2.1 Thin and fat slices

By definition, the polytope $\mathscr {R}_{k,\textbf {c}}$ introduced in equation (1.1) is the intersection between an n-dimensional rectangular prism and a hyperplane. It is an $(n-1)$ -dimensional polytope, unless $k = 0$ or $k = c_1+\cdots +c_n$ : in these two cases, it collapses to a point; also, it will be certainly empty if $k> c_1+\cdots +c_n$ . The combinatorics of slices of prisms, i.e., their face lattices, are much richer than for hypersimplices. For an elementary treatment of the face enumeration of certain slices of cubes, we refer to a beautiful article by Chakerian and Logothetti [Reference Chakerian and Logothetti8].

Example 2.1 Consider the vector $\textbf {c}=(6,3,4)$ and take $k = 7$ . The polytope $\mathscr {R}_{k,\textbf {c}}$ is obtained as the intersection of a rectangular prism in $\mathbb {R}^3$ and the hyperplane $x+y+z=7$ , as depicted on the left in Figure 1. We can see that it corresponds to a pentagon in $\mathbb {R}^3$ with vertices $(6,0,1)$ , $(6,1,0)$ , $(4,3,0)$ , $(0,3,4)$ , and $(3,0,4)$ , shown on the right of Figure 1. Furthermore, notice that the combinatorics of this polytope is genuinely different to that of all hypersimplices having dimension 2 (no hypersimplex is a pentagon).

Figure 1: The polytope $\mathscr {R}_{7,(6,3,4)}$ .

We will sometimes restrict to the case in which $0 < k < c_1+\cdots +c_n$ ; under this assumption, the Ehrhart polynomial has degree $n-1$ . Slices of prisms are always integral polytopes because they are discrete polymatroids (see Proposition 2.3 below).

As we mentioned in the Introduction, the Ehrhart theory of the “thin” slices $\mathscr {R}_{k,\textbf {c}}$ defined in (1.1) and the “fat” slices $\mathscr {R}^{\prime }_{a,b,\textbf {c}}$ defined in (1.4) is essentially the same. Let us state this more precisely. Two lattice polytopes $\mathscr {P}_1\subseteq {\mathbb {R}}^n$ and $\mathscr {P}_2\subseteq {\mathbb {R}}^m$ are said to be integrally equivalent when there is an affine map $\varphi :{\mathbb {R}}^n\to {\mathbb {R}}^m$ such that its restriction to $\mathscr {P}_1$ induces a bijection $\varphi :\mathscr {P}_1\to \mathscr {P}_2$ which preserves the lattice, i.e., the image under $\varphi $ of $\mathbb {Z}^n \cap \operatorname {aff}\mathscr {P}_1$ is $\mathbb {Z}^m \cap \operatorname {aff}\mathscr {P}_2$ , where $\operatorname {aff}\mathscr {P}_1$ denotes the affine space spanned by $\mathscr {P}_1$ and analogously for $\mathscr {P}_2$ . Something immediate from the definitions is that integrally equivalent polytopes have the same Ehrhart polynomial.

Proposition 2.2 Let $\textbf {c}=(c_1,\ldots ,c_n)\in \mathbb {Z}_{>0}^n$ and $0\leq a < b$ . There is an integral equivalence between $\mathscr {R}^{\prime }_{a,b,\textbf {c}}$ (cf. equation (1.4)) and $\mathscr {R}_{b,\textbf {c}'}$ , where $\textbf {c}'\in \mathbb {Z}_{>0}^{n+1}$ is given by $\textbf {c}' = (\textbf {c}, b-a)$ . In particular,

$$\begin{align*}\operatorname{\mathrm{ehr}}(\mathscr{R}^{\prime}_{a,b,\textbf{c}}, t) = \operatorname{\mathrm{ehr}}(\mathscr{R}_{b,\textbf{c}'},t).\end{align*}$$

Proof Let us prove that, using the notation of the statement, the two polytopes $\mathscr {R}^{\prime }_{a,b,\textbf {c}}$ and $\mathscr {R}_{b,\textbf {c}'}$ are integrally equivalent. Notice that the polytope $\mathscr {R}_{b,\textbf {c}'}$ lies in $\mathbb {R}^{n+1}$ , whereas $\mathscr {R}^{\prime }_{a,b,\textbf {c}}\subseteq \mathbb {R}^n$ . Let us consider the map $\pi : \mathbb {R}^{n+1}\to \mathbb {R}^n$ that forgets the last coordinate. The image of the polytope $\mathscr {R}_{b,\textbf {c}'}\subseteq \mathbb {R}^{n+1}$ is given by

$$\begin{align*}\pi(\mathscr{R}_{b,\textbf{c}'}) = \left\{ x\in \mathscr{R}_{\textbf{c}} : a\leq \sum_{i=1}^{n} x_i \leq b \right\} = \mathscr{R}^{\prime}_{a,b,\textbf{c}}.\end{align*}$$

Moreover, the restriction of $\pi $ to $\mathscr {R}_{b,\textbf {c}'}$ is in fact a bijection which preserves the lattice. To see this, notice that every lattice point in $\mathbb {R}^{n+1}$ having sum of coordinates equal to b can be recovered by knowing the first n coordinates, and its preimage has to be a lattice point.

It is because of this property that we will focus only on the Ehrhart theory of the “thin” slices of prisms.

2.2 Polymatroids with the strong exchange property

In [Reference Herzog and Hibi23], Herzog and Hibi introduced the notion of polymatroid of Veronese type. For the necessary background on polymatroids, we refer to their article, in particular [Reference Herzog and Hibi23, Example 2.6].

In particular, since base polytopes of polymatroids are generalized permutohedra, we obtain that the edge directions of any slice of a prism is of the form $e_i-e_j$ . On the other hand, in light of the form of the inequalities that describe a slice of a prism, it follows that they are alcoved polytopes (as in [Reference Lam and Postnikov30]), hence:

Undertaking the discussion on polymatroids of Veronese type, as we mentioned in the Introduction, they admit a very neat characterization. They are (up to an affinity) precisely the discrete polymatroids that satisfy the strong exchange property (we refer to [Reference Herzog, Hibi and Vladoiu24] for the details).

2.3 The Ehrhart polynomial explicitly

Toward a proof of Theorem 1.2, the first step is to give a somewhat explicit expression for the Ehrhart polynomial of $\mathscr {R}_{k,\textbf {c}}$ . To simplify the statement, we introduce some notation. From now on, whenever $\textbf {c}=(c_1,\ldots ,c_n)\in \mathbb {Z}_{> 0}^n$ is fixed, for each $0\leq j \leq n$ , we will denote

(2.1) $$ \begin{align} \rho_{\textbf{c},j}(s) := \#\left\{ I\in \binom{[n]}{j} : \sum_{i \in I} c_i = s\right\}.\end{align} $$

In other words, $\rho _{\textbf {c},j}(s)$ is the number of ways of choosing exactly j of the $c_i$ ’s in such a way that their sum is exactly s. If $j=0$ , we define $\rho _{\textbf {c},j}(s)$ to be $1$ if $s=0$ and $0$ otherwise. The following result, also appearing in [Reference Katzman27, Theorem 2.1], is only a preliminary step toward the main proofs of this article.

Theorem 2.5 Let $\textbf {c}=(c_1,\dots ,c_n)\in \mathbb {Z}_{>0}^n$ . For each positive integer k, the Ehrhart polynomial of the polytope $\mathscr {R}_{k,\textbf {c}}$ is given by

$$\begin{align*}\operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t)= \sum_{j=0}^{k-1}(-1)^j\sum_{v=0}^{k-1} \binom{t(k-v)+n-1-j}{n-1} \rho_{\textbf{c},j}(v). \end{align*}$$

Proof By using the definition, we can write

$$\begin{align*}\mathscr{R}_{k,\textbf{c}} = \left\{ x\in \mathbb{R}^n_{\geq 0} : \sum_{i=1}^n x_i = k \text{ and } x_i \leq c_i \text{ for all }1\leq i \leq n \right\}.\end{align*}$$

Therefore, it follows that

$$ \begin{align*} \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}}, t) &= \#(t \mathscr{R}_{k,\textbf{c}} \cap \mathbb{Z}^n)\\ &= \# \left\{ x\in \mathbb{Z}_{\geq 0}^n: \sum_{i=1}^n x_i = kt \text{ and } x_i \leq c_it \text{ for all }1\leq i \leq n \right\}\\ &= [x^{kt}] \prod_{i=1}^n (1+x+x^2+\cdots+x^{c_it}). \end{align*} $$

Using the identity $1+x+\cdots +x^{c_it} = \dfrac {1-x^{c_it+1}}{1-x}$ , we can further reduce

(2.2) $$ \begin{align} \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t) = [x^{kt}] \left( \frac{1}{(1-x)^n} \prod_{i=1}^n (1-x^{c_it+1})\right). \end{align} $$

Now, let us consider a generic coefficient of the factor $\prod _{i=1}^n (1-x^{c_it+1})$ ; we can write

(2.3) $$ \begin{align} [x^u] \prod_{i=1}^n (1-x^{c_it+1}) = \sum_{j=0}^n(-1)^j\#\left\{i_1<\cdots<i_j : \sum_{\ell=1}^j (c_{i_\ell}t+1) = u\right\}. \end{align} $$

Thus, we can use what we obtained in equation (2.3) to simplify our formula in (2.2). Recall the classical generating function identity: $\frac {1}{(1-x)^n} = \sum _{i=0}^{\infty } \binom {n-1+i}{n-1} x^i$ .

$$ \begin{align*} \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t) &= [x^{kt}] \left( \frac{1}{(1-x)^n} \prod_{i=1}^n (1-x^{c_it+1})\right)\\ &= [x^{kt}] \left( \sum_{i=0}^{\infty} \binom{n-1+i}{n-1}x^i \cdot \prod_{i=1}^n (1-x^{c_it+1})\right)\\ &= \sum_{u=0}^{kt} \binom{n-1+kt-u}{n-1}\left([x^u] \prod_{i=1}^n (1-x^{c_it+1})\right)\\ &= \sum_{u=0}^{kt} \binom{n-1+kt-u}{n-1} \sum_{j=0}^{k-1}(-1)^j\#\left\{i_1<\cdots<i_j : \sum_{\ell=1}^j (c_{i_\ell}t+1) = u\right\}\\ &= \sum_{j=0}^{k-1} (-1)^j \underbrace{\sum_{u=0}^{kt} \binom{n-1+kt-u}{n-1}\#\left\{i_1<\cdots<i_j : \sum_{\ell=1}^j (c_{i_\ell}t+1) = u\right\}}_{(\star)}. \end{align*} $$

Let us focus on the sum labeled by $(\star )$ . If $j=0$ , then it reduces to $\binom {n-1+kt}{n-1}$ , since only $u=0$ contributes, as the sum of the “empty choice” of $c_i$ ’s is zero by definition. On the other hand, if we assume that $j\geq 1$ , then the condition $\sum _{\ell =1}^j (c_{i_\ell }t+1) = u$ implies that $(c_{i_1}+\cdots +c_{i_j})t + j = u$ , and hence $u\equiv j\pmod {t}$ . Therefore, since $0\leq u\leq kt$ , we look only at the values $u = j$ , $u=t+j$ , $u=2t+j$ , …, $u = (k-1)t+j$ . Namely,

$$ \begin{align*} (\star) &= \sum_{v=0}^{k-1} \binom{n-1+kt-(vt+j)}{n-1} \# \left\{i_1<\cdots<i_j : c_{i_1}+\cdots+c_{i_j} = v\right\}\\ &= \sum_{v=0}^{k-1} \binom{t(k-v)+n-1-j}{n-1} \rho_{\textbf{c},j}(v). \end{align*} $$

Thus, we obtain

$$\begin{align*}\operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t) = \sum_{j=0}^{k-1}(-1)^j \sum_{v=0}^{k-1}\binom{t(k-v)+n-1-j}{n-1}\rho_{\textbf{c},j}(v), \end{align*}$$

and the proof is complete.

As Katzman notes in [Reference Katzman27], the particular case $\textbf {c} = (1,\ldots ,1)$ yields

$$\begin{align*}\rho_{\textbf{c},j}(v) = \left\{ \begin{matrix} \binom{n}{j}, & \text{ if }v=j,\\ 0, & \text{ if }v\neq j,\end{matrix} \right.\end{align*}$$

and the formula reduces to

$$\begin{align*}\operatorname{\mathrm{ehr}}(\Delta_{k,n}, t) = \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,(1,\ldots,1)},t)= \sum_{j=0}^{k-1}(-1)^j \binom{n}{j}\binom{t(k-j)+n-1-j}{n-1},\end{align*}$$

which is the well-known formula for the Ehrhart polynomial of the hypersimplex appearing in Stanley’s book [Reference Stanley41, Exercise 4.62]. Notice that even in this very particular case this sum is alternating in sign and that the variable t appears inside a binomial coefficient, which is actually a polynomial with some negative coefficients when $j\geq 2$ . The positivity of the coefficients for hypersimplices was classified as an open problem in Stanley’s book, and was settled in [Reference Ferroni14].

3.1 Weighted Lah Numbers Revisited

First, recall that the Lah number $L(n,m)$ is defined as the number of ways of partitioning $[n]$ into a set of exactly m blocks, each of which is internally ordered. For example, if we partition the set $[4]$ into two blocks, then we distinguish between $\{(1,3),(2,4)\}$ and $\{(1,3),(4,2)\}$ . Let us denote by $\mathscr {L}(n,m)$ the set of all such partitions. It is not difficult to show that $L(n,m)=|\mathscr {L}(n,m)| = \frac {n!}{m!}\binom {n-1}{m-1}$ .

Definition 3.1 [Reference Ferroni14]

Let $\pi \in \mathscr {L}(n,m)$ . We define the weight of $\pi $ by

$$\begin{align*}w(\pi) := \sum_{b\in\pi} w(b),\end{align*}$$

where $w(b)$ is the number of elements in the block b that are smaller (as positive integers) than the first element in b.

As a quick example, consider the partition $\pi =\{(5,3,7),(6,2,4,1)\}\in \mathscr {L}(7,2)$ , which has weight $w(\pi )=1+3=4$ .

In [Reference Ferroni14, Remark 3.11], it is provided a linear recurrence that the weighted Lah numbers satisfy and that may be used to compute them.

The reason why they are relevant in this paper is because we are going to provide a vast generalization for them. Essentially, apart from the three parameters $\ell $ , n, and m, we will introduce an additional input $\textbf {c}$ consisting of n integer numbers. When all the entries of the vector $\textbf {c}$ are ones, we will recover the weighted Lah numbers.

3.2 Weights for permutations

In [Reference Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez and Yin22], a different approach to the weighted Lah numbers is provided. We will review that version of their definition here. One particular advantage that it has is that these numbers can be thought as the quantity of “weighted permutations” satisfying a certain property. This suggests how to provide the right generalization for the main result of this section.

Definition 3.3 [Reference Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez and Yin22]

Let $\sigma \in \mathfrak {S}_n$ . A weight for $\sigma $ is a map $w:C(\sigma ) \to \mathbb {Z}_{\geq 0}$ . The total weight of $\sigma $ with respect to w, denoted by $w(\sigma )$ , is defined by

$$\begin{align*}w(\sigma) = \sum_{\mathfrak{c}\in C(\sigma)} w(\mathfrak{c}).\end{align*}$$

The pair $(\sigma ,w)$ will be called a weighted permutation.

Proof The bijection is constructed explicitly as follows. Take $(\sigma ,w)$ as in the statement. To each cycle $\mathfrak {c}\in C(\sigma )$ , we assign a linearly ordered set b consisting of the elements of $\mathfrak {c}$ as follows: choose the first element of b to be the $(w(\mathfrak {c})+1)$ th smallest element in $\mathfrak {c}$ and complete the ordering of b according to the order induced by the cycle $\mathfrak {c}$ . For example, if

$$\begin{align*}\sigma = (1\; 2\; 6)(3\; 5\; 7)(4\; 8),\end{align*}$$

where $w((1\; 2\; 6)) = 1$ , $w((3\; 5\; 7))=2$ and $w((4\; 8)) = 0$ , then we take the partition

$$\begin{align*}\pi = \{(2,6,1),(7,3,5),(4,8)\}.\end{align*}$$

It is straightforward to verify that this is indeed a bijection.

3.3 Compatibility

The bijection mentioned in Remark 3.6 suggests the following definition. Essentially, we use a certain bound for the weight of the cycles that depends on the elements that the cycles contain and not on their lengths as in Proposition 3.5.

Definition 3.7 Let $(\sigma , w)$ be a weighted permutation, $\sigma \in \mathfrak {S}_n$ . Let us fix the vector $\textbf {c}=(c_1,\ldots ,c_n)\in \mathbb {Z}_{>0}^n$ . We say that $(\sigma ,w)$ is $\textbf {c}$ -compatible if

$$\begin{align*}w(\mathfrak{c}) < \sum_{i\in \mathfrak{c}} c_i \end{align*}$$

for every cycle $\mathfrak {c}\in C(\sigma )$ .

Let us consider $\textbf {c} = \textbf {1} = (1,\ldots ,1)\in \mathbb {Z}_{> 0}^n$ . A weighted permutation $\sigma \in \mathfrak {S}_n$ is $\textbf {1}$ -compatible if and only if each cycle of $\sigma $ is assigned a number smaller than its length. In other words, by Proposition 3.5, we have that $\mathscr {L}(n,m)$ is in bijection with the set of all $\textbf {1}$ -compatible weighted permutations $(\sigma ,w)$ such that $\sigma \in \mathfrak {S}_n$ has exactly m cycles.

If we use the notation

$$ \begin{align*} L(n,m,\textbf{c}) &:= |\mathscr{L}(n,m,\textbf{c})|,\\ W(\ell,n,m,\textbf{c}) &:= |\mathscr{W}(\ell,n,m,\textbf{c})|, \end{align*} $$

then by taking $\textbf {c} = \textbf {1}$ , we have that $L(n,m,\textbf {1}) = L(n,m)$ and $W(\ell ,n,m,\textbf {1}) = W(\ell ,n,m)$ .

3.4 A counting formula for weighted permutations

The main goal now is to provide a formula for $W(\ell ,n,m,\textbf {c})$ that is as concrete as possible. We introduce some notation that will be useful not only in this section but later in the paper.

If $[a,b]$ is an interval of (possibly negative) integers, we will denote

$$\begin{align*}P_{a,b}^s := \sum_{I\in \binom{[a,b]}{s}} \prod_{i \in I} i = \sum_{a\leq i_1 < \cdots < i_s \leq b} i_1\cdots i_s.\end{align*}$$

This is the sth elementary symmetric polynomial in $b-a+1$ variables, evaluated in the integers of the interval $[a,b]$ .

Remark 3.10 Observe that when $a=1$ , the number $P_{a,b}^s$ reduces to a Stirling number of the first kind,

$$\begin{align*}P_{1,b}^s = \biggl[\genfrac{}{}{0pt}{}{b+1}{b+1-s}\biggr],\end{align*}$$

that is, the number of permutations in $\mathfrak {S}_{b+1}$ with $b+1-s$ cycles. Also, if $a> 0$ and $b> 0$ , we have

(3.1) $$ \begin{align} P_{-a,b}^s &= \sum_{j=0}^s P_{-a,-1}^j P_{1,b}^{s-j}\nonumber\\ &= \sum_{j=0}^s (-1)^j P_{1,a}^j P_{1,b}^{s-j}\nonumber\\ &= \sum_{j=0}^s (-1)^j \biggl[\genfrac{}{}{0pt}{}{a+1}{a+1-j}\biggr]\biggl[\genfrac{}{}{0pt}{}{b+1}{b+1-s+j}\biggr]. \end{align} $$

Theorem 3.11 For every $\textbf {c}\in \mathbb {Z}_{> 0}^n$ , the following formula holds:

$$\begin{align*}W(\ell,n,m+1,\textbf{c}) = \sum_{j=0}^{n}(-1)^jP^{n-1-m}_{-j+1,n-1-j}\sum_{i=0}^\ell\rho_{\textbf{c},j}(i)\binom{m+\ell-i}{m}. \end{align*}$$

Proof First, by using (3.1), we can rewrite the expression on the right as follows:

$$ \begin{align*} &\sum_{j=0}^{n}\sum_{u=0}^j(-1)^{j-u} \biggl[\genfrac{}{}{0pt}{}{j}{j-u}\biggr]\biggl[\genfrac{}{}{0pt}{}{n-j}{m+1+u-j}\biggr]\sum_{i=0}^\ell\rho_{\textbf{c},j}(i)\binom{m+\ell-i}{m}\\ &=\sum_{j=0}^{n}\sum_{u=0}^j(-1)^{j-u} \biggl[\genfrac{}{}{0pt}{}{j}{j-u}\biggr]\biggl[\genfrac{}{}{0pt}{}{n-j}{m+1+u-j}\biggr]\sum_{i=0}^\ell\sum_{\substack{A\in \binom{[n]}{j}\\ \sum_{a\in A} c_a=i}}\binom{m+\ell-i}{m}\\ &=\sum_{j=0}^{n}\sum_{u=0}^j\sum_{i=0}^\ell\sum_{\substack{A\in \binom{[n]}{j}\\ \sum_{a\in A} c_a=i}}(-1)^{j-u} \biggl[\genfrac{}{}{0pt}{}{j}{j-u}\biggr]\biggl[\genfrac{}{}{0pt}{}{n-j}{m+1+u-j}\biggr]\binom{m+\ell-i}{m}. \end{align*} $$

If we look at a fixed $A\in \binom {[n]}{j}$ such that $\sum _{a\in A}c_a=i$ , the quantity

$$\begin{align*}\biggl[\genfrac{}{}{0pt}{}{j}{j-u}\biggr]\biggl[\genfrac{}{}{0pt}{}{n-j}{m+1+u-j}\biggr]\binom{m+\ell-i}{m}\end{align*}$$

can be seen as the number of weighted permutations $(\sigma ,w)$ that simultaneously satisfy the following properties:

• • $\sigma \in \mathfrak {S}_n$ has exactly $m+1$ cycles.

• • $w(\sigma ) =\ell $ .

• • Exactly $j-u$ of the cycles consist only of elements that are contained in A.

• • The remaining $m+1-j+u$ cycles consist only of elements that are contained in $[n]\smallsetminus A$ .

• • $w(\mathfrak {c}) \geq \sum _{h \in \mathfrak {c}} c_h$ for each of the $j-u$ cycles $\mathfrak {c}\in C(\sigma )$ that consist only of elements contained in A.

The last condition explains the factor $\binom {m+\ell -i}{m}$ , which is exactly the number of ways of putting $\ell -i$ balls into $m+1$ boxes. This is because we put at least $i = \sum _{a\in A} c_a$ weight in the cycles consisting of elements in A, and then we can assign the remaining weight $\ell - i$ in $\binom {m+\ell -i}{m}$ different ways.

Let us call $\widehat {\mathscr {W}}(n,m+1,\ell ,A,j-u,\textbf {c})$ the set of all weighted permutations $(\sigma , w)$ satisfying the five conditions above (recall that i is determined from A and $\textbf {c}$ ). Further, notice that

$$\begin{align*}\widehat{\mathscr{W}}(\ell,n,m+1,\varnothing,0,\textbf{c}) = \mathscr{W}(\ell,n,m+1,\textbf{c}),\end{align*}$$

which is a direct consequence of the definitions.

The statement that we want to prove is therefore equivalent to showing that

$$\begin{align*}|\mathscr{W}(\ell,n,m+1,\textbf{c})| =\sum_{j=0}^{n}\sum_{u=0}^j\sum_{i=0}^\ell\sum_{\substack{A\in \binom{[n]}{j}\\ \sum_{a\in A} c_a=i}}(-1)^{j-u} |\widehat{\mathscr{W}}(\ell,n,m+1,A,j-u,\textbf{c})|.\end{align*}$$

This equality is rather a consequence of an inclusion–exclusion argument that we explain now. In what follows, whenever $\mathscr {S}$ is a set, we use the notation $\mathscr {S}[x]$ to denote $0$ or $1$ according to whether $x\notin \mathscr {S}$ or $x\in \mathscr {S}$ , respectively.

If $(\sigma ,w)\in \mathscr {W}(\ell ,n,m+1,\textbf {c})$ , then $(\sigma ,w)$ is present only in the set $\widehat {\mathscr {W}}(\ell ,n,m+1, \varnothing ,0,\textbf {c})$ , which corresponds to the case in which $u = j = 0$ , and appears with a plus sign.

On the contrary, let us fix some weighted permutation in $\mathfrak {S}_n$ , with $m+1$ cycles and total weight $\ell $ but is not $\textbf {c}$ -compatible; in other words, $(\sigma ,w)\notin \mathscr {W}(\ell ,n,m+1,\textbf {c})$ . Let $\mathfrak {c}_1,\dots ,\mathfrak {c}_b$ be the cycles $\mathfrak {c}\in C(\sigma )$ for which $w(\mathfrak {c})\geq \sum _{h\in \mathfrak {c}} c_h$ . For $B\subseteq [b]$ , let $A_B:= \bigcup _{s\in B} \mathfrak {c}_s$ (where we are regarding the cycles $\mathfrak {c}_s$ as sets). Then $(\sigma ,w)$ is contained in precisely the sets $\widehat {\mathscr {W}}(\ell ,n,m+1,A_B,|B|,\textbf {c})$ for $B\subseteq [b]$ . Therefore,

$$ \begin{align*} &\sum_{j=0}^{n}\sum_{u=0}^j\sum_{i=0}^\ell\sum_{\substack{A\in \binom{[n]}{j}\\ \sum_{a\in A} c_a=i}}(-1)^{j-u}\cdot \widehat{\mathscr{W}}(\ell,n,m+1,A,j-u,\textbf{c})[(\sigma,w)]\\ &=\sum_{B\subseteq [b]} (-1)^{|B|}\cdot \widehat{\mathscr{W}}(n,m+1,\ell,A_B,|B|,\textbf{c})[(\sigma,w)]\\ &=\sum_{B\subseteq [b]}(-1)^{|B|}=0. \end{align*} $$

Hence, the proof is complete.

4.1 The proof of Theorem 1.2

We start with a lemma that will be used later.

Lemma 4.1 The following equality holds:

$$\begin{align*}[t^m] \binom{t(k-v)+n-1-j}{n-1} =\frac{1}{(n-1)!} \, (k-v)^m P_{-j+1,n-1-j}^{n-1-m}. \end{align*}$$

Proof By definition, we have that

$$\begin{align*}\binom{t(k-v)+n-1-j}{n-1} = \frac{1}{(n-1)!} \prod_{s=1}^{n-1} \left(t(k-v)+n-j-s\right).\end{align*}$$

Therefore, apart from the factor $\frac {1}{(n-1)!}$ , we see that the coefficient of degree m consists of the product between $(k-v)^m$ and the sum of all products of $n-m-1$ numbers chosen in the interval of integers $[-j+1, n-1-j]$ , namely $P_{-j+1,n-1-j}^{n-1-m}$ .

We are now ready to prove Theorem 1.2.

Theorem 4.2 (Theorem 1.2)

Let $\textbf {c}=(c_1,\dots ,c_n)\in \mathbb {Z}_{>0}^n$ . Let $0\leq m\leq n-1$ . The coefficient of $t^m$ in $\operatorname {\mathrm{ehr}}(\mathscr {R}_{k,\textbf {c}},t)$ is given by

$$\begin{align*}[t^m]\operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t) = \frac{1}{(n-1)!}\sum_{\ell = 0}^{k-1}W(\ell,n,m+1,\textbf{c})A(m,k-\ell-1). \end{align*}$$

Proof By using the formula that we obtained in Theorem 2.5 and Lemma 4.1, we have the following chain of equalities:

$$ \begin{align*} [t^m]\operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t) &= \sum_{j=0}^{k-1}(-1)^j\sum_{v=0}^{k-1} [t^m]\binom{t(k-v)+n-1-j}{n-1} \rho_{\textbf{c},j}(v)\\ &=\frac{1}{(n-1)!}\sum_{j=0}^{k-1}\sum_{v=0}^{k-1}(-1)^{j} \,(k-v)^m\, P^{n-1-m}_{-j+1,n-1-j}\, \rho_{\textbf{c},j}(v)\\ &=\frac{1}{(n-1)!}\sum_{v=0}^k (k-v)^m\, \sum_{j=0}^{k-1}(-1)^j\, P^{n-1-m}_{-j+1,n-1-j} \,\rho_{\textbf{c},j}(v)\\ &=\frac{1}{(n-1)!}\sum_{v=0}^k (k-v)^m\, \sum_{j=0}^{n}(-1)^j\, P^{n-1-m}_{-j+1,n-1-j} \,\rho_{\textbf{c},j}(v), \end{align*} $$

where in the second to last step we changed the summation order and the upper limit for v because for $v=k$ we are just adding a zero due to the factor $(k-v)^m$ , and in the last step we changed the upper limit for j, as $\rho _{\textbf {c},j}(v)$ is zero whenever $j\geq k$ , as v is always at most k and the $c_i$ ’s are strictly positive. Let us introduce the following notation:

$$ \begin{align*} F_{n,m}(x) &:= \sum_{v=0}^{\infty} \left( \sum_{j=0}^{n}(-1)^j P^{n-1-m}_{-j+1,n-1-j} \rho_{\textbf{c},j}(v)\right) x^v,\\ G_{m}(x) &:= \sum_{v = 0}^{\infty} v^m \cdot x^v. \end{align*} $$

Using these names, the preceding chain of equalities reduces to

(4.1) $$ \begin{align} [t^m]\operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t) = \frac{1}{(n-1)!}[x^k] \left(F_{n,m}(x)\cdot G_{m}(x)\right).\end{align} $$

It is well known that $G_m(x)$ satisfies the following identity:

(4.2) $$ \begin{align} G_m(x) = \frac{1}{(1-x)^{m+1}} \sum_{i=0}^{m} A(m,i) x^{i+1},\end{align} $$

where $A(m,i)$ is an Eulerian number (see [Reference Stanley41, p. 40] for instance). On the other hand, notice that

(4.3) $$ \begin{align} \frac{1}{(1-x)^{m+1}} F_{n,m}(x) &= \sum_{\ell=0}^{\infty} \left(\sum_{i=0}^{\ell} \sum_{j=0}^{n}(-1)^j P^{n-1-m}_{-j+1,n-1-j} \rho_{\textbf{c},j}(i) \binom{m+\ell-i}{m}\right) x^{\ell}\nonumber \\ &= \sum_{\ell = 0}^{\infty} W(\ell,n,m+1,\textbf{c}) x^{\ell}, \end{align} $$

where in the last step we used Theorem 2.5. Now the equality of the statement follows from equations 4.1–4.3, just by writing

$$ \begin{align*} [t^m]\operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t) &=\frac{1}{(n-1)!} [x^k] \left(\sum_{\ell=0}^{\infty} W(\ell,n,m+1,\textbf{c}) x^{\ell} \cdot \sum_{i=1}^m A(m,i-1) x^i\right)\\ &= \frac{1}{(n-1)!}\sum_{\ell = 0}^{k-1}W(\ell,n,m+1,\textbf{c})A(m,k-\ell-1).\\[-36pt] \end{align*} $$

Remark 4.3 If we change the last coordinate of $\textbf {c}=(c_1,\ldots ,c_{n-1},c_n)$ by $\textbf {c}'=(c_1,\ldots ,c_{n-1},c_n+1)$ , it is immediate by definition that $\mathscr {W}(\ell ,n,m+1,\textbf {c})\subseteq \mathscr {W}(\ell ,n,m+1,\textbf {c}')$ . More generally, by induction, it follows that whenever the vector $\textbf {c}'-\textbf {c}$ has nonnegative coefficients, one has that

$$\begin{align*}\mathscr{W}(\ell,n,m+1,\textbf{c}) \subseteq \mathscr{W}(\ell,n,m+1,\textbf{c}').\end{align*}$$

Due to the preceding result, this monotonicity property has a counterpart for the Ehrhart polynomials.

Corollary 4.4 If $\textbf {c},\textbf {c}'\in \mathbb {Z}^n_{>0}$ are vectors such that $\textbf {c}'-\textbf {c}\in \mathbb {Z}^n_{\geq 0}$ , then

$$\begin{align*}\operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t) \preceq \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}'},t),\end{align*}$$

where $\preceq $ denotes coefficientwise inequality.

4.2 A unit-circle-rootedness conjecture

While verifying computationally the main results of the present paper, the following intriguing problem, that we pose here as a conjecture, arose.

Conjecture 4.5 For each $\textbf {c}=(c_1,\ldots ,c_n)\in \mathbb {Z}_{>0}^n$ and each $0\leq m\leq n - 1$ , the polynomial defined by

$$\begin{align*}p_{n,m,\textbf{c}}(x) = \sum_{\ell = 0}^{\infty} W(\ell,n,m+1,\textbf{c})\, x^{\ell},\end{align*}$$

has all of its complex roots lying on the unit circle $|z| = 1$ .

Notice that although we used the upper limit $\infty $ in the sum, it actually yields a polynomial, as for $\ell> c_1+\cdots +c_n$ , one will certainly have $W(\ell ,n,m+1,\textbf {c}) = 0$ . A reasonable question is whether some techniques regarding $h^*$ -polynomials having roots on the unit circle can be applied to these particular polynomials (notice that they may not be the $h^*$ -polynomial of a polytope, as they can have a constant term larger than $1$ ). See [Reference Braun and Liu3] for results regarding polytopes having a unit-circle-rooted $h^*$ -polynomial.

5.1 A review of terminology

In the same way that the $\textbf {c}$ -compatible weighted permutations play a key role in the description of the coefficients of the Ehrhart polynomial of $\mathscr {R}_{k,\textbf {c}}$ , for the $h^*$ -polynomial, we have to introduce a different object, already studied in both [Reference Early12, Reference Kim28] and endow it with an extra condition that we call again “ $\textbf {c}$ -compatibility.”

Recall that a cyclically ordered partition of $[n]$ is a partition of $[n]$ into disjoint blocks that are ordered cyclically. For example, the partitions of $[5]$ given by $(\{1,2\}, \{3,5\}, \{4\})$ and $(\{3,5\}, \{4\}, \{1,2\})$ are considered as equal. The set of blocks of such a partition $\xi $ will be customarily denoted by $P(\xi )$ .

Definition 5.1 A decorated ordered set partition of type $(k,n)$ consists of a cyclically ordered partition $\xi $ of $[n]$ and a function $w:P(\xi ) \to \mathbb {Z}_{\geq 0}$ such that

$$\begin{align*}\sum_{\mathfrak{p}\in P(\xi)} w(\mathfrak{p}) = k.\end{align*}$$

For a vector $\textbf {c}=(c_1,\ldots ,c_n)\in \mathbb {Z}_{>0}^n$ , we say that a decorated ordered set partition $\xi $ is $\textbf {c}$ -compatible if

$$\begin{align*}w(\mathfrak{p}) < \sum_{i\in\mathfrak{p}} c_i\end{align*}$$

for all $\mathfrak {p}\in P(\xi )$ .

A decorated ordered set partition of type $(k,n)$ , $\xi =(L_1,\ldots ,L_m)$ , can be represented as a set of k points in a circle ordered in a clockwise fashion. The blocks $L_i$ are placed among these points in such a way that the clockwise distance between $L_i$ and $L_{i+1}$ is $w(L_i)$ . Observe that this only depends on the cyclic order and not in the particular choice of the first block $L_1$ .

Notice that in the case in which $\textbf {c}=(1,\ldots ,1)$ the notion of being $\textbf {c}$ -compatible was named as “hypersimplicial” in [Reference Early12], whereas in the case $\textbf {c}=(r,\ldots ,r)$ it was called “r-hypersimplicial” in [Reference Kim28].

Definition 5.2 The winding number of a decorated ordered set partition of type $(k,n)$ is defined as the only number d such that

$$\begin{align*}dk = \lambda_1 + \cdots + \lambda_n,\end{align*}$$

where $\lambda _i$ denotes the clockwise distance between the block containing i and the block containing $i+1$ (modulo n).

Having set all these names and notations, we are ready to state the main result of this section. The proof is carried out in the next subsection.

5.2 A proof à la Kim

We will outline a proof of Theorem 1.7. Some details are omitted as the proofs are carried out verbatim from [Reference Kim28]; in particular, to improve readability, we have decided to use the same (if not, very similar) notation.

The family of all the partitions of a set A will be henceforth denoted by $\Pi (A)$ . Also, we will use the notation

$$\begin{align*}\binom{n}{a}_b := [x^a] (1+x+x^2+\cdots+x^{b-1})^n,\end{align*}$$

so that in particular when $b=2$ , we recover the classical binomial numbers.

Lemma 5.4 The following formula holds:

$$\begin{align*}\binom{n}{a}_b = \sum_{j=0}^{\lfloor\frac{a}{b}\rfloor} (-1)^j \binom{n}{j}\binom{n-1+a-bj}{n-1}.\end{align*}$$

Proof The proof is a standard argument with generating functions.

$$ \begin{align*} \binom{n}{a}_b &= [x^a]\left(\sum_{i=0}^{b-1} x^i\right)^n\\ &= [x^a]\frac{(1-x^b)^n}{(1-x)^n}\\ &= [x^a]\left(\sum_{j=0}^n (-1)^j \binom{n}{j} x^{bj} \right)\left(\sum_{j=0}^{\infty}\binom{n-1+j}{n-1} x^j\right)\\ &= \sum_{j=0}^{\lfloor\frac{a}{b}\rfloor} (-1)^j\binom{n}{j}\binom{n-1+a-bj}{n-1}.\\[-34pt] \end{align*} $$

Definition 5.5 Let $\xi $ be a decorated ordered set partition of type $(k,n)$ , and let $\textbf {c}=(c_1,\ldots ,c_n)\in \mathbb {Z}_{>0}^n$ . The set of $\textbf {c}$ -bad blocks of $\xi $ is the family

$$\begin{align*}I_{\textbf{c}}(\xi) = \left\{ \mathfrak{p}\in P(\xi) : w(\mathfrak{p}) \geq \sum_{i\in \mathfrak{p}} c_i\right\}.\end{align*}$$

If S is a family of pairwise disjoint subsets of $[n]$ , we define

$$\begin{align*}K_{\textbf{c}}(S) := \{\xi : I_{\textbf{c}}(\xi) \supseteq S\},\end{align*}$$

and

$$\begin{align*}H_{\textbf{c}}(T) = \sum_{S\in \Pi(T)} (-1)^{|S|} |K_{\textbf{c}}(S)|.\end{align*}$$

Proposition 5.6 The number of $\textbf {c}$ -compatible decorated ordered set partitions of type $(k,n)$ with winding number d is given by

$$\begin{align*}\sum_{T\subseteq [n]} H_{\textbf{c}}(T). \end{align*}$$

From now on, if we consider a cyclically ordered partition $\xi = (L_1,\ldots ,L_m)$ , we will consider the indices of the blocks modulo m. For instance, $L_0=L_m$ and $L_{m+2} = L_{2}$ .

Fix $T=\{t_1<\cdots <t_m\}\subseteq [n]$ with $n\notin T$ . For each $S\in \Pi (T)$ , Lemma 5.8 provides a map

$$\begin{align*}i_S: K_{\textbf{c}}(S) \hookrightarrow K_{\textbf{c}}(\{t_1\},\ldots,\{t_m\}). \end{align*}$$

If we name $\chi _S$ the characteristic map of $i_S(K_{\textbf {c}}(S))$ , in other words,

$$\begin{align*}\chi_s(\xi) = \left\{ \begin{matrix} 1, & \text{ if }\xi\in i_S(K_{\textbf{c}}(S)),\\ 0, & \text{ if }\xi\notin i_S(K_{\textbf{c}}(S)), \end{matrix} \right.\end{align*}$$

then we have that

$$ \begin{align*} H_{\textbf{c}}(T) &= \sum_{S\in \Pi(T)}(-1)^{|S|} |K_{\textbf{c}}(S)| = \sum_{S\in \Pi(T)}(-1)^{|S|} |i_S(K_{\textbf{c}}(S))|\\ &=\sum_{S\in \Pi(T)}(-1)^{|S|} \sum_{\xi\in K_{\textbf{c}}(\{\{t_1\},\dots,\{t_m\}\})}\chi_S(\xi)\\ &=\sum_{\xi\in K_{\textbf{c}}(\{\{t_1\},\dots,\{t_m\}\})}\sum_{S\in \Pi(T)}(-1)^{|S|}\chi_S(\xi). \end{align*} $$

Proposition 5.9 motivates the following definition.

Notice by Proposition 5.9 and the short calculation preceding it, we have that

$$\begin{align*}H_{\textbf{c}}(T)=(-1)^{|T|}\hat{K}_{\textbf{c}}(T). \end{align*}$$

Proposition 5.11 The set $\hat {K}_{\textbf {c}}(T)$ is in bijection with the following set:

$$\begin{align*}\left\{(v_1,\ldots,v_n)\in \mathbb{Z}^n : \sum_{i=1}^n v_i = (k-c_T)d \text{ and }\; \begin{matrix} 0&\leq v_i\leq& k-c_T-1 & \text{for each }i\notin T\\ 1&\leq v_i\leq &k-c_T & \text{for each }i\in T \end{matrix}\right\}, \end{align*}$$

where we use the notation $c_T:=\sum _{i\in T} c_i$ .

The idea is to provide a formula for $H_{\textbf {c}}$ . Since $H_{\textbf {c}} = (-1)^{|T|} \hat {K}_{\textbf {c}}(T)$ , it is enough to enumerate the elements in the set of Proposition 5.11. Its size can be calculated as follows. Let us consider new variables $v^{\prime }_1,\ldots ,v^{\prime }_n$ . For each $i\notin T$ , we set $v_i'=v_i$ ; otherwise, we put $v_i'=v_i-1$ . Then the cardinality of the set in Proposition 5.11 can be seen as the number of nonnegative integer solutions to $v^{\prime }_1+\cdots +v^{\prime }_n=(k-c_T)d-|T|$ under the constraints $0\leq v_i'\leq k-c_T-1$ for each $1\leq i\leq n$ . This coincides with the coefficient of $x^{(k-c_T)d-|T|}$ in $(1+x+\cdots +x^{k-c_T-1})^n$ , which is the number $\binom {n}{(k-c_T)d-|T|}_{k-c_T}$ .

Putting all the pieces together, we obtain

(5.1) $$ \begin{align} \sum_{T\subseteq [n]}H_{\textbf{c}}(T) &=\sum_{T\subseteq [n]}(-1)^{|T|}|\hat{K}_{\textbf{c}}(T)|\nonumber\\ &=\sum_{T\subseteq [n]}(-1)^{|T|}\binom{n}{(k-c_T)d-|T|}_{k-c_T}\nonumber\\ &=\sum_{m=0}^n\sum_{|T|=m}(-1)^{|T|}\binom{n}{(k-c_T)d-|T|}_{k-c_T}\nonumber\\ &=\sum_{m=0}^{k-1}\sum_{|T|=m}(-1)^{m}\binom{n}{(k-c_T)d-m}_{k-c_T}\nonumber\\ &=\sum_{m=0}^{k-1}\sum_{v=0}^{k-1}\sum_{\substack{|T|=m\nonumber\\ c_T=v}}(-1)^{m}\binom{n}{(k-v)d-m}_{k-v}\nonumber\\ &=\sum_{m=0}^{k-1}\sum_{v=0}^{k-1}(-1)^{m}\rho_{\textbf{c},m}(v)\binom{n}{(k-v)d-m}_{k-v} \nonumber\\ &=\sum_{m=0}^{k-1}\sum_{v=0}^{k-1}(-1)^{m}\rho_{\textbf{c},m}(v)\sum_{i=0}^{d}(-1)^i\binom{n}{i}\binom{n-1+(k-v)(d-i)-m}{n-1} \end{align} $$

(5.2) $$ \begin{align} &=\sum_{i=0}^{d}(-1)^i \binom{n}{i} \left(\sum_{m=0}^{k-1}\sum_{v=0}^{k-1}(-1)^{m}\rho_{\textbf{c},m}(v)\binom{n-1+(k-v)(d-i)-m}{n-1}\right)\nonumber\\ &=\sum_{i=0}^{d}(-1)^i \binom{n}{i} \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},d-i) \end{align} $$

(5.3) $$ \begin{align} &= [x^d] \left((1-x)^n\, \sum_{j=0}^{\infty} \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}}, j) x^j\right)\nonumber\\ &=[x^d] h^*(\mathscr{R}_{k,\textbf{c}},x), \end{align} $$

where in (5.1) we used Lemma 5.4, in (5.2) we leveraged the formula of Theorem 2.5, and in (5.2) we just used the definition of the $h^*$ -polynomial. Now, combining this with Proposition 5.6, we have proved Theorem 1.7.

5.3 A real-rootedness conjecture

In the context of the study of the $h^*$ -polynomials of the polytopes $\mathscr {R}_{k,\textbf {c}}$ , we pose the following conjecture.

Observe that this would automatically provide a proof for the unimodality of the coefficients of the $h^*$ -polynomial of all hypersimplices, a long-standing open problem regarding the combinatorics of the hypersimplex (see [Reference Braun2]). In particular, if we use the notation $p(x)\lessdot q(x)$ to denote that p and q interlace, after fixing k and n, our conjecture implies that one can produce the sequence of polynomials

$$\begin{align*}h^*(\mathscr{R}_{k,(n)},x) \lessdot h^*(\mathscr{R}_{k,(n-1,1)},x)\lessdot h^*(\mathscr{R}_{k,(n-2,1,1)},x) \lessdot \cdots \lessdot h^*(\mathscr{R}_{k,(1,\ldots,1)},x),\end{align*}$$

and $h^*(\mathscr {R}_{k,(1,\ldots ,1)},x)=h^*(\Delta _{k,n},x)$ . It is reasonable to search for recurrences that these polynomials satisfy and use techniques as those used in [Reference Haglund and Zhang20] or [Reference Gustafsson and Solus19].

6.1 Volumes

As we mentioned in the Introduction, the leading coefficient of $\operatorname {\mathrm{ehr}}(\mathscr {P},t)$ is $\operatorname {vol}(\mathscr {P})$ . In particular, by using the formula of Theorem 1.2, for $m=n-1$ , we obtain Corollary 1.5. We restate it and prove it now.

Corollary 6.1 Let $\textbf {c}=(c_1,\dots ,c_n)\in \mathbb {Z}_{>0}^n$ and $0 < k < c_1+\cdots +c_n$ . Then the volume of $\mathscr {R}_{k,\textbf {c}}$ is given by

$$\begin{align*}\operatorname{vol}(\mathscr{R}_{k,\textbf{c}}) = \frac{1}{(n-1)!} \sum_{\ell = 0}^{k-1} B(\ell,\textbf{c})A(n-1,k-\ell-1),\end{align*}$$

where $B(\ell ,\textbf {c})$ is defined as the number of ways of placing $\ell $ indistinguishable balls into n boxes of capacities $c_1-1, \ldots , c_n-1$ , respectively.

Proof Given that we assume that $0 < k < c_1+\cdots +c_n$ , the Ehrhart polynomial has degree $n-1$ and hence the volume is obtained by taking $m=n-1$ in the statement of our main theorem.

$$ \begin{align*} \operatorname{vol}(\mathscr{R}_{k,\textbf{c}}) &= [t^{n-1}] \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}},t)\\ &= \frac{1}{(n-1)!} \sum_{\ell = 0}^{k-1} W(\ell, n,n, \textbf{c}) A(n-1,k-\ell-1). \end{align*} $$

Notice that we have a factor $W(\ell ,n,n,\textbf {c})$ in each summand. The only permutation $\sigma \in \mathfrak {S}_n$ with n cycles is the identity. Therefore, in order for the permutation to admit a $\textbf {c}$ -compatible weight, we have to ensure that each of the numbers $i\in [n]$ is assigned a value between $0$ and $c_i-1$ . To guarantee that the total weight is $\ell $ , we can think that we are putting $\ell $ balls inside boxes with capacities $c_1-1,\ldots , c_n-1$ , as is described in the statement.

6.2 Flag Eulerian numbers

In this subsection, we will extend and generalize some of the results of [Reference Han and Josuat-Vergès21]. Let $\textbf {c}=(c_1,\ldots ,c_n)\in \mathbb {Z}^n_{>0}$ . In what follows, we will define objects that coincide with those defined in [Reference Han and Josuat-Vergès21] when $c_1=\cdots =c_n=r$ . A $\textbf {c}$ -colored permutation on $[n]$ consists of a permutation $\sigma \in \mathfrak {S}_n$ and a function $\textbf {s}:[n]\to \mathbb {Z}_{\geq 0}$ such that $s_i:=\textbf {s}(i) \leq c_i-1$ for each $i\in [n]$ . The set of all $\textbf {c}$ -colored permutations for a fixed value of n is denoted by $\mathfrak {S}_n^{(\textbf {c})}$ . The descent number of a $\textbf {c}$ -colored permutation $(\sigma , \textbf {s})\in \mathfrak {S}_n^{(\textbf {c})}$ is defined as $\operatorname {des}(\sigma ,\textbf {s}) := \#\operatorname {Des}(\sigma ,\textbf {s})$ where

$$\begin{align*}\operatorname{Des}(\sigma,\textbf{s}) := \left\{ i\in [n-1] : s_i> s_{i+1}\, \text{ or }s_i=s_{i+1}\text{ and }\sigma_i>\sigma_{i+1}\right\}.\end{align*}$$

The flag descent number of a $\textbf {c}$ -colored permutation $(\sigma ,\textbf {s})\in \mathfrak {S}_n^{(\textbf {c})}$ is defined by

$$\begin{align*}\operatorname{fdes}(\sigma,\textbf{s}) := s_n+ \sum_{i\in \operatorname{Des}(\sigma,\textbf{s})} c_{i+1}. \end{align*}$$

The flag Eulerian numbers are defined by

$$\begin{align*}A_{n,k}^{(\textbf{c})} := \#\left\{ (\sigma,\textbf{s}) \in \mathfrak{S}^{(\textbf{c})}_n: \operatorname{fdes}(\sigma,\textbf{s}) = k \right\}. \end{align*}$$

In [Reference Han and Josuat-Vergès21], Han and Josuat-Vergès approached the case in which $c_1=\cdots =c_n=r$ Footnote ¹ . They provided formulas and interpretations for $A_{n,k}^{(r,\ldots ,r)}$ . In particular, to obtain an explicit formula for such numbers, they studied certain half-open polytopes having these numbers as their volumes. We can extend this technique to arbitrary vectors $\textbf {c}=(c_1,\ldots ,c_n)$ .

First, for each $v=(v_1,\ldots ,v_n)\in \mathbb {R}^n$ , we call $\operatorname {Des}(v)=\{i\in [n-1]:v_i>v_{i+1}\}$ . Let us use the name

$$\begin{align*}\operatorname{fdes}_{\textbf{c}}(v):=v_n+\sum_{i\in \operatorname{Des}(v)} c_{i+1},\end{align*}$$

and define the following two (half-open) polytopes:

$$ \begin{align*} \mathcal{F}_{n,k}^{(\textbf{c})} &:= \left\{v\in\mathscr{R}_{\textbf{c}} : k \leq \operatorname{fdes}_{\textbf{c}}(v) < k+1\right\},\\ \mathcal{A}_{n,k}^{(\textbf{c})} &:= \left\{ v\in \mathscr{R}_{\textbf{c}} : k \leq \sum_{i=1}^n v_i < k+1 \right\}. \end{align*} $$

The key observation is that the above two (half-open) polytopes have the same volume, as there is a measure-preserving map between them that we shall describe. In [Reference Stanley39], Stanley studied the case $c_1=\cdots =c_n=1$ , and in [Reference Han and Josuat-Vergès21], Han and Josuat-Vergès extended this to the case $c_1=\cdots =c_n=r$ . The latter case had also been approached before by Steingrímsson in [Reference Steingrímsson42, Section 4.4]. We outline here how to show it for an arbitrary vector $\textbf {c}$ . Consider the map $\varphi :[0,c_1)\times \cdots \times [0,c_n)\to [0,c_1)\times \cdots \times [0,c_n)$ defined everywhere, except in a measure zero set, by $(a_1,\ldots ,a_n)\mapsto (b_1,\ldots ,b_n)$ where for each $1\leq i\leq n$ we have

$$\begin{align*}b_i := \left\{ \begin{matrix} a_i - a_{i-1}, & \text{ if }a_{i-1}< a_i,\\ a_i - a_{i-1} + c_i, & \text{ if }a_{i-1}>a_i, \end{matrix} \right.\end{align*}$$

where we use the convention $a_0:=0$ . This map is injective where defined, as if we assume that $(a_1,\ldots ,a_n)$ and $(a_1',\ldots ,a_n')$ map to the same vector $(b_1,\ldots ,b_n)$ , then looking at the first coordinates yields $a_1=a_1'$ . Next, looking at the second coordinates, we must have

$$\begin{align*}a_2 - a_1 + \varepsilon c_2 = a_2'-a_1' + \varepsilon' c_2,\end{align*}$$

where $\varepsilon , \varepsilon '\in \{0,1\}$ depend on how $a_1$ and $a_2$ compare (resp. $a_1'$ and $a_2'$ ). Canceling the summand $a_1=a_1'$ from both sides gives $a_2 = a_2' + (\varepsilon '-\varepsilon ) c_2$ . Since we are assuming that both vectors $(a_1,\ldots ,a_n)$ and $(a_1',\ldots ,a_n')$ lie in $[0,c_1)\times \cdots \times [0,c_n)$ , the only possibility is that $\varepsilon =\varepsilon '$ and that $a_2=a_2'$ . Inductively we can continue using the same reasoning.

Notice that the map $\varphi $ is indeed measure-preserving, as if we consider for each $\sigma \in \mathfrak {S}_n$ the subset of $\mathscr {R}_{\textbf {c}}$ defined by

$$\begin{align*}\mathscr{R}_{\textbf{c},\sigma} := \left\{v\in \mathscr{R}_{\textbf{c}} : v_{\sigma(1)} < v_{\sigma(2)} < \cdots < v_{\sigma(n)}\right\},\end{align*}$$

thenFootnote ² $\varphi |_{\mathscr {R}_{\textbf {c},\sigma }}$ is just a map of the form $\varphi (\textbf {x}) = A\textbf {x} + \boldsymbol {\varepsilon }$ , where A is a matrix with $1$ ’s in the main diagonal and $-1$ ’s in second diagonal, whereas $\boldsymbol {\varepsilon }$ is a vector having $c_i$ ’s or zeros as entries.

Moreover, if we take a vector $v\in \mathcal {F}_{n,k}^{(\textbf {c})}$ having distinct coordinates and we call $(v_1',\ldots ,v_n'):=\varphi (v)$ , we have that

$$\begin{align*}v_1'+\cdots+v_n' = (v_1-v_0)+(v_2-v_1)+\cdots+(v_n-v_{n-1}) + \sum_{i\in \operatorname{Des}(v)} c_{i+1} = \operatorname{fdes}_{\textbf{c}}(v),\end{align*}$$

and hence $\varphi $ maps (up to a measure zero subset) all of $\mathcal {F}_{n,k}^{(\textbf {c})}$ to $\mathcal {A}_{n,k}^{(\textbf {c})}$ , and hence the volumes of these two (half-

On one hand, we have that $\operatorname {vol}(\mathcal {A}_{n,k}^{(\textbf {c})})=\operatorname {vol}(\mathcal {F}_{n,k}^{(\textbf {c})})$ , whereas, on the other hand, it is not difficult to show that $\operatorname {vol}(\mathcal {F}^{(\textbf {c})}_{n,k}) = \frac {1}{n!}A_{n,k}^{(\textbf {c})}$ . In other words,

$$\begin{align*}\operatorname{vol}(\mathcal{A}_{n,k}^{(\textbf{c})}) = \frac{1}{n!}\,A_{n,k}^{(\textbf{c})}.\end{align*}$$

We can use that $\mathcal {A}_{n,k}^{(\textbf {c})}$ is a slice of a prism with some missing facets. In particular, its volume will not change if we add these facets. This trick leads to a proof of the following result.

Corollary 6.2 The flag Eulerian number $A_{n,k}^{(\textbf {c})}$ is given by

$$\begin{align*}A_{n,k}^{(\textbf{c})} = \sum_{\ell = 0}^{k} B(\ell, \textbf{c}) A(n, k-\ell),\end{align*}$$

where $B(\ell ,\textbf {c})$ is defined as in Corollary 1.5.

Proof Consider the polytope $\mathscr {R}_{k+1, \textbf {c}'}$ for $\textbf {c}' = (c_1,\ldots ,c_n, 1)\in \mathbb {Z}_{>0}^{n+1}$ . By Corollary 1.5, we obtain

(6.1) $$ \begin{align} A_{n,k}^{(\textbf{c})} = n! \operatorname{vol}\left(\mathcal{A}^{(\textbf{c})}_{n,k}\right) = n! \operatorname{vol}\left(\mathscr{R}_{k+1,\textbf{c}'}\right) =\sum_{\ell = 0}^{k} B(\ell, \textbf{c}') A(n, k-\ell),\end{align} $$

where $B(\ell , \textbf {c}')$ is in this case the number of ways of placing $\ell $ balls into $n+1$ boxes of capacities $c_1-1,\ldots ,c_n-1$ and $0$ or, in other words, $B(\ell ,\textbf {c}')=B(\ell ,\textbf {c})$ , which completes the proof.

The identity in the preceding corollary provides a new combinatorial formula for the version of the flag Eulerian numbers studied in [Reference Han and Josuat-Vergès21], by just particularizing $c_1=\cdots =c_n=r$ . We do not know of a direct combinatorial way of deriving the preceding equality.

Additionally, since the (normalized) volume of a polytope can be obtained by evaluating the $h^*$ -polynomial in $x=1$ , we have a refinement of $A_{n,k}^{(\textbf {c})}$ by considering the coefficients of the $h^*$ -polynomial of the polytope $\mathscr {R}_{k+1,\textbf {c}'}$ by considering the winding numbers.

Corollary 6.3 The flag Eulerian number $A_{n,k}^{(\textbf {c})}$ is given by

$$\begin{align*}A_{n,k}^{(\textbf{c})} = \#\{\textbf{c}'\text{-compatible decorated ordered set partitions of type }(k+1,n+1)\},\end{align*}$$

where $\textbf {c}'=(\textbf {c},1)\in \mathbb {Z}^{n+1}_{>0}$ .

Compare this result to [Reference Kim28, Remark 1.2]. A version of this fact for hypersimplicial permutations (i.e., $\textbf {c}=(1,\ldots ,1)$ ) appeared also in [Reference Ocneanu36].

6.3 Independence polytopes of uniform matroids

The uniform matroid $\mathsf{U}_{k,n}$ of rank k and cardinality n has the hypersimplex $\Delta _{k,n}$ as its base polytope. Its independence polytope, namely, the convex hull of the indicator vectors of all the independent subsets, denoted $\mathscr {P}_I(\mathsf{U}_{k,n})$ is given by

(6.2) $$ \begin{align} \mathscr{P}_I(\mathsf{U}_{k,n}) = \left\{ x\in [0,1]^n : \sum_{i=1}^n x_i \leq k\right\}. \end{align} $$

In his Ph.D. thesis [Reference Duna11], Duna conjectured that these polytopes are Ehrhart positive, and he proved that they indeed are when $k=2$ . Later, Ferroni proved in [Reference Ferroni15] that Ehrhart positivity holds for all k and n, but relies on complicated inequalities.

Observe that this polytope is a fat slice of the unit cube. Therefore, using Proposition 2.2, we see that

$$\begin{align*}\operatorname{\mathrm{ehr}}(\mathscr{P}_I(\mathsf{U}_{k,n}), t) = \operatorname{\mathrm{ehr}}(\mathscr{R}_{k,\textbf{c}'},t),\end{align*}$$

where $\textbf {c}' = (1,\ldots , 1, k)$ . This provides an alternative and much more conceptual proof of the Ehrhart positivity of these polytopes. Moreover, we can retrieve an explicit formula for their Ehrhart polynomial, also described in [Reference Duna11, Theorem 4.9].

Corollary 6.4 The Ehrhart polynomial of $\mathscr {P}_I(\mathsf{U}_{k,n})$ is given by

$$\begin{align*}\operatorname{\mathrm{ehr}}(\mathscr{P}_I(\mathsf{U}_{k,n}), t) = \sum_{j=0}^{k-1}(-1)^j \binom{n}{j} \binom{(k-j)t+n-j}{n}.\end{align*}$$

This polynomial has positive coefficients.

Proof Since our polytope is integrally equivalent to $\mathscr {R}_{k,\textbf {c}'}$ for $\textbf {c}'=(1,\ldots ,1,k)\in \mathbb {Z}_{>0}^{n+1}$ , using Theorem 2.5, we have

$$\begin{align*}\operatorname{\mathrm{ehr}}(\mathscr{P}_I(\mathsf{U}_{k,n}), t) = \sum_{j=0}^{k-1}(-1)^j \sum_{v=0}^{k-1} \binom{(k-v)t+(n+1)-1-j}{(n+1)-1}\rho_{\textbf{c},j}(v).\end{align*}$$

Recalling from equation (2.1) the definition of $\rho _{\textbf {c},j}(v)$ , we have that it is the number of ways of summing v using exactly j numbers among the $c_i$ ’s. However, the only number v that can be obtained as the sum of j values among the $c_i$ ’s is just $v=j$ , in exactly $\binom {n}{j}$ ways. In other words, $\rho _{\textbf {c},j}(v) = \binom {n}{j}$ if $j=v$ , and it is zero otherwise. This yields the result.

6.4 Hilbert series of algebras of Veronese type

The following algebraic structures arise naturally in combinatorics and commutative algebra.

In [Reference De Negri and Hibi10], De Negri and Hibi studied the algebras of Veronese type and characterized the scenarios in which they are Gorenstein. In particular, in [Reference De Negri and Hibi10, Corollary 2.2], they prove that these algebras are Cohen–Macaulay and normal. An important step in such proofs consists of showing that there is an isomorphism between $\mathscr {V}(\textbf {c},k)$ and the Ehrhart ring of $\mathscr {R}_{k,\textbf {c}}$ (see [Reference Bruns and Herzog4] for detailed definitions on Ehrhart rings of polytopes). In particular, we obtain as a consequence of Theorem 1.2 the following result.

Proof By definition, the Hilbert function of $\mathscr {A}=\mathscr {V}(\textbf {c},k)$ is the map:

$$\begin{align*}m \mapsto \dim \mathscr{A}^m. \end{align*}$$

By the discussion above, the dimension of the graded component $\mathscr {A}^m$ equals $\operatorname {\mathrm{ehr}}(\mathscr {R}_{k,\textbf {c}},m)$ , which we know has positive coefficients.

Notice that Hilbert functions very rarely possess such property. In [Reference Katzman27, p. 1145], Katzman referred to the combinatorial descriptions of the h-vector of $\mathscr {V}(\textbf {c},k)$ as “forbidding.” As a consequence of Theorem 1.7, we now state an interpretation for each of these entries.