\mathcal{P}_{\mathcal{Z}}(n,n-k)$ denote the set of paths in $\mathcal{Z}$ from $\emptyset$ to a partition of $n$ with $n-k$ parts. In this section, we define a weight-preserving bijection $\Phi:\mathcal{C}(B_{n},k)\rightarrow\mathcal{P}_{\mathcal{Z}}(n,n-k)$.

Proposition 3.6**.**

Let $n\geq 1$ and $k=0,\ldots,n-1$. Let $C\in\mathcal{C}(B_{n},k)$ be a rook placement with columns $C_{1},\ldots,C_{n}$. There exists a unique path $P\in\mathcal{P}_{\mathcal{Z}}(n,n-k)$ with edge weights $(\epsilon_{1}(q),\ldots,\epsilon_{n}(q))=(C_{1}(q),\ldots,C_{n}(q))$.

Proof.

Proceed by induction on $n+k$. When $n=1$ and $k=0$, there is a unique rook placement on the empty board $B_{1}$ with no rooks having weight one, corresponding to the unique path $P=(\emptyset,(1))$ in $\mathcal{Z}$ with the same weight.

Assume the result holds for all rook placements in $\mathcal{C}(B_{n-1},k)$ and $\mathcal{C}(B_{n-1},k-1)$. Given a rook placement $C\in\mathcal{C}(B_{n},k)$, let $C^{\prime}$ be the sub-placement consisting of the first $n-1$ columns of $C$. By induction, the sequence $(C_{1}(q),\ldots,C_{n-1}(q))$ determines a unique path $(\emptyset,\pi^{(1)},\ldots,\pi^{(n-1)})$ in $\mathcal{Z}$ such that $\epsilon_{i}(q)=C_{i}(q)$ for $i=1,\ldots,n-1$.

There are now two cases two consider. The first case is if $C^{\prime}\in\mathcal{C}(B_{n-1},k)$, so that $\ell(\pi^{(n-1)})=n-k-1$. There are $k$ rooks in $C^{\prime}$, so the $n$th column of $C$ does not contain any rooks, and $C_{n}(q)=q^{k}$. By Lemma 3.4, there exists a unique edge in the graph $\mathcal{Z}$ originating at $\pi^{(n-1)}$ with weight $q^{k}$. Thus $C$ corresponds to the path $P=(\emptyset,\pi^{(1)},\ldots,\pi^{(n-1)},\pi^{(n)})$ where $\pi^{(n)}$ is obtained from $\pi^{(n-1)}$ by adding a box to the first column, and $\epsilon_{n}(q)=q^{k}$. Moreover, $\ell(\pi^{(n)})=n-k$.

The second case is if $C^{\prime}\in\mathcal{C}(B_{n-1},k-1)$, so that $\ell(\pi^{(n-1)})=n-k$. There must be $k-1$ ‘northeast’ squares in the $n$th column of $C$, and there are $n-k$ remaining squares in that column where a rook may be placed. Label these available squares $a_{0},a_{1},\ldots,a_{n-k-1}$ from the top to the bottom. Observe that $C_{n}(q)=(q-1)q^{k-1+i}$ if a rook is placed in the square $a_{i}$, for $0\leq i\leq n-k-1$. Again by Lemma 3.4, there exists $n-k$ edges in the graph $\mathcal{Z}$ originating at $\pi^{(n-1)}$ with the weights $(q-1)q^{h}$ for $k-1\leq h\leq n-2$. Thus if the $k$th rook of $C$ is placed in the square $a_{i}$, then $C$ corresponds to the path $P=(\emptyset,\pi^{(1)},\ldots,\pi^{(n-1)},\pi^{(n)})$ with $\epsilon_{n}(q)=(q-1)q^{k-1+i}$, and $\ell(\pi^{(n)})=n-k$. ∎

Given $C\in\mathcal{C}(B_{n},k)$, let $\Phi(C)$ be the path in $\mathcal{P}_{\mathcal{Z}}(n,n-k)$ with edge weights $(\epsilon_{1}(q),\ldots,\epsilon_{n}(q))=(C_{1}(q),\ldots,C_{n}(q))$.

Theorem 3.7**.**

The map $\Phi:\mathcal{C}(B_{n},k)\rightarrow\mathcal{P}_{\mathcal{Z}}(n,n-k)$ is a weight-preserving bijection.

Proof.

Proposition 3.6 shows that the map $\Phi$ is an injective weight-preserving map, since each column of the rook placement determines each edge of the path $\Phi(C)$:

[TABLE]

In fact, the proof of the Proposition also shows that $\Phi$ is surjective because the number of possible ways to add a column to an existing rook placement is equal to the number of possible ways to extend a path in $\mathcal{Z}$ by one edge. Therefore, $\Phi$ is a weight-preserving bijection. ∎

A sequence of nonnegative integers is $\mathcal{C}$-admissible if it is the degree sequence of a rook placement. That is, $(d_{1},\ldots,d_{n})=(\deg C_{1}(q)),\ldots,\deg C_{n}(q))$ for a $C\in\mathcal{C}(B_{n},k)$. The next Corollary follows easily from Theorem 3.7.

Corollary 3.8**.**

A $\mathcal{C}$-admissible sequence determines a unique rook placement. ∎

It follows from Theorem 3.7 that we may associate a partition type to each rook placement on $B_{n}$. The partition type of a rook placement $C$ is the partition at the endpoint of the path $\Phi(C)$ in $\mathcal{Z}$. Let $\mathcal{C}(\lambda)=\Phi^{-1}\left(P_{\mathcal{Z}}(\lambda)\right)$ denote the set of rook placements of partition type $\lambda$.

Corollary 3.9**.**

Let $\lambda\vdash n$ be a partition with $\ell(\lambda)=n-k$ parts. Then

[TABLE]

Proof.

The result follows from Equation 20 and the bijection $\Phi$. ∎

Remark 3.10**.**

The polynomial $G_{\lambda}(q)\in\mathbb{N}[q]$ defined in Equation (12) is simply a sum over the rook placements of type $\lambda$ involving the north-east statistic.

4. A connection with set partitions

The results of the previous section naturally leads to a decomposition of $F_{T}(q)$, indexed by some tableau $T$, into a sum of polynomials indexed by set partitions, which we explain below.

A set partition is a set $S=\{s_{1},\ldots,s_{k}\}$ of nonempty disjoint subsets of $[n]$ such that $\bigcup_{i=1}^{k}s_{i}=[n]$. The $s_{i}$’s are the blocks of $\sigma$. Let $\ell(S)$ denote the number of blocks of $S$, and let $\mathcal{S}(n,n-k)$ denote the set of set partitions of $[n]$ with $n-k$ blocks. We adopt the convention of listing the blocks in order so that

[TABLE]

This allows us to represent a set partition with a diagram similar to that of a standard Young tableau; the $i$th row of the diagram consists of the elements in the block $s_{i}$ listed in increasing order, but there are no restrictions on the entries in each column of the diagram. A set partition $S=(s_{1},\ldots,s_{m})$ has partition type $\lambda$ if $\lambda=(|s_{1}|,\ldots,|s_{m}|)$.

For $i=1,\ldots,n$, let $S^{(i)}$ denote the sub-diagram of $S$ consisting of the boxes containing $1,\ldots,i$, with rows ordered according to the convention set forth in Equation (21). If the box containing $i$ is not in the first column of the diagram, let $u$ be the least element in the same row as $i$ in $S^{(i)}$, and suppose $u$ is in the $r$th row of $S^{(i-1)}$ for some $1\leq r\leq\ell(S^{(i-1)})$. The weight arising from the $i$th box is

[TABLE]

We define the weight of $S$ as $F_{S}(q)=\prod_{i=1}^{n}S^{(i)}(q)$.

A sequence of nonnegative integers is $\mathcal{S}$-admissible if it is the degree sequence of a set partition. That is, $(d_{1},\ldots,d_{n})=(\deg S^{(1)}(q)),\ldots,\deg S^{(n)}(q))$ for a $S\in\mathcal{S}(n)$.

Lemma 4.1**.**

An $\mathcal{S}$-admissible sequence determines a unique set partition.

Proof.

Induct on $n$. When $n=1$, the only set partition is $\{\{1\}\}$, and its degree sequence is $(0)$.

Given an $\mathcal{S}$-admissible sequence $(d_{1},\ldots,d_{n})$, the subsequence $(d_{1},\ldots,d_{n-1})$ determines a unique set partition $S^{(n-1)}=(S^{(n-1)}_{1},\ldots,S^{(n-1)}_{m})$. By Equation (22), $n-1-m\leq d_{n}\leq n-1$, and each of the $m+1$ choices for $d_{n}$ determines the block of $S^{(n-1)}$ into which $n$ should be inserted. ∎

We have already constructed a weight-preserving bijection $\Phi$ between rook placements and paths in $\mathcal{Z}$. We now construct a weight-preserving bijection $\Psi$ between rook placements and set partitions, effectively showing that paths in $\mathcal{Z}$ are equivalent to set partitions, so that $F_{Z}(q)=F_{C}(q)=F_{S}(q)$ if $Z\longleftrightarrow C\longleftrightarrow S$ for $Z\in\mathcal{P}_{\mathcal{Z}}(n,n-k)$, $C\in\mathcal{C}(n,k)$, and $S\in\mathcal{S}(n,n-k)$.

Remark 4.2**.**

There is a classically known bijection (see [Sta99]) between the set of rook placements on the staircase board $B_{n}$ with $k$ rooks and the set of set partitions of $[n]=\{1,\ldots,n\}$ with $n-k$ blocks: the placement $C$ corresponds to the set partition where the integers $i$ and $j$ are in the same block if and only if there is a rook in the square $(i,j)\in C$. This bijection is different from the one described in Theorem 4.3. For example, the classical bijection associates the rook placement

[TABLE]

to the set partition $(\{1,2\},\{3,4\})$ and so has partition type $(2,2)$, but as we shall see below, this placement is associated to the set partition $(\{1,2,4\},\{3\})$ under the bijection in Theorem 4.3 and has partition type $(3,1)$.

4.1. The construction of $\Psi$

Let $C\in\mathcal{C}(B_{n},k)$ be a rook placement. The main idea is that the degree of $C_{i}(q)$ arising from the $i$th column of $C$ determines the block of the set partition in which we place $i$. In the construction of the set partition $\Psi(C)$, we will create a sequence of intermediate set partitions $S^{(i)}$ of $[i]$ for $i=1,\ldots,n$.

The initial case is always $\deg(C_{1}(q))=\deg(1)=0$, so $S^{(1)}=\{\{1\}\}$. Assume that $S^{(i-1)}=\{S^{(i-1)}_{1},\ldots,S^{(i-1)}_{m}\}$ is the set partition which corresponds to the first $i-1$ columns of $C$, so that $m=\ell(S^{(i-1)})$. Observe that there are $m+1$ possible blocks in which to insert $i$ to obtain $S^{(i)}$. By Corollary 3.5, we know that

[TABLE]

so we construct $S^{(i)}$ by placing $i$ in the $j$th block of $S^{(i-1)}$, where $j=i-\deg(C_{i}(q))$, and then rearranging the blocks to fit the convention in Equation (21) if necessary.

Theorem 4.3**.**

The map $\Psi:\mathcal{C}(n,k)\rightarrow\mathcal{S}(n,n-k)$ is a weight-preserving bijection.

Proof.

Let $S=\Psi(C)$. The map $\Psi$ is weight-preserving, as $C_{i}(q)=S^{(i)}(q)$ by construction, for each $i=1,\ldots,n$. Now, since the degrees $\deg C_{i}(q)=\deg S^{(i)}(q)$, and by Corollary 3.8 and Lemma 4.1 the sequences of degrees completely determine $C$ and $S$ respectively, then $\Psi$ is injective. Finally, we note that $|\mathcal{C}(n,k)|=|\mathcal{S}(n,n-k)|$, so $\Psi$ is a bijection. ∎

Corollary 4.4**.**

Let $\mathcal{S}(\lambda)$ denote the set of all set partitions of partition type $\lambda$. Then

[TABLE]

∎

Example 4.5**.**

Let $C$ be

[TABLE]

The associated sequence of set partition diagrams associated to $C$ is \Yboxdim12pt

[TABLE]

so the set partition associated to the rook placement $C$ is

[TABLE]

Remark 4.6**.**

An intriguing question is to ask for a geometric interpretation of the polynomials $F_{C}(q)$, indexed by rook placements (or set partitions or paths in $\mathcal{Z}$).

The problem of determining the number of adjoint $G_{n}(\mathbb{F}_{q})$ orbits on $\mathfrak{g}_{n}(\mathbb{F}_{q})$ remains open. In the case $q=2$, this number has been computed for $n\leq 16$ by Pak and Soffer [PS15, Appendix B]. Let $\mathcal{O}_{n}(k)$ denote the orbits of rank $k$ matrices. When $k=1$, it turns out that the polynomials $F_{C}(q)$ indexed by rook placements with exactly one rook gives the sizes of the ${n\choose 2}$ orbits in $\mathcal{O}_{n}(1)$. For $2\leq i<j\leq n$, each orbit contains a unique matrix $E_{ij}$ whose $ij$th entry is $1$, and is zero everywhere else. The orbit containing $E_{ij}$ is associated to the rook placement $C(i,j)$ with a single rook in the $ij$th square, and the size of the associated orbit is $F_{C(i,j)}(q)=(q-1)q^{n-1-(j-i)}$.

In particular, the formula in Proposition 2.13 applied to the partition $\lambda=(2,1^{n-2})$ gives the generating function

[TABLE]

for rank one orbits of $G_{n}(\mathbb{F}(q))$ on $\mathfrak{g}_{n}(\mathbb{F}(q))$.

Remark 4.7**.**

To close, we mention a related problem which may provide a geometrical interpretation of $F_{C}(q)$ for every rook placement $C$. Let $N$ be an $n\times n$ nilpotent matrix with entries in an algebraically closed field $k$ containing $\mathbb{F}_{q}$, and suppose $N$ has Jordan type $\lambda\vdash n$. A complete flag $f=(f_{1},\ldots,f_{n})$ is a sequence of subspaces in $k^{n}$ such that $f_{1}\subset\cdots\subset f_{n}$ and $\dim f_{i}=i$ for all $i$. A flag is $N$-stable if $N(f_{i})\subseteq f_{i}$ for all $i$. Spaltenstein [Spa76] showed that the variety $X_{\lambda}$ of $N$-stable flags is a disjoint union of $f^{\lambda}$ smooth irreducible subvarieties $X_{T}$ indexed by the standard Young tableaux of shape $\lambda$. Moreover, the closures $\overline{X}_{T}$ are the irreducible components of $X_{\lambda}$, each of which has dimension $n_{\lambda}$. The number of $\mathbb{F}_{q}$-rational points in $X_{\lambda}$ is given by Green’s polynomials $Q_{(1^{n})}^{\lambda}(q)$. Evidently,

[TABLE]

with $m=\min_{C\in\mathcal{C}(\lambda)}\mathrm{ne}(C)$. Based on some computations for small values of $n$, we expect that $F_{C}(q)$ plays a role in counting points in certain intersections of the irreducible components $\overline{X}_{T}$.