3.1 Last passage hypermatrix

A lattice path in $\mathbb {Z}^d$ is called directed if it uses only steps of the form $\mathbf {i} \to \mathbf {i} + \mathbf {e}_\ell $ for $\mathbf {i} \in \mathbb {Z}^d$ and $\ell \in [d]$ . Given a d-dimensional $\mathbb {N}$ -hypermatrix $A = (a_{i_1, \ldots , i_d})$ , define the last passage times Footnote ²

$$ \begin{align*} G_{i_1, \ldots, i_d} := \max_{\Pi\, :\, (i_1, \ldots, i_d) \to (\infty,\ldots,\infty)} \sum_{(j_1, \ldots, j_d) \in \Pi} a_{j_1, \ldots, j_d}, \end{align*} $$

where the maximum is over directed lattice paths $\Pi $ which start at $(i_1, \ldots , i_d) \in \mathbb {Z}^d_+$ . It is easy to see that the following recurrence relation holds:

(1) $$ \begin{align} G_{\mathbf{i}} = a_{\mathbf{i}} + \max_{\ell \in [d]} G_{\mathbf{i} + \mathbf{e}_\ell}, \quad \mathbf{i} \in \mathbb{Z}^d_+. \end{align} $$

Notice that the hypermatrix $G = (G_{\mathbf {i}})_{\mathbf {i} \in \mathbb {Z}^d_+} \in \mathcal {P}^{(d)}$ is a d-dimensional partition.

3.2 The bijection

Define the map $\Phi : \mathcal {M}^{(d)} \to \mathcal {P}^{(d)}$ as follows:

(2) $$ \begin{align} \Phi : A \longmapsto G \end{align} $$

Let $\rho \subset \mathbb {Z}^d_+$ be a shape of some d-dimensional partition (or a diagram of a $(d-1)$ -dimensional partition). Let

$$ \begin{align*} \mathcal{P}(\rho, n) := \left \{ \pi \in \mathcal{P}^{(d)} : \mathrm{sh}(\pi) \subseteq \rho,\, \pi_{1, \ldots, 1} \le n \right \} \end{align*} $$

be the set of d-dimensional partitions whose shape is a subset of $\rho $ and the largest entry is at most n. Let

$$ \begin{align*} \mathcal{M}(\rho, n) := \left \{A = (a_{\mathbf{i}}) \in \mathcal{M}^{(d)} : a_{\mathbf{i}}> 0 \implies \mathbf{i} \in \rho,\, G_{1,\ldots, 1} \le n \right \} \end{align*} $$

be the set of d-dimensional $\mathbb {N}$ -hypermatrices whose support (i.e., the set of indices corresponding to positive entries) lies inside $\rho $ and whose largest last passage time is at most n.

Proof. Let $A = (a_{\mathbf {i}}) \in \mathcal {M}(\rho , n)$ . By construction of the map, it is not difficult to see that $\pi = \Phi (A) \in \mathcal {P}(\rho , n)$ . Indeed, we have the largest last passage time $\pi _{1,\ldots , 1} \le n$ , and $\mathrm {sh}(\pi ) \subseteq \rho $ since if $a_{\mathbf {i}}> 0$ , then $\mathbf {i} \in \rho $ .

Conversely, given $\pi \in \mathcal {P}(\rho , n)$ , to reconstruct the inverse map $\Phi ^{-1}$ , using the recurrence (1) we define the hypermatrix $A = (a_{\mathbf {i}})$ given by

(3) $$ \begin{align} a_{\mathbf{i}} = \pi_{\mathbf{i}} - \max_{\ell \in [d]} \pi_{\mathbf{i} + \mathbf{e}_\ell} \ge 0, \quad \mathbf{i} \in \mathbb{Z}^d_+. \end{align} $$

Let $G = (G_{\mathbf {i}}) = \Phi (A)$ . Let us check that $G = \pi $ and $A \in \mathcal {M}(\rho , n)$ . Since $\mathrm {sh}(\pi ) \subseteq \rho $ , we have $a_{\mathbf {i}} = 0$ for all $\mathbf {i} \not \in \rho $ (in particular, $A \in \mathcal {M}(\rho , \infty )$ ). Hence, $G_{\mathbf {i}} = \pi _{\mathbf {i}} = 0$ for all $\mathbf {i} \not \in \rho $ . Consider the directed graph $\Gamma $ on the vertex set $\rho $ and edges $\mathbf {i} \to \mathbf {i} + \mathbf {e}_{\ell }$ (when $\mathbf {i} + \mathbf {e}_{\ell } \in \rho $ ) for $\ell \in [d]$ . Then $\Gamma $ is acyclic (i.e., has no directed cycles). Notice that $a_{\mathbf {i}} = \pi _{\mathbf {i}} = G_{\mathbf {i}}$ if a vertex $\mathbf {i} \in \Gamma $ has no outgoing edges. Since $\Gamma $ is acyclic, we can sort its vertices in linear order $(\mathbf {i}^{(1)}, \ldots , \mathbf {i}^{(m)})$ so that the edges go only in one direction $\mathbf {i}^{(\ell )} \to \mathbf {i}^{(k)}$ for $\ell < k$ . We already noticed that $\pi _{\mathbf {i}^{(m)}} = G_{\mathbf {i}^{(m)}}$ . Then inductively on $\ell = m-1,\ldots , 1$ , we have

$$ \begin{align*}\pi_{\mathbf{i}^{(\ell)}} = a_{\mathbf{i}^{(\ell)}} + \max_{\mathbf{i}^{(\ell)} \to \mathbf{i}^{(k)}} \pi_{\mathbf{i}^{(k)}} = a_{\mathbf{i}^{(\ell)}} + \max_{\mathbf{i}^{(\ell)} \to \mathbf{i}^{(k)}} G_{\mathbf{i}^{(k)}} = G_{\mathbf{i}^{(\ell)}}. \end{align*} $$

Therefore, $\pi = G$ . In particular, $G_{1,\ldots , 1} \le n$ and hence, $A \in \mathcal {M}(\rho , n)$ .

3.3 Corners and the inverse map $\Phi ^{-1}$

Now we are going to describe the inverse map $\Phi ^{-1}$ more concretely using a structure of d-dimensional partitions.

Given a partition $\pi \in \mathcal {P}^{(d)}$ , define the set of corners as follows:

$$ \begin{align*}\mathrm{Cor}(\pi) := \left\{\mathbf{i} \in \mathbb{Z}^{d+1}_+ : \mathbf{i} \in D(\pi),\, \mathbf{i} + \mathbf{e}_{\ell} \not\in D(\pi) \text{ for all } \ell \in [d] \right\} \subseteq D(\pi). \end{align*} $$

(Here $\{\mathbf {e}_{\ell }\}$ is the standard basis in $\mathbb {Z}^{d+1}$ .) Let $\mathrm {cor}(\pi ) := |\mathrm {Cor}(\pi )|$ be the number of corners of $\pi $ . Define also the set of top corners as follows:

$$ \begin{align*}\mathrm{Cr}(\pi) := \left\{\mathbf{i} \in \mathbb{Z}^{d+1}_+ : \mathbf{i} \in D(\pi),\, \mathbf{i} + \mathbf{e}_{\ell} \not\in D(\pi) \text{ for all } \ell \in [d + 1] \right \}\, \subseteq\, \mathrm{Cor}(\pi). \end{align*} $$

Let $\mathrm {cr}(\pi ) := |\mathrm {Cr}(\pi )|$ be the number of top corners of $\pi $ . More intuitively, a top corner (resp. corner) of $\pi $ is an element removable from the diagram of (resp. the shape of) $\pi $ . Note that the set of top corners $\mathrm {Cr}(\pi )$ uniquely determines the partition $\pi $ .

Example 3.3. Let $d = 2$ and $\pi $ be the plane partition given in Figure 1. We then have

$$ \begin{align*} \mathrm{Cor}(\pi) &= \{(i,j,k) \in D(\pi) : (i+1,j,k), (i,j+1,k) \not\in D(\pi) \} \\ &= \{(1,1,4), (1,3,1), (1,3,2), (2,2,1), (2,2,2), (2,2,3) \}\\ \mathrm{Cr}(\pi) &= \{(i,j,k) \in D(\pi) : (i+1,j,k), (i,j+1,k),(i,j,k+1) \not\in D(\pi) \} \\ &= \{(1,1,4), (1,3,2), (2,2,3)\}, \end{align*} $$

where corners in Figure 1 correspond to local configurations and top corners correspond to the configurations .

Figure 1 A plane partition $\pi \in \mathcal {P}^{(2)}$ whose $\mathrm {sh}(\pi )$ is the diagram of the partition $(3,2)$ ; its boxed diagram presentation as a pile of cubes in $\mathbb {R}^3$ ; and boxes of this diagram which correspond to corners.

Consider the corner projection map $\varphi : \mathcal {P}^{(d)} \to \mathcal {M}^{(d)}$ given by $\pi \mapsto (a_{\mathbf {i}})$ , where

$$ \begin{align*} a_{\mathbf{i}} = |\{i_{d+1} : (\mathbf{i}, i_{d+1}) \in \mathrm{Cor}(\pi) \}|, \quad \mathbf{i} \in \mathbb{Z}^{d}_+. \end{align*} $$

Proof. The key observation is that $(\mathbf {i}, i_{d+1}) \in \mathrm {Cor}(\pi )$ if and only if $\pi _{\mathbf {i}} \ge i_{d+1}> \pi _{\mathbf {i} + \mathbf {e}_\ell }$ for all $\ell \in [d]$ . Hence,

$$ \begin{align*}a_{\mathbf{i}} = |\{i_{d+1} : (\mathbf{i}, i_{d+1}) \in \mathrm{Cor}(\pi) \}| = \pi_{\mathbf{i}} - \max_{\ell \in [d]} \pi_{\mathbf{i} + \mathbf{e}_{\ell}}, \quad \mathbf{i} \in \mathbb{Z}^{d}_+, \end{align*} $$

which gives $\Phi ^{-1} : \pi \mapsto (a_{\mathbf {i}})$ .

We will also use the following properties relating shapes of partitions and top corners.

4.1 Main formulas

Let $(x_{i_1,\ldots , i_d})$ be indeterminate variables.

Theorem 4.1. Let $\rho \subset \mathbb {Z}^d_+$ be a fixed shape of a d-dimensional partition. We have the following multivariate generating function identities:

(4) $$ \begin{align} \sum_{{\pi \in \mathcal{P}^{(d)}, \atop \, \mathrm{sh}(\pi) \subseteq \rho} }\, \prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x_{i_1,\ldots, i_d} &= \prod_{(i_1,\ldots, i_d) \in \rho} {\left(1 - x_{i_1,\ldots, i_d} \right)^{-1}} ,\qquad\qquad\qquad\ \ \qquad \end{align} $$

(5) $$ \begin{align} \sum_{ {\pi \in \mathcal{P}^{(d)},\atop \, \mathrm{sh}(\pi) = \rho} }\, \prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x_{i_1,\ldots, i_d} &= \prod_{(i_1,\ldots, i_d) \in \mathrm{Cr}(\rho)} x_{i_1,\ldots, i_d} \prod_{(i_1,\ldots, i_d) \in \rho} {\left(1 - x_{i_1,\ldots, i_d} \right)^{-1}}, \end{align} $$

where $\mathrm {Cr}(\rho ) := \mathrm {Cr}(\text {the partition whose diagram is } \rho )$ .

It is convenient to define weights of hypermatrices and partitions as follows. Given a hypermatrix $A = (a_{i_1,\ldots ,i_d})\in \mathcal {M}^{(d)}$ , we associate to it a multivariable monomial weight

$$ \begin{align*}w_A := \prod_{(i_1,\ldots,i_d) \in \mathbb{Z}^d_+} \left( x_{i_1,\ldots, i_d} \right)^{a_{i_1,\ldots,i_d}}.\end{align*} $$

Given a partition $\pi \in \mathcal {P}^{(d)}$ , we associate to it a multivariable monomial weight

$$ \begin{align*}w(\pi) := \prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x_{i_1,\ldots, i_d}. \end{align*} $$

The following lemma shows that the bijection $\Phi $ is weight-preserving.

Proof. Using the corner projection map $\varphi = \Phi ^{-1}$ , by Lemma 3.4, we have

$$ \begin{align*} w_A &= \prod_{(i_1,\ldots,i_d) \in \mathbb{Z}^d_+} \left(x_{i_1,\ldots, i_d} \right)^{a_{i_1,\ldots,i_d}} \\ &=\prod_{(i_1,\ldots,i_d) \in \mathbb{Z}^d_+} \left(x_{i_1,\ldots, i_d} \right)^{|\{i_{d+1} : (i_1,\ldots,i_d, i_{d+1}) \in \mathrm{Cor}(\pi) \}|} \\ &=\prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x_{i_1,\ldots, i_d} \\ &= w(\pi), \end{align*} $$

which gives what is needed.

Proof of Theorem 4.1.

First, note that

$$ \begin{align*} \sum_{A = (a_{\mathbf{i}}) \in \mathcal{M}(\rho, \infty)} w_A &= \sum_{A = (a_{\mathbf{i}}) \in \mathcal{M}(\rho, \infty)} \, \prod_{(i_1,\ldots,i_d) \in \rho} \left( x_{i_1,\ldots, i_d} \right)^{a_{i_1,\ldots,i_d}} \\ &= \prod_{(i_1,\ldots,i_d) \in \rho} \left(1 - x_{i_1,\ldots, i_d} \right)^{-1}. \end{align*} $$

On the other hand, using Corollary 3.2 (iii) and Lemma 4.2, we have

$$ \begin{align*}\sum_{A = (a_{\mathbf{i}}) \in \mathcal{M}(\rho, \infty)} w_A = \sum_{\pi \in \mathcal{P}(\rho, \infty)} w(\pi) = \sum_{\pi \in \mathcal{P}^{(d)},\, \mathrm{sh}(\pi) \subseteq \rho} \, \prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x_{i_1,\ldots, i_d} \end{align*} $$

and hence the identity (4) follows.

Let $\overline {\mathcal {M}}(\rho , \infty ) = \{A \in \mathcal {M}(\rho , \infty ) : \mathbf {i} \in \mathrm {Cr}(\rho ) \implies a_{\mathbf {i}}> 0\}$ . Similarly, note that

$$ \begin{align*} \sum_{A = (a_{\mathbf{i}}) \in \overline{\mathcal{M}}(\rho, \infty)} w_A &= \sum_{A = (a_{\mathbf{i}}) \in \mathcal{M}(\rho, \infty)} \, \prod_{(i_1,\ldots, i_d) \in \mathrm{Cr}(\rho)} x_{i_1,\ldots, i_d} \prod_{(i_1,\ldots,i_d) \in \rho} \left(x_{i_1,\ldots, i_d} \right)^{a_{i_1,\ldots,i_d}} \\ &= \prod_{(i_1,\ldots, i_d) \in \mathrm{Cr}(\rho)} x_{i_1,\ldots, i_d} \prod_{(i_1,\ldots,i_d) \in \rho} \left(1 - x_{i_1,\ldots, i_d} \right)^{-1}. \end{align*} $$

On the other hand, using Lemma 3.5 we have

$$ \begin{align*}\sum_{A = (a_{\mathbf{i}}) \in \overline{\mathcal{M}}(\rho, \infty)} w_A = \sum_{\pi \in \mathcal{P}(\rho, \infty), \mathrm{sh}(\pi) = \rho} w(\pi) = \sum_{\pi \in \mathcal{P}^{(d)},\, \mathrm{sh}(\pi) = \rho} \, \prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x_{i_1,\ldots, i_d} \end{align*} $$

and hence, the identity (5) follows.

4.2 Some special cases

Let us list a few immediate special cases of the above formulas.

Corollary 4.3 (Boxed case).

For any $n_1,\ldots , n_d \ge 0$ , we have

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{}(n_1,\ldots,n_d, \infty)}\, \prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x_{i_1,\ldots, i_d} &= \prod_{i_1 = 1}^{n_1} \cdots \prod_{i_d = 1}^{n_d} {\left(1 - x_{i_1,\ldots, i_d} \right)^{-1}}. \end{align*} $$

Corollary 4.4 (Solid partitions, $d = 3$ ).

Let $\rho $ be a plane partition. We have

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{(3)},\, \mathrm{sh}(\pi) \subseteq D(\rho)}\, \prod_{(i,j,k,\ell) \in \mathrm{Cor}(\pi)} x^{}_{i j k} &= \prod_{(i,j,k) \in D(\rho)} {\left(1 - x^{}_{i j k} \right)^{-1}}\\ \sum_{\pi \in \mathcal{P}^{(3)},\, \mathrm{sh}(\pi) = D(\rho)}\, \prod_{(i,j,k,\ell) \in \mathrm{Cor}(\pi)} x^{}_{i j k} &= \prod_{(i,j,k) \in D(\rho)} {\left(1 - x^{}_{i j k} \right)^{-1}} \prod_{(i, j, k) \in \mathrm{Cr}(\rho)} x^{}_{i j k}. \end{align*} $$

For $d = 2$ we obtain the following new identity for plane partitions.

Corollary 4.5 (Plane partitions, $d = 2$ ).

Let $\lambda $ be a partition. We have

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{(2)},\, \mathrm{sh}(\pi) \subseteq D(\lambda)}\, \prod_{(i,j,k) \in \mathrm{Cor}(\pi)} x^{}_{i j} &= \prod_{(i,j) \in D(\lambda)} {\left(1 - x^{}_{i j} \right)^{-1}}\\ \sum_{\pi \in \mathcal{P}^{(2)},\, \mathrm{sh}(\pi)= D(\lambda)}\, \prod_{(i,j,k) \in \mathrm{Cor}(\pi)} x^{}_{i j} &= \prod_{(i,j) \in D(\lambda)} {\left(1 - x^{}_{i j} \right)^{-1}} \prod_{(i, j) \in \mathrm{Cr}(\lambda)} x^{}_{i j}. \end{align*} $$

5.1 Corner-hook volume

Let $\pi \in \mathcal {P}^{(d)}$ be a d-dimensional partition. For each point $(i_1,\ldots ,i_d)$ , define the cohook length

$$ \begin{align*}\mathrm{ch}(i_1,\ldots,i_d) := i_1 + \ldots + i_d - d + 1. \end{align*} $$

Define now the corner-hook volume statistic $|\cdot |_{ch} : \mathcal {P}^{(d)} \to \mathbb {N}$ , computed as follows:

$$ \begin{align*}|\pi|_{ch} := \sum_{(\mathbf{i}, i_{d+1})\, \in\, \mathrm{Cor}(\pi)} \mathrm{ch}(\mathbf{i}). \end{align*} $$

Example 5.1. Let $d = 2$ and $\pi $ be the plane partition given in Figure 1. Recall that

$$ \begin{align*} \mathrm{Cor}(\pi) &= \{(i,j,k) \in D(\pi) : (i+1,j,k), (i,j+1,k) \not\in D(\pi) \} \\ &= \{(1,1,4), (1,3,1), (1,3,2), (2,2,1), (2,2,2), (2,2,3) \}, \end{align*} $$

and hence, we have

$$ \begin{align*}|\pi|_{ch} = (1 + 1 - 1) + (1 + 3 - 1) + (1 + 3 - 1) + (2 + 2 - 1) + (2 + 2 - 1) + (2 + 2 - 1) = 16. \end{align*} $$

Theorem 5.2. Let $\rho \subset \mathbb {Z}^d_+$ be a fixed shape of a d-dimensional partition. We have the following generating functions:

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{(d)},\, \mathrm{sh}(\pi) \subseteq \rho} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= \prod_{(i_1,\ldots,i_d) \in \rho} {\left( 1 - t q^{i_1 + \cdots + i_d - d + 1} \right)^{-1}},\\ \sum_{\pi \in \mathcal{P}^{(d)},\, \mathrm{sh}(\pi) = \rho} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= t^{\mathrm{cr}(\rho)} q^{|\rho|_{cr}} \prod_{(i_1,\ldots,i_d) \in \rho} {\left( 1 - t q^{i_1 + \cdots + i_d - d + 1} \right)^{-1}}, \end{align*} $$

where

$$ \begin{align*}|\rho|_{cr} := \sum_{(i_1,\ldots, i_d) \in \mathrm{Cr}(\rho)} \mathrm{ch}(i_1,\ldots,i_d). \end{align*} $$

Corollary 5.3 (Boxed version).

We have

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{}(n_1,\ldots, n_d, \infty)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= \prod_{i_1 = 1}^{n_1} \cdots \prod_{i_d = 1}^{n_d} {\left( 1 - t q^{i_1 + \cdots + i_d - d + 1} \right)^{-1}}. \end{align*} $$

Corollary 5.4 (Full generating function).

We have

$$ \begin{align*}\sum_{\pi \in \mathcal{P}^{(d)}} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} = \prod_{n \ge 1} (1 - t q^{n})^{-\binom{n + d - 2}{d - 1}}. \end{align*} $$

Corollary 5.5 (Interpretation of MacMahon’s numbers).

We have

$$ \begin{align*}\sum_{\pi \in \mathcal{P}^{(d)}} q^{|\pi|_{ch}} = \prod_{n \ge 1} (1 - q^{n})^{-\binom{n + d - 2}{d - 1}} = \sum_{n = 0}^{\infty} m_d(n) q^n, \end{align*} $$

and hence,

$$ \begin{align*}m_d(n) = |\{ \pi \in \mathcal{P}^{(d)} : |\pi|_{ch} = n \}|\end{align*} $$

(i.e., $m_d(n)$ is the number of d-dimensional partitions whose corner-hook volume is n).

Corollary 5.6 (Pyramid partitions).

Let $\Delta _d({m})$ be a d-dimensional partition whose diagram is $D(\Delta _{d}({m})) = \{(i_1,\ldots , i_{d+1}) : \mathbb {Z}^{d+1}_+ : i_1 + \cdots + i_{d+1} - d \le m\}$ . We have

$$ \begin{align*}\sum_{\pi \in \mathcal{P}^{}(D(\Delta_{d-1}({m})), \infty)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} = \prod_{n = 1}^m (1 - t q^{n})^{-\binom{n + d - 2}{d - 1}}. \end{align*} $$

Corollary 5.7 ( $q = 1$ specialization).

We have

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{(d)},\, \mathrm{sh}(\pi) \subseteq \rho} t^{\mathrm{cor}(\pi)} &= {\left( 1 - t \right)^{-|\rho|}},\\ \sum_{\pi \in \mathcal{P}^{(d)},\, \mathrm{sh}(\pi) = \rho} t^{\mathrm{cor}(\pi)} &= t^{\mathrm{cr}(\rho)} {\left( 1 - t \right)^{-|\rho|}}. \end{align*} $$

Then the number of $\pi \in \mathcal {P}^{(d)}$ of shape $\rho $ with k corners is equal to $\binom {k - \mathrm {cr}(\rho ) + |\rho | - 1}{|\rho | - 1}$ .

5.2 Solid partitions, $d = 3$

Let us restate some of these results for solid partitions. Let $\pi \in \mathcal {P}^{(3)}$ be a solid partition. We then have

$$ \begin{align*}|\pi|_{ch} = \sum_{(i,j,k,\ell)\in \mathrm{Cor}(\pi)} (i + j + k - 2). \end{align*} $$

Corollary 5.8. Let $\rho $ be a fixed plane partition. We have

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{(3)},\, \mathrm{sh}(\pi) \subseteq D(\rho)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= \prod_{(i, j, k) \in D(\rho)} {\left( 1 - t q^{i + j + k - 2} \right)^{-1}},\\ \sum_{\pi \in \mathcal{P}^{(3)},\, \mathrm{sh}(\pi) = D(\rho)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= t^{\mathrm{cr}(\rho)} q^{|\rho|_{cr}} \prod_{(i, j, k) \in D(\rho)} {\left( 1 - t q^{i + j + k - 2} \right)^{-1}}, \end{align*} $$

and, in particular, the boxed version

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{}(n_1,n_2, n_3, \infty)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= \prod_{i = 1}^{n_1} \prod_{j = 1}^{n_2} \prod_{k = 1}^{n_3} {\left( 1 - t q^{i + j + k - 2} \right)^{-1}}. \end{align*} $$

5.3 Plane partitions, $d = 2$

Similarly, let us restate some of these results for plane partitions. Let $\pi \in \mathcal {P}^{(2)}$ be a plane partition. We then have

$$ \begin{align*}|\pi|_{ch} = \sum_{(i,j,k)\in \mathrm{Cor}(\pi)} (i + j - 1). \end{align*} $$

Corollary 5.9. Let $\lambda $ be a fixed partition. We have

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{(2)},\, \mathrm{sh}(\pi) \subseteq D(\lambda)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= \prod_{(i, j) \in D(\lambda)} {\left( 1 - t q^{i + j - 1} \right)^{-1}},\\ \sum_{\pi \in \mathcal{P}^{(2)},\, \mathrm{sh}(\pi) = D(\lambda)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= t^{\mathrm{cr}(\lambda)} q^{|\lambda|_{cr}} \prod_{(i, j) \in D(\lambda)} {\left( 1 - t q^{i + j - 1} \right)^{-1}}, \end{align*} $$

and, in particular, the boxed version

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{}(n_1,n_2, \infty)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} &= \prod_{i = 1}^{n_1} \prod_{j = 1}^{n_2} {\left( 1 - t q^{i + j - 1} \right)^{-1}}. \end{align*} $$

Let us compare the last boxed formula with known results. The following trace generating function is known for plane partitions (see, for example, [Reference StanleySta99, Thm 7.20.1]):

$$ \begin{align*}\prod_{i = 1}^{n_1} \prod_{j = 1}^{n_2} {\left( 1 - t q^{i + j - 1} \right)^{-1}} = \sum_{\pi \in \mathcal{P}^{}(n_1,n_2, \infty)} t^{\mathrm{tr}(\pi)} q^{|\pi|}, \end{align*} $$

where $\mathrm {tr}(\pi ) := \sum _{i} \pi _{i,i}$ is the trace of a plane partition. Therefore, in this case, we actually have the following equidistribution result.

Theorem 5.10 (Equidistribution of (tr, vol) and (cor, ch-vol) for plane partitions).

We have

$$ \begin{align*}\sum_{\pi \in \mathcal{P}^{}(n_1,n_2, \infty)} t^{\mathrm{cor}(\pi)} q^{|\pi|_{ch}} = \sum_{\pi \in \mathcal{P}^{}(n_1,n_2,\infty)} t^{\mathrm{tr}(\pi)} q^{|\pi|}. \end{align*} $$

Remark 5. The formulas in Theorem 5.2 can be viewed as higher-dimensional analogues of the well-known formula

$$ \begin{align*}\sum_{\mathrm{sh}(\pi) = \lambda} q^{|\pi|} = \prod_{(i,j) \in D(\lambda)} \left(1 - q^{h_{\lambda}(i,j)} \right)^{-1}, \end{align*} $$

where $\lambda $ is a (usual) partition, $h_{\lambda }(i,j) := \lambda _i - i + \lambda ^{\prime }_j - j + 1$ are hook lengths and the sum runs over weak reverse plane partitions $\pi $ ; see [Reference StanleySta99, Ch. 7.22]. Its combinatorial proof is known as the Hillman–Grassl correspondence [Reference Hillman and GrasslHG76]. Now, setting $x_{i j} = q^{h_{\lambda }(i,j)}$ in Corollary 4.5 gives

$$ \begin{align*}\sum_{\mathrm{sh}(\pi) \subseteq \lambda} q^{|\pi|_h} = \prod_{(i,j) \in D(\lambda)} \left(1 - q^{h_{\lambda}(i,j)} \right)^{-1}, \end{align*} $$

where the sum runs over plane partitions $\pi $ and

$$ \begin{align*}|\pi|_h := \sum_{(i,j,k) \in \mathrm{Cor}(\pi)} h_{\lambda}(i,j). \end{align*} $$

These formulas give another interesting equidistribution result.

5.4 Other statistics

Theorem 4.1 is a source for many statistics over d-dimensional partitions, whose generating functions can be computed explicitly by taking appropriate specializations. For instance, another interesting statistic $|\cdot |_{c} : \mathcal {P}^{(d)} \to \mathbb {N}$ is given by

$$ \begin{align*}|\pi|_c := \sum_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} i_1, \quad \pi \in \mathcal{P}^{(d)}. \end{align*} $$

Then via the substitution $x_{i_1,\ldots , i_d} = q^{i_1}$ we obtain the following generating function:

$$ \begin{align*} \sum_{\pi \in \mathcal{P}^{}(n_1,\ldots, n_{d}, \infty)} q^{|\pi|_c} = \prod_{i = 1}^{n_1} (1 - q^{i})^{-n_2\cdots n_d}. \end{align*} $$

Another curious statistic is given by

$$ \begin{align*}|\pi|_p := \sum_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} (i_1 + 2 \, i_2 + \ldots + d \, i_d), \quad \pi \in \mathcal{P}^{(d)} \end{align*} $$

for which via the substitution $x_{i_1,\ldots , i_d} = q^{i_1 + 2 i_2 + \ldots + d i_d}$ we obtain the following generating function:

$$ \begin{align*}\sum_{\pi \in \mathcal{P}^{(d)}} q^{|\pi|_p} = \prod_{n = 1}^{\infty} (1 - q^{n})^{-p(n, d)}, \end{align*} $$

where $p(n, d)$ is the number of integer partitions of n into d distinct parts.

6.1 Definitions

Let $\pi $ be a d-dimensional partition. Define the set

$$ \begin{align*}\mathrm{sh}_1(\pi) := \{(i_2,\ldots, i_{d+1}) : (i_1,\ldots, i_{d+1}) \in D(\pi) \} = \{(i_2,\ldots, i_{d+1}) : (1, i_2, \ldots, i_{d+1}) \in D(\pi) \}, \end{align*} $$

which can be viewed as a shape of $\pi $ with respect to the first coordinate. Note that if $\pi \in \mathcal {P}(n_1,\ldots , n_{d+1})$ , then $\mathrm {sh}_1(\pi )$ is a diagram of $(d-1)$ -dimensional partition from $\mathcal {P}(n_2,\ldots , n_{d+1}).$ Concretely, $\mathrm {sh}_1(\pi )$ is the diagram of the partition $(\pi _{1,i_2,\ldots ,i_d})$ . For example, if $\pi $ is the plane partition in Figure 1, then $\mathrm {sh}_1(\pi )$ corresponds to the partition $(4,3,2)$ , which is the first row of $\pi $ .

Throughout this section, let us assume that $n_1, \ldots , n_d$ are fixed and we have the sets of variables

$$ \begin{align*}\mathbf{x}^{(i)} := (x^{(i)}_1,\ldots, x^{(i)}_{n_i}), \quad i \in [d].\end{align*} $$

Definition 6.1. Let $\rho $ be a $(d-1)$ -dimensional partition from the set $\mathcal {P}(n_2,\ldots ,n_{d+1})$ . Define the d-dimensional Grothendieck polynomials in d sets of variables as follows:

(6) $$ \begin{align} g_{\rho}(\mathbf{x}^{(1)}; \ldots; \mathbf{x}^{(d)}) := \sum_{\pi\, :\, \mathrm{sh}_1(\pi) = \rho}\, \prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x^{(1)}_{i_1} \cdots x^{(d)}_{i_d}, \end{align} $$

where the sum runs over d-dimensional partitions $\pi \in \mathcal {P}(n_1,\ldots , n_{d+1})$ with $\mathrm {sh}_1(\pi ) = \rho $ (here $\rho $ is identified with its diagram).

In the specialization $x^{(k)}_{i} = 1$ for all $k \ge 2$ , we simply denote these polynomials by $g_{\rho }(\mathbf {x}) = g_{\rho }(x_1,x_2,\ldots )$ in one set of variables $\mathbf {x}^{(1)} = \mathbf {x} = (x_1,\ldots , x_{n_1})$ so that

(7) $$ \begin{align} g_{\rho}(\mathbf{x}^{}) = \sum_{\pi\, :\, \mathrm{sh}_1(\pi) = \rho}\, \prod_{i = 1}^{n_1} x_i^{c_i(\pi)}, \text{ where } c_i(\pi) := |\{\mathbf{i} : (i, \mathbf{i}) \in \mathrm{Cor}(\pi) \}| \end{align} $$

and the sum runs over $\pi \in \mathcal {P}(n_1,\ldots , n_{d+1})$ .

6.2 Examples

Example 6.2. Consider the case $d = 2$ . Let $\lambda \in \mathcal {P}(n_2, n_3)$ be a partition and $\mathbf {x}^{(1)} = \mathbf {x}, \mathbf {x}^{(2)} = \mathbf {y}$ . Then (7) becomes

$$ \begin{align*}g_{\lambda}(\mathbf{x}) = \sum_{\pi\, :\, \mathrm{sh}_1(\pi) = \lambda} \prod_{i = 1}^{n_1} x_i^{c_i(\pi)}, \text{ where } c_i(\pi) = |\{(j,k) : (i, j, k) \in \mathrm{Cor}(\pi) \}|, \end{align*} $$

and the sum runs over plane partitions $\pi \in \mathcal {P}(n_1,n_2,n_3)$ . Let us transpose $\pi $ to $\pi '$ via cyclic shift of the diagram so that $(i,j,k) \in D(\pi ) \iff (j,k, i) \in D(\pi ')$ . Note that $\mathrm {sh}_1(\pi ) = \mathrm {sh}(\pi ') = \lambda $ and $c_i(\pi )$ is equal to the number of columns of $\pi '$ (viewed as a 2d array) containing the entry $i \in [n_1]$ ; see Figure 2. This shows that $\{g_{\lambda }(\mathbf {x})\}$ are the dual stable Grothendieck polynomials introducedFootnote ³ in [Reference Lam and PylyavskyyLP07], but phrased in a slightly different yet equivalent form.

Figure 2 A plane partition $\pi $ and its transpose $\pi '$ of the shape $\lambda = (432)$ with the corresponding diagrams in which the corner boxes of $\pi $ are highlighted.

More generally, (6) becomes

$$ \begin{align*}g_{\lambda}(\mathbf{x}; \mathbf{y}) = \sum_{\pi\, :\, \mathrm{sh}_1(\pi) = \lambda} \prod_{(i,j,k) \in \mathrm{Cor}(\pi)} x_i y_j, \end{align*} $$

which by rescaling $\tilde g_{\lambda }(\mathbf {x}; \mathbf {y}) = \mathbf {y}^{\lambda } g_{\lambda }(\mathbf {x}; \mathbf {y}^{-1})$ gives the refined version of dual stable Grothendieck polynomials introduced in [Reference Galashin, Grinberg and LiuGGL16], where it was shown that these polynomials are symmetric in the variables $\mathbf {x}$ .

Example 6.3. Let $d = 3$ , $(n_1,n_2,n_3,n_4) = (3,2,2,2)$ and $\mathbf {x}^{(1)} = \mathbf {x} = (x_1,x_2,x_3)$ , $\mathbf {x}^{(2)} = \mathbf {y} = (y_1,y_2)$ , $\mathbf {x}^{(3)} = \mathbf {z} = (z_1,z_2)$ . Note that in this case, $3$ -dimensional Grothendieck polynomials are indexed by plane partitions and defined as sums over solid partitions. Consider a few examples.

(a) Let . Then we have

$$ \begin{align*} g_\rho(\mathbf{x};\mathbf{y};\mathbf{z}) = (x_{1}^{2}x_{2} &+ x_{1}^{2}x_{3} + x_{1}x_{2}^{2} + x_{1}x_{3}^{2} + x_{2}^{2}x_{3} + x_{2}x_{3}^{2} + 2 x_{1}x_{2}x_{3}) \cdot y_{1}^{3}z_{1}^{2}z_{2}\\ &+ (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3}) \cdot y_{1}^{2}z_{1}z_{2}, \end{align*} $$

which coincides with the ordinary dual stable Grothendieck polynomial indexed by the partition $\lambda = (2,1)$ (i.e., in this case, we have $g_{\lambda }(\mathbf {x}) = g_\rho (\mathbf {x}, \mathbf {1}, \mathbf {1})$ ).

(b) Let . Then we have

$$ \begin{align*} g_{\rho}(\mathbf{x}; \mathbf{y}; \mathbf{z}) = (x_{1}^{2}x_{2} &+ x_{1}^{2}x_{3} + x_{2}^{2}x_{3}) \cdot y_{1}^{2}y_{2}z_{1}^{2}z_{2} + 2 x_{1}x_{2}x_{3} \cdot y_{1}^{2}y_{2}z_{1}^{2}z_{2} \\ & + (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + 2 x_{1}x_{2} + 2 x_{1}x_{3} + 2 x_{2}x_{3}) \cdot y_{1}y_{2}z_{1}z_{2}, \end{align*} $$

and in particular,

$$ \begin{align*} g_{\rho}(\mathbf{x}) = x_{1}^{2}x_{2} + x_{1}^{2}x_{3} + x_{2}^{2}x_{3} + 2 x_{1}x_{2}x_{3} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + 2 (x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3}). \end{align*} $$

(c) Let . Then we have

$$ \begin{align*} g_{\rho}(\mathbf{x},\mathbf{y},\mathbf{z}) &= (3 x_{1}^{2}x_{2}x_{3} + 3 x_{1}x_{2}^{2}x_{3} + 2 x_{1}x_{2}x_{3}^{2} + x_{1}^{2}x_{2}^{2} + x_{1}^{2}x_{3}^{2} + x_{2}^{2}x_{3}^{2} + x_{1}^{3}x_{2} + x_{1}^{3}x_{3} + x_{2}^{3}x_{3})\cdot y_{1}^{3}y_{2}z_{1}^{3}z_{2}\\ &+ (4 x_{1}x_{2}x_{3} + 2 x_{1}^{2}x_{2} + 2 x_{1}^{2}x_{3} + 2 x_{2}^{2}x_{3} + 3 x_{1}x_{2}^{2} + 3 x_{1}x_{3}^{2} + 3 x_{2}x_{3}^{2} + x_{1}^{3} + x_{2}^{3} + x_{3}^{3}) \cdot y_{1}^{2}y_{2}z_{1}^{2}z_{2}. \end{align*} $$

Figure 3 illustrates a few examples of solid partitions contributing to the last expansion.

Figure 3 Each picture here represents a solid partition as a filling of a diagram of some plane partition with numbers written on top of each box (to make entries of inner boxes visible, some facets are removed). On the left, we have $\mathrm {sh}_{1}\!(\pi ) = \rho $ . The next two are solid partitions $\pi ^{(1)}$ and $\pi ^{(2)}$ represented as fillings of diagrams of plane partitions and ; each has the weight $w(\pi ^{(i)}) = x_2^2x_3 \cdot y_1^2 y_2 z_1^2 z_2$ ; and both have the same $\mathrm {sh}_1(\pi ^{(i)}) =\rho $ ( $i = 1,2$ ) displayed on the left.

6.3 Properties

We now prove some properties of d-dimensional Grothendieck polynomials.

Theorem 6.4 (Cauchy–Littlewood-type identity).

Let $\eta \in \mathcal {P}(n_2,\ldots , n_d)$ be a $(d-2)$ -dimensional partition. Let $n \times \eta $ be a $(d-1)$ -dimensional partition with the diagram $D(n \times \eta ) = \{(i, \mathbf {i}) : i \in [n], \mathbf {i} \in D(\eta ) \}$ . Then we have the following generating series:

$$ \begin{align*}\sum_{\rho \in \mathcal{P}(\eta, \infty)} g_{\rho}(\mathbf{x}^{(1)}; \ldots; \mathbf{x}^{(d)}) = \prod_{(i_1,\ldots, i_d) \in D(n_1 \times \eta)} \left(1 - x^{(1)}_{i_1} \cdots x^{(d)}_{i_d} \right)^{-1}. \end{align*} $$

Proof. Notice that we have

$$ \begin{align*}\sum_{\rho \in \mathcal{P}(\eta, \infty)} g_{\rho}(\mathbf{x}^{(1)}; \ldots; \mathbf{x}^{(d)}) = \sum_{\rho \in \mathcal{P}(\eta, \infty)} \sum_{\mathrm{sh}_1(\pi) = \rho} w(\pi) = \sum_{\pi \in \mathcal{P}(n_1 \times \eta, \infty)} w(\pi). \end{align*} $$

On the other hand, from Theorem 4.1, we have

$$ \begin{align*} \sum_{\pi \in \mathcal{P}(n_1 \times \eta, \infty)} w(\pi) = \prod_{(i_1,\ldots, i_d) \in D(n_1 \times \eta)} \left(1 - x^{(1)}_{i_1} \cdots x^{(d)}_{i_d} \right)^{-1} \end{align*} $$

which gives the result.

Corollary 6.5. We have

$$ \begin{align*}\sum_{\rho \in \mathcal{P}(n_2,\ldots, n_d, \infty)} g_{\rho}(\mathbf{x}^{(1)}; \ldots; \mathbf{x}^{(d)}) = \prod_{i_1 = 1}^{n_1} \cdots \prod_{i_d = 1}^{n_d} \left(1 - x^{(1)}_{i_1} \cdots x^{(d)}_{i_d} \right)^{-1}. \end{align*} $$

Lemma 6.6 (Simple branching rule).

We have

$$ \begin{align*}g_{\pi}(1, x_1,\ldots, x_n) = \sum_{\rho \subseteq \pi} g_{\rho}(x_1,\ldots, x_n). \end{align*} $$

Proof. Given a d-dimensional partition $\tau $ with $\mathrm {sh}_1(\tau ) = \pi $ , it contributes to the l.h.s. the weight $\prod _{i = 1}^{n} x_i^{c_{i+1}(\tau )}$ (see Equation (7)). Let us form the new partition $\eta \subseteq \pi $ with the diagram

$$ \begin{align*}\{(i, \mathbf{i}) : (i+1, \mathbf{i}) \in D(\tau) \}\end{align*} $$

so that $\prod _{i = 1}^{n} x_i^{c_{i+1}(\tau )} = \prod _{i = 1}^{n} x_i^{c_{i}(\eta )}$ , which contributes to the r.h.s. In other words, remove from $D(\tau )$ the points with the first coordinate $1$ , then decrease by $1$ the first coordinates for the remaining points. It is not difficult to see that this defines a proper weight-preserving bijection between both sides of the equation.

Denote $1^k = (1,\ldots , 1)$ with k ones.

Proposition 6.7 (Boxed specialization).

We have

$$ \begin{align*}g_{[n_2] \times \cdots \times [n_{d+1}]}(1^{n_1 + 1}) = g_{[n_2] \times \cdots \times [n_{d+1}]}(1^{n_1 + 1}; 1^{n_2}; \ldots; 1^{n_d}) = |\mathcal{P}(n_1,\ldots, n_{d+1})|. \end{align*} $$

Proof. Denote $B = [n_2] \times \cdots \times [n_{d+1}]$ . Let $\rho $ be a partition diagram inside B. From the definition of g, we immediately obtain that

$$ \begin{align*}g_{\rho}(1^{n_1}; \ldots; 1^{n_d}) = |\{\pi \in \mathcal{P}(n_1,\ldots, n_{d+1})\, :\, \mathrm{sh}_1(\pi) = \rho \}|.\end{align*} $$

Therefore, using the branching formula above, we get

$$ \begin{align*} g_{B}(1^{n_1 + 1}; 1^{n_2}; \ldots; 1^{n_d}) &= \sum_{\rho \subseteq B} g_{\rho}(1^{n_1}; 1^{n_2}; \ldots; 1^{n_d}) \\ &= \sum_{\rho \subseteq B} |\{\pi \in \mathcal{P}(n_1,\ldots, n_{d+1})\, :\, \mathrm{sh}_1(\pi) = \rho \}| \\ &=|\mathcal{P}(n_1,\ldots, n_{d+1})|, \end{align*} $$

which gives what is needed.

6.4 Quasisymmetry

It is known that the dual stable Grothendieck polynomials $g_{\lambda }(\mathbf {x})$ are symmetric in $\mathbf {x}$ (in the case $d = 2$ ). As Example 6.3 shows, the generalized polynomials $g_{\rho }$ are not necessarily symmetric for $d \ge 3$ . However, as we show in this subsection, these polynomials are always quasisymmetric.

Definition 6.8. A polynomial $f \in \mathbb {Z}[x_1,\ldots , x_n]$ is called quasisymmetric if for all $1 \le \ell _1 < \cdots < \ell _k \le n$ , $1 \le j_1 < \cdots < j_k \le n$ , and $a_1,\ldots , a_k \in \mathbb {Z}_+$ , we have

$$ \begin{align*}[x^{a_1}_{\ell_1} \cdots x^{a_{k}}_{\ell_k}]\, f = [x^{a_1}_{j_1} \cdots x^{a_{k}}_{j_k}]\, f, \end{align*} $$

where $[\mathbf {x}^{\alpha }] f$ denotes the coefficient of the monomial $\mathbf {x}^{\alpha }$ in f.

Proof. To simplify notation, let us denote $\mathbf {x}^{(1)} = \mathbf {x} = (x_1, x_2,\ldots )$ . We need to show that for all $a_1,\ldots , a_k \in \mathbb {Z}_+$ , $1 \le \ell _1 < \cdots < \ell _k \le n_1$ , $1 \le j_1 < \cdots < j_k \le n_1$ , we have

$$ \begin{align*}[x^{a_1}_{\ell_1} \cdots x^{a_{k}}_{\ell_k}]\, g_{\rho} = [x^{a_1}_{j_1} \cdots x^{a_{k}}_{j_k}]\, g_{\rho} \end{align*} $$

(where the indeterminates $\mathbf {x}^{(2)}, \ldots , \mathbf {x}^{(d)}$ are regarded as constants). Let L and R be the sets of d-dimensional partitions which contribute to the l.h.s. and r.h.s., respectively. We are going to construct a weight-preserving bijection $\phi : L \to R$ .

Let $\pi \in L$ for which we have $\pi \in \mathcal {P}(n_1,\ldots ,n_{d+1})$ with $\mathrm {sh}_1(\pi ) = D(\rho )$ and $w(\pi ) = x^{a_1}_{\ell _1} \cdots x^{a_{k}}_{\ell _k} \times w'$ , where $w'$ is the remaining product which does not contain the variables $\mathbf {x}$ .

For a hypermatrix $X = ({x}_{\mathbf {i}})_{\mathbf {i} \in \mathbb {Z}^{d}_+}$ , define the slices $X^{(\ell )} = (x_{\ell ,\mathbf {i}})_{\mathbf {i} \in \mathbb {Z}^{d-1}_+}$ . Let $|X|$ denote the sum of the entries of X.

Let $A = (a_{\mathbf {i}}) = \Phi ^{-1}(\pi ) \in \mathcal {M}(n_1,\ldots , n_d)$ . Note that $A^{(\ell )} \in \mathcal {M}(n_2,\ldots , n_d)$ for $\ell \in [n_1]$ . Since $\Phi $ preserves weights (i.e., $w_A = w(\pi )$ ; see Lemma 4.2), we must have $A^{(\ell )} \ne \mathbf {0}$ iff $\ell \in \{\ell _1,\ldots , \ell _k \}$ . We then have

$$ \begin{align*}w(\pi) = w_A = \prod_{i = 1}^{n_1} x_i^{|A^{(i)}|} \prod_{\mathbf{i} = (i_2,\ldots, i_d)} (x^{(2)}_{i_2} \cdots x^{(d)}_{i_d})^{a_{i,\mathbf{i}}} = \prod_{i = 1}^{k} x_{\ell_i}^{a_i} \times w'. \end{align*} $$

Let us now construct another hypermatrix $B = (b_{\mathbf {i}}) \in \mathcal {M}(n_1,\ldots , n_d)$ so that $B^{(j)} \ne \mathbf {0}$ iff ${j \in \{j_1,\ldots ,j_k\}}$ and $B^{(j_i)} = A^{(\ell _i)}$ for all $i \in [k]$ . Let $\pi ' = \Phi (B)$ . We then clearly have

$$ \begin{align*}w(\pi') = w_B = \prod_{i = 1}^{k} x_{j_i}^{a_i} \times w'. \end{align*} $$

Let us show that $\mathrm {sh}_1(\pi ') = D(\rho ) = \mathrm {sh}_1(\pi )$ . Recall that $\mathrm {sh}_1(\pi ')$ is the diagram of the partition $(\pi ^{\prime }_{1,i_2,\ldots ,i_d})$ . By definition of $\Phi $ , each entry $\pi ^{\prime }_{1,i_2,\ldots ,i_d} $ is the largest weight of a directed path from $(1,i_2,\ldots , i_d)$ to $(n_1,\ldots , n_d)$ through the hypermatrix B. This holds as the maximum of $\sum _{\mathbf {j} \in \Pi } b_{\mathbf {j}}$ among all paths $\Pi : (1,i_2,\ldots , i_d) \to \infty ^d$ is achieved for some path that passes through $(n_1,\ldots , n_d)$ , as any other path could be redirected to $(n_1,\ldots , n_d)$ from the point where it first leaves the box $[n_1] \times \cdots \times [n_d]$ without lowering $\sum _{\mathbf {j} \in \Pi } b_{\mathbf {j}}$ . Similarly, each entry $\pi _{1,i_2,\ldots ,i_d} $ is the largest weight of a directed path from $(1,i_2,\ldots , i_d)$ to $(n_1,\ldots , n_d)$ through the hypermatrix A. In addition, note that when taking the maximum over lattice paths, we can ‘skip’ zero slices $A^{(\ell )} = \mathbf {0}$ . We then have

$$ \begin{align*} \pi_{1,i_2,\ldots,i_d} &= \max_{\Pi : (1,i_2,\ldots, i_d) \to (n_1,\ldots, n_d)} \sum_{(\ell,\mathbf{i}) \in \Pi, \, \ell \in \{\ell_1,\ldots, \ell_k \}} a_{(\ell, \mathbf{i}) } \\ &= \max_{\Pi : (1,i_2,\ldots, i_d) \to (n_1,\ldots, n_d)} \sum_{(j,\mathbf{i}) \in \Pi, \, j \in \{j_1,\ldots, j_k \}} b_{(j, \mathbf{i}) } \\ &= \pi^{\prime}_{1,i_2,\ldots,i_d}. \end{align*} $$

Hence $\pi ' \in R$ , we can set $\phi : \pi \mapsto \pi '$ and it is a well-defined bijection between L and R.

Note also that d-dimensional Grothendieck polynomials satisfy the stability: $g_{\rho }(\mathbf {x}^{(1)}; \ldots ;\mathbf {x}^{(d)})$ does not change for $n_1 \to n_1 + 1$ and $x^{(1)}_{n_1+1} = 0$ (i.e., if we add an extra $0$ at the end of $\mathbf {x}^{(1)}$ ). Therefore, $g_{\rho }(\mathbf {x}^{(1)}; \ldots ;\mathbf {x}^{(d)})$ can be treated as a quasisymmetric function in infinitely many variables $\mathbf {x}^{(1)}$ (The stability is just the projective limit of quasisymmetric functions).

Let us define the boxed polynomials

$$ \begin{align*}F_{(n_1,\ldots, n_{d+ 1})}(\mathbf{x}^{(1)}; \ldots; \mathbf{x}^{(d)}) := \sum_{\pi \in \mathcal{P}(n_1,\ldots, n_{d+1})} \prod_{(i_1,\ldots, i_{d+1}) \in \mathrm{Cor}(\pi)} x^{(1)}_{i_1} \cdots x^{(d)}_{i_d}, \end{align*} $$

which are bounded versions of the Cauchy product as by Corollary 4.3; we have

$$ \begin{align*}\lim_{n_{d+1} \to \infty} F_{(n_1,\ldots, n_{d+ 1})}(\mathbf{x}^{(1)}; \ldots; \mathbf{x}^{(d)}) = \prod_{i_1 = 1}^{n_1} \cdots \prod_{i_d = 1}^{n_d} \left(1 - x^{(1)}_{i_1} \cdots x^{(d)}_{i_d} \right)^{-1}. \end{align*} $$

These polynomials can also be expanded as follows:

$$ \begin{align*}F_{(n_1,\ldots, n_{d+ 1})}(\mathbf{x}^{(1)}; \ldots; \mathbf{x}^{(d)}) = \sum_{\rho \in \mathcal{P}(n_2,\ldots,n_{d+1})} \sum_{\mathrm{sh}_{1}(\pi) = \rho} w(\pi) = \sum_{\rho \in \mathcal{P}(n_2,\ldots,n_{d+1})} g_{\rho}(\mathbf{x}^{(1)}; \ldots; \mathbf{x}^{(d)}). \end{align*} $$

Definition 6.11. Let $A = (a_{i_1,\ldots ,i_d})\in \mathcal {M}^{}(n_1,\ldots , n_d)$ . For each $\ell \in [d]$ , consider the hypermatrices $B^{(\ell )}_{i} = (a_{i_1,\ldots ,i_d})_{i_{\ell } = i}$ (i.e., slices of A with fixed $\ell $ -th coordinate). Define the vectors

$$ \begin{align*}s_{\ell}(A) := (|B^{(\ell)}_1|, |B^{(\ell)}_2|, \ldots),\end{align*} $$

where $|B|$ denotes the sum of entries of B. For example, if $d = 2$ , then $s_{1}(A)$ is the vector of row sums of A, and $s_{2}(A)$ is the column sums of A. Let us also say that A is a packed hypermatrix if for each $\ell \in [d]$ , the sequence $s_{\ell }(A)$ does not contain zeros between its positive entries. Denote by $\mathrm {pack}(A)$ the packed hypermatrix formed from A by removing its zero slices $B^{(\ell )}_i = \mathbf {0}$ .

Note that for any $A \in \mathcal {M}(n_1,\ldots , n_d)$ , we have

$$ \begin{align*}w_A = \prod_{\ell = 1}^d \left(\mathbf{x}^{(\ell)} \right)^{s_{\ell}(A)}, \end{align*} $$

where for a set of variables $\mathbf {x} = (x_1,x_2,\ldots )$ and a vector $s = (s_1, s_2,\ldots )$ , we use the notation ${\mathbf {x}^{s} = x_1^{s_1} x_2^{s_2} \cdots }$ .

For a composition $\alpha = (\alpha _1,\ldots , \alpha _k) \in \mathbb {Z}^k_+$ , recall the monomial quasisymmetric functions

$$ \begin{align*}M_{\alpha}(\mathbf{x}) := \sum_{i_1 < \ldots < i_k} x^{\alpha_1}_{i_1} \cdots x^{\alpha_k}_{i_k}. \end{align*} $$

Note that they form a basis of the algebra of quasisymmetric functions.

It is easy to see that

$$ \begin{align*}F_{(n_1,\ldots,n_d, \infty)} = \sum_{A \in \mathcal{M}(n_1,\ldots,n_d)} w_A = \sum_{{\alpha^{(1)}, \ldots, \alpha^{(d)} \atop \text{compositions}} } m^{}_{\alpha^{(1)},\ldots,\alpha^{(d)}}\, (\mathbf{x}^{(1)})^{\alpha^{(1)}} \cdots (\mathbf{x}^{(d)})^{\alpha^{(d)}}, \end{align*} $$

where $m^{}_{\alpha ^{(1)},\ldots ,\alpha ^{(d)}}$ is the number of $A \in \mathcal {M}(n_1,\ldots , n_d)$ with $s_{\ell }(A) = \alpha ^{(\ell )} \in \mathbb {N}^{n_{\ell }}$ . The following result is a finite boxed version of this expansion.

Theorem 6.12 (Monomial basis expansion of boxed polynomials).

We have

$$ \begin{align*}F_{(n_1,\ldots,n_{d+1})} = \sum_{\alpha^{(1)}, \ldots, \alpha^{(d)}} m^{(n_{d+1})}_{\alpha^{(1)},\ldots,\alpha^{(d)}}\, M_{\alpha^{(1)}}(\mathbf{x}^{(1)}) \cdots M_{\alpha^{(d)}}(\mathbf{x}^{(d)}), \end{align*} $$

where the sum runs over compositions $\alpha ^{(1)}, \ldots , \alpha ^{(d)}$ such that $|\alpha ^{(i)}| = |\alpha ^{(j)}|$ for all $i,j$ , and the coefficient $m^{(n_{d+1})}_{\alpha ^{(1)},\ldots ,\alpha ^{(d)}}$ is equal to the number of packed hypermatrices $A \in \mathcal {M}([n_1] \times \cdots \times [n_d], n_{d+1})$ such that $s_{\ell }(A) = \alpha ^{(\ell )}$ for all $\ell \in [d]$ .

Proof. Note that replacing a hypermatrix A by $\mathrm {pack}(A)$ does not change its last passage time $G_{1,\ldots , 1}$ . Let $P \in \mathcal {M}([n_1] \times \cdots \times [n_d], n_{d+1})$ be a packed hypermatrix and let $M(P)$ be the set of hypermatrices $A \in \mathcal {M}([n_1] \times \cdots \times [n_d], n_{d+1})$ such that $\mathrm {pack}(A) = P$ . Let $s_{\ell }(P) = \alpha ^{(\ell )}$ . Then (by an argument as in Proposition 6.9) it is not difficult to obtain that we have

$$ \begin{align*}\sum_{A \in M(P)} w_A = M_{\alpha^{(1)}}(\mathbf{x}^{(1)}) \cdots M_{\alpha^{(d)}}(\mathbf{x}^{(d)}). \end{align*} $$

Therefore, we obtain

$$ \begin{align*} F_{(n_1,\ldots,n_{d+1})} &= \sum_{\pi \in \mathcal{P}(n_1,\ldots, n_{d+1})} w(\pi) \\ &= \sum_{A \in \mathcal{M}([n_1] \times \cdots \times [n_d], n_{d+1})} w_A \\ &= \sum_{P \text{ packed}} \sum_{A \in M(P)} w_A \\ &= \sum_{\alpha^{(1)}, \ldots, \alpha^{(d)}} m^{(n_{d+1})}_{\alpha^{(1)},\ldots,\alpha^{(d)}}\, M_{\alpha^{(1)}}(\mathbf{x}^{(1)}) \cdots M_{\alpha^{(d)}}(\mathbf{x}^{(d)}) \end{align*} $$

as needed.

6.5 Some remarks

Remark 8 (Dual stable Grothendieck polynomials, $d = 2$ ).

Recall that in this case (see Example 6.2), we get the following definition of polynomials $g_{\lambda }(\mathbf {x}; \mathbf {y})$ indexed by partitions $\lambda $ . We define

$$ \begin{align*}g_{\lambda}(\mathbf{x}; \mathbf{y}) := \sum_{\pi\, :\, \mathrm{sh}_1(\pi) = \lambda} \prod_{(i,j,k) \in \mathrm{Cor}(\pi)} x_i y_j, \end{align*} $$

where the sum runs over plane partitions $\pi $ . The polynomials $g_{\lambda }(\mathbf {x}; \mathbf {y})$ are generalizations of dual stable Grothendieck polynomials which correspond to the specialization $g_{\lambda }(\mathbf {x}) = g_{\lambda }(\mathbf {x}; \mathbf {1})$ . In fact, $g_{\lambda }(\mathbf {x}; \mathbf {y})$ is symmetric in $\mathbf {x}$ . The Cauchy–Littlewood-type identity in Corollary 6.5 becomes

$$ \begin{align*}\sum_{\lambda \in \mathcal{P}(n_2, \infty)} g_{\lambda}(\mathbf{x}; \mathbf{y}) = \prod_{i = 1}^{n_1} \prod_{j = 1}^{n_2} \frac{1}{1 - x_i y_j}, \end{align*} $$

which was proved in [Reference YeliussizovYel21a, Reference YeliussizovYel21b]. The boxed specialization formula in Proposition 6.7 becomes the following:

$$ \begin{align*}g_{[n_2] \times [n_3]}(1^{n_1 + 1}) = |\mathcal{P}(n_1, n_2, n_3)|, \end{align*} $$

the number of plane partitions inside the box $[n_1] \times [n_2] \times [n_3]$ , for which there is also the famous MacMahon boxed product formula

$$ \begin{align*}|\mathcal{P}(n_1, n_2, n_3)| = \prod_{i = 1}^{n_1} \prod_{j = 1}^{n_2} \prod_{k = 1}^{n_3} \frac{i + j + k - 1}{i + j + k - 2}. \end{align*} $$

Using determinantal formulas for dual stable Grothendieck polynomials [Reference YeliussizovYel17], we also have the following ‘coincidence’ formula (see [Reference YeliussizovYel21a, Lemma 3.4], [Reference YeliussizovYel21b, Lemma 6.9]) connecting them with the Schur polynomials $\{s_{\lambda } \}$ as follows:

$$ \begin{align*}g_{[n_2] \times [n_3]}(\mathbf{x}) = s_{[n_2] \times [n_3]}(\mathbf{x}, 1^{n_2 - 1}). \end{align*} $$

Remark 9 (3d Grothendieck polynomials, $d = 3$ ).

In this case, we get the following definition of polynomials $g_{\rho }(\mathbf {x}; \mathbf {y}; \mathbf {z})$ indexed by plane partitions $\rho $ . We define

(8) $$ \begin{align} g_{\rho}(\mathbf{x}; \mathbf{y}; \mathbf{z}) := \sum_{\pi\, :\, \mathrm{sh}_1(\pi) = \rho} \prod_{(i,j,k,\ell) \in \mathrm{Cor}(\pi)} x_{i} y_{j} z_{k}, \end{align} $$

where the sum runs over solid partitions $\pi \in \mathcal {P}(n_1,n_2,n_3,n_4)$ . Note also that if $\rho $ satisfies $D(\rho ) = \{(1,i,j) : (i,j) \in D(\lambda )\}$ where $\lambda $ is a partition, we then have $g_{\rho }(\mathbf {x}; \mathbf {1}; \mathbf {z}) = g_{\lambda }(\mathbf {x}; \mathbf {z})$ reduces to the 2d case discussed above. The polynomials $g_{\rho }(\mathbf {x}; \mathbf {y}; \mathbf {z})$ are quasisymmetric in $\mathbf {x}$ . The Cauchy–Littlewood-type identity in Corollary 6.5 becomes

$$ \begin{align*}\sum_{\rho \in \mathcal{P}(n_2, n_3, \infty)} g_{\rho}(\mathbf{x}^{}; \mathbf{y}; \mathbf{z}) = \prod_{i = 1}^{n_1} \prod_{j = 1}^{n_2} \prod_{k = 1}^{n_3} \left(1 - x^{}_{i} y_j z_k \right)^{-1}. \end{align*} $$

The boxed specialization formula becomes the following:

$$ \begin{align*}g_{[n_2] \times [n_3] \times [n_4]}(1^{n_1 + 1}) = |\mathcal{P}(n_1, n_2, n_3, n_4)|, \end{align*} $$

the number of solid partitions inside the box $[n_1] \times [n_2] \times [n_3] \times [n_4]$ .

8.1 Asymptotics

MacMahon’s numbers $m_d(n)$ have the following asymptotics [Reference Balakrishnan, Govindarajan and PrabhakarBGP12]:

$$ \begin{align*}\lim_{n \to \infty} n^{-d/(1 + d)}\log m_d(n) = \frac{1 + d}{d} \left(d\, \zeta(1 + d) \right)^{1/(1 + d)}, \end{align*} $$

where $\zeta $ is the Riemann zeta function (which is computed based on the explicit formula for the generating function). Let $p_d(n)$ be the number of d-dimensional partitions of volume n. Supported by numerical experiments for solid partitions, it was conjectured in [Reference Mustonen and RajeshMR03] that $p_3(n)$ has the same asymptotics as $m_3(n)$ . However, later computations reported in [Reference Destainville and GovindarajanDG15] suggest that this is not the case and that $p_3(n)$ is asymptotically larger than $m_3(n)$ , despite the fact that $m_3(n) = p_3(n)$ for $n \le 5$ and $m_3(n)> p_3(n)$ for the next many values of n [Reference Atkin, Bratley, Macdonald and McKayABMM67, Reference Destainville and GovindarajanDG15]; cf. the sequences A000293, A000294 in [OEIS]). See also [Reference EkhadEkh12] and a useful resource [Reference GovindarajanGov] for more related data. Using our interpretation for $m_d(n)$ (Corollary 5.5) and bounds on the corner-hook volume, it was shown in [Reference YeliussizovYel23] that $p_d(n) < dn \cdot m_d(dn)$ , and obtaining a more accurate comparison between these sequences (e.g., by showing that $\log p_d(n) \sim \log m_d(\alpha n)$ for some $\alpha $ ) will be important for understanding the asymptotics of $p_d(n)$ .

8.2 d-dimensional Grothendieck polynomials

Are there any (algebraic, determinantal) formulas for d-dimensional Grothendieck polynomials? They will be important for at least two applications: enumeration of boxed higher-dimensional partitions and computing distribution formulas (or performing asymptotic analysis) for the last passage percolation problem discussed above. Note that for $d = 2$ , there are several determinantal formulas (Jacobi-Trudi, bialternant types) known; see, for example, [Reference YeliussizovYel17, Reference Amanov and YeliussizovAY22, Reference IwaoIwa20, Reference KimKim22, Reference Hwang, Jang, Kim, Song and SongHJKSS21, Reference IwaoIwa21].