Partitions

A partition of an integer $n$ is a sequence $(a_1, a_2, ... )$ of weakly decreasing nonnegative integers $a_i$ with $\sum_i a_i = n$.

Ex: $(5, 3, 2, 2, 1) = (5, 3, 2, 2, 1, 0, 0, ...)$ is a partition of $13$.

The Young diagram corresponding to a partition $(a_i)$ is the set $\{(i, j) \in \mathbb{N}^2 \mid 1 \leq j \leq a_i\}$. (We use the convention of $\mathbb{N} = \{1, 2, ... \}$).

Young diagrams are top-left-justified sets of squares in a grid, where $a_i$ is the length of row $i$. We freely identify partitions with their Young diagrams.

Asymptotic Young Diagrams

A Young diagram asymptotic to a partition $\lambda$ is the collection of boxes $\mathbb{N}^2 \setminus \lambda$.

The hook corresponding to a box in a Young diagram is the right-angled collection of all boxes directly below and to the right of it. The hook length $h(i, j)$ of a box is the total number of boxes in its hook, and the corner box $(i, j)$ is called the pivot of the hook.

In an asymptotic Young diagram, hooks extend up and left (i.e. always toward the boundary).

Plane Partitions

A reverse plane partition (or RPP) is a placement of nonnegative integers into a Young diagram, such that the rows and columns weakly increase.

A skew plane partition (or SPP) is identical, but we place nonnegative integers into an asymptotic Young diagram $\mathbb{N}^2 \setminus \lambda$ that weakly decrease along rows and columns. When $\lambda = \emptyset$, we just call the object a plane partition.

The weight of an RPP or SPP $\pi$, written $|\pi|$, is the sum of its entries.

An RPP Generating Function

A famous result (due to Stanley in his Ph.D. thesis) expresses the generating function for RPPs of shape $\lambda$:

Combinatorially, RPPs of shape $\lambda$ are in bijection with hook-length-weighted tableaux of shape $\lambda$.

An SPP Generating Function

An identical result holds for SPPs (recall that hook length for these is measured up and left instead of down and right):

Thm: $\displaystyle \sum_\sigma q^{|\sigma|} = \prod_{\square \in \mathbb{N}^2 \setminus \lambda} \frac{1}{1 - q^{h(\square)}}.$

This result is most famously presented as a formula for MacMahon's generating function $M(q)$ for plane partitions:

$\displaystyle M(q) = \prod_{\square \in \mathbb{N}^2} \frac{1}{1 - q^{h(\square)}} = \prod_{n \in \mathbb{N}} \left( \frac{1}{1 - q^n} \right)^n.$

Hillman–Grassl

Instead, each hook is reshaped in an invertible manner before it is placed down. We denote their map $\operatorname{HG}$ and can view it operating on a plane partition.

Pak–Sulzgruber

In 2002, Igor Pak introduced a different bijection. In contrast to $\operatorname{HG}$, it operates on the level of diagonals rather than hooks.

Pak's bijection performs a sequence of invertible local moves on diagonals of an SPP/RPP that change its shape and each produce one element of a hook-length-weighted tableau.

In 2017, Robin Sulzgruber found an equivalent formulation of this bijection that operates on the level of hooks, analogously to $\operatorname{HG}$. We collectively refer to the map as $\operatorname{PS}$.

Other Bijections

Garver and Patrias introduced a universal formulation of both $\operatorname{HG}$ and $\operatorname{PS}$: given an ordering $\mathcal{O}$ of the boxes of a Young diagram $\lambda$ and a set of distinct labels $\mathcal{L}$, they produce an algorithm $\operatorname{GP}^\mathcal{O}_\mathcal{L}$ that is sometimes a well-defined bijection, and coincides with $\operatorname{HG}$ and $\operatorname{PS}$ for particular choices of $\mathcal{O}$ and $\mathcal{L}$.

The question of other possible bijections for choices of $\mathcal{O}$ and $\mathcal{L}$ remained open; we resolved it using integer programming for $3 \times 3$ tableau. Out of $9!^2 \approx 131$ billion possible algorithms, only 192 are well-defined bijections, and there are only 8 distinct maps.

The PT–DT Correspondence

RPPs of shape $\lambda$ and SPPs of shape $\mathbb{N}^2 \setminus \lambda$ are used to compute the Calabi–Yau topological vertex in Pandharipande–Thomas (PT) theory and Donaldson–Thomas (DT) theory.

i.e. SPPs $\sigma$ of shape $\mathbb{N}^2 \setminus \lambda$ correspond to pairs $(\rho, \pi)$ of RPPs $\rho$ of shape $\lambda$ and plane partitions $\pi$, where $|\sigma| = |\rho| + |\pi|$.

One-Leg Objects

This theorem was stated in the PT paper; we supply omitted details and bijectivize it.

Due to only one partition being nonempty in the tuple $(\emptyset, \emptyset, \lambda)$, the RPPs and SPPs we have defined so far are called one-leg objects.

Our approach to bijectivizing the one-leg (and two-leg) correspondence is to bijectivize local moves that we iterate to decompose both RPPs and SPPs into more malleable objects.

Interlacing

While the rows and columns of RPPs and SPPs contain valuable information, the diagonals of both turn out to be much more useful.

Given partitions $\lambda$ and $\mu$, we say $\lambda$ interlaces $\mu$, written $\lambda \succ \mu$, if $\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \cdots$.

In other words, $\lambda$ and $\mu$ satisfy the plane partition inequalities when placed next to one another as diagonal slices with $\lambda$ coming first.

Vertex Operators

We can use the interlacing relation to build the generating functions for RPPs and SPPs diagonal by diagonal.

We define a vector space $\Lambda$ over $\mathbb{Q}$ whose basis consists of vectors $\ket{\lambda}$ for partitions $\lambda$, and we denote the corresponding dual basis elements by $\bra{\lambda}$.

Weighing and Interlacing Operators

We define a weighing operator $Q$ by $Q(q)\,\ket{\lambda} = q^{|\lambda|}\,\ket{\lambda}$.

A Generating Function for Plane Partitions

To convert this into a more useful formula, we need to commute the $Q$ and $\Gamma$ operators.

Commutation Relations

Toggles

Given partitions $\lambda$ and $\mu$, the set of partitions $\nu$ with $\lambda \succ \nu \succ \mu$ is a poset product of intervals.

More Toggles

More broadly, we define the toggle of $\nu$ when it has any interlacing relationship with $\lambda$ and $\mu$.

When $\lambda \prec \nu \succ \mu$, we need to handle $\nu_1$ differently, since its poset of possible entries is unbounded above. We instead “pop off” the value $\nu_1 - \max\left\{ \lambda_1, \mu_1 \right\}$ in addition to producing the toggled partition $T(\nu)$, which then satisfies $\lambda \succ T(\nu) \prec \mu$.

Similarly, if $\lambda \succ \nu \prec \mu$, then the toggle map takes in both $\nu$ and a nonnegative integer $n$ to produce a partition $T(\nu, n)$, where $T(\nu, n)_1 = n + \max\left\{ \lambda_1, \mu_1 \right\}$. This new partition then satisfies $\lambda \prec T(\nu, n) \succ \mu$.

Bijectivizing $\Gamma$ Commutation

This result bijectivizes the following established commutation relations for $\Gamma$ operators:

$\displaystyle \bra{\mu}\, \Gamma_-(a)\Gamma_+(b)\, \ket{\lambda} = \frac{1}{1 - ab}\bra{\mu}\, \Gamma_+(b)\Gamma_-(a)\, \ket{\lambda}$

$\displaystyle \bra{\mu}\, \Gamma_+(a)\Gamma_+(b) \, \ket{\lambda} = \bra{\mu}\, \Gamma_+(b)\Gamma_+(a)\, \ket{\lambda}$

$\displaystyle \bra{\mu}\, \Gamma_-(a)\Gamma_-(b) \, \ket{\lambda} = \bra{\mu}\, \Gamma_-(b)\Gamma_-(a)\, \ket{\lambda}$

Lemma (G.): In a generating function $z_1(q)$ equal to a product of $\Gamma$ operators, commuting any two adjacent operators in $z_1$ results in a generating function $z_2$ that counts objects given by toggling the corresponding diagonal of the objects counted by $z_1$.

Producing $M(q)$

This corresponds to toggling the main diagonal of the plane partitions counted by $M(q)$ and recording the popped number separately.

The result is a weight-preserving bijection $\tau$ from plane partitions to hook-length-weighted tableaux, which is functionally identical to Pak's description of $\operatorname{PS}$.

What is new with this presentation is a more clear explanation of why it is independent of the order of toggles, as well as the ability to extend to generalizations of plane partitions.

Extending the bijection $\tau$ to accept SPPs requires first expressing their generating function $V_{(\emptyset, \emptyset, \lambda)}$ in terms of vertex operators.

Thm (G.): Let $\lambda$ be a partition and $(i, j) \in \mathbb{N}^2 \setminus \lambda$. If the hook of $(i, j)$ meets the boundary of $\mathbb{N}^2 \setminus \lambda$ at edges corresponding to $\Gamma_-\left( q^k \right)$ and $\Gamma_+\left( q^l \right)$, then $h(i, j) = k + l$.

One-Leg Bijections

This lets us extend $\tau$ to RPPs and SPPs; it outputs a hook-length-weighted tableau of the same shape. The result is a bijective proof of the existing results

$\displaystyle V_{(\emptyset, \emptyset, \lambda)}(q) = \prod_{\square \in \mathbb{N}^2 \setminus \lambda} \frac{1}{1 - q^{h(\square)}}, \quad W_{(\emptyset, \emptyset, \lambda)}(q) = \prod_{\square \in \lambda} \frac{1}{1 - q^{h(\square)}}.$

Delightfully, this is true on the level of hooks: for any Young diagram $\lambda$ and any $n \in \mathbb{N}$, there are exactly $n$ distinct $n$-hooks in $\mathbb{N}^2$, and if there are $k$ distinct $n$-hooks in $\lambda$, then there are $n + k$ distinct $n$-hooks in $\mathbb{N}^2 \setminus \lambda$, and they are in bijection nearly-canonically. (This is an established result via $n$-quotients — we omit it here for brevity).

Finishing the One-Leg Bijection

Thm (G.): There is a bijection between SPPs $\sigma$ of shape $\mathbb{N}^2 \setminus \lambda$ and pairs $(\rho, \pi)$ of RPPs $\rho$ of shape $\lambda$ and plane partitions $\pi$, where $|\sigma| = |\rho| + |\pi|$.

We iteratively toggle the diagonals of $\sigma$ until there are no more nonzero entries in it (which only requires a finite number of toggles). The result is a hook-length-weighted tableau of shape $\mathbb{N}^2 \setminus \lambda$, whose entries we distribute between tableaux of shape $\lambda$ and $\mathbb{N}^2$ while preserving hook length. We then untoggle those tableaux to produce $\rho$ and $\pi$.

Other Legs

The notation $(\emptyset, \emptyset, \lambda)$ for SPPs of shape $\mathbb{N}^2 \setminus \lambda$ and RPPs of shape $\lambda$ suggests further generalizations, and this is the case.

SPPs with more legs are direct generalizations of their one-leg counterparts when viewed as stacks of boxes. Given partitions $\lambda,$ $\mu,$ and $\nu$, an SPP $\sigma$ of shape $(\lambda, \mu, \nu)$ has the appearance of boxes stacked in a room with dents in all three edges.

RPP Generalizations

Three-leg RPPs are unfortunately extremely complicated — much more so than their one- or even two-leg counterparts. We have proven several structural results and made progress toward a complete bijection, but it remains elusive.

We instead focus only on two-leg RPPs of a specific shape: $(\lambda, \mu, \emptyset)$.

Rather than treating the entries of a one-leg RPP as the number of boxes stacked on a Young diagram, a generalizable interpretation is to treat them as the number of boxes removed from an infinite tower.

Ex: a one-leg RPP of shape $(4, 3, 1)$ and weight 19, visualized as 19 boxes removed from an infinite vertical tower.

Two-Leg RPPs

In this interpretation, the single leg is vertical, we remove boxes from it instead of adding them, and the resulting object satisfies the regular plane partition inequalities, not the reverse ones.

A two-leg RPP of shape $(\lambda, \mu, \emptyset)$ is now exactly the analogue of this object with two intersecting horizontal legs instead of a single vertical one.

Ex: an RPP of weight 7 and shape $(\lambda, \mu, \emptyset)$ for $\lambda = (3, 1)$ and $\mu = (2, 2)$.

Two-Leg Vertex Operators

We can produce $W$ from $V$ by commuting every $\Gamma_+$ past every $\Gamma_-$ as before, but also inverting the order of the same-sign $\Gamma$ operators relative to one another.

Bijectivization Difficulty

Given an SPP $\sigma$, we toggle its diagonals until we only pop off zeros. However, this time there is an RPP-like remnant effectively at infinity. To convert it into a two-leg RPP, we “palindromically” commute the same-sign $\Gamma$ operators.

The challenges with this bijection arise in showing it is well-defined. While only finitely many steps are required for any specific SPP, there was substantial work involved in showing that that remains true for a map defined on all SPPs.

Edge Sequences

For a partition $\lambda$, we introduce the doubly-infinite edge sign sequence $e_\lambda(n)$ and edge power sequence $p_\lambda(n)$ for the edges of the boundary of $\mathbb{N}^2 \setminus \lambda$, labeled with integers from bottom-left to top right so that the main diagonal has edges with labels $-1$ and $0$.

$e_\lambda(n)$ is $1$ if the edge labeled $n$ is horizontal and $-1$ if it is horizontal.

$p_\lambda(n) = \begin{cases} \left| n + \frac{1}{2} \right|, & e_\lambda(n) = \operatorname{sign}\left(n + \frac{1}{2}\right) \\ - \left| n + \frac{1}{2} \right|, & e_\lambda(n) \neq \operatorname{sign}\left(n + \frac{1}{2}\right) \end{cases}$.

A Generating Function for SPPs

Producing $V_{(\emptyset, \emptyset, \lambda)}(q)$

Thm (G.): Let $\lambda$ be a Young diagram. If $(i, j) \in \mathbb{N}^2 \setminus \lambda$ and $k$ and $l$ are the labels of the edges where the hook of $(i, j)$ meets the boundary of $\mathbb{N}^2 \setminus \lambda$, then $h(i, j) = p_\lambda(k) + p_\lambda(l)$.

Similarly, if $(i, j) \in \lambda$, then $h(i, j) = -\left( p_\lambda(k) + p_\lambda(l) \right)$.

This theorem guarantees that when we commute the $\Gamma$ operators in the expression for $V_{(\emptyset, \emptyset, \lambda)}(q)$, each factor we produce is exactly $\left( 1 - q^{h(\square)} \right)^{-1}$, where $\square$ is the box we pop off in the corresponding toggle.

More Legs

The PT–DT correspondence holds in different and more general settings. Two-leg SPPs and RPPs are largely analogous to their one-leg counterparts, and our methods of bijectivizing the two-leg PT–DT correspondence extend with only moderate difficulty.

Three-leg objects are substantively more complicated. Given partitions $\lambda$, $\mu$, and $\nu$, an SPP $\sigma$ of shape $(\lambda, \mu, \nu)$ is an SPP of shape $\mathbb{N}^2 \setminus \nu$ so that $\sigma(i, j) \geq \max \left\{ \lambda_j, \mu_i \right\}$ for all $(i, j)$.

Three-Leg RPPs

Three-leg RPPs are much more complicated than their one- or even two-leg counterparts. Rather than thinking of the entries of one-leg RPPs as the number of boxes stacked on a Young diagram, a more generalizable interpretation is to treat them as the number of boxes removed from an infinite tower.

Generalizing this interpretation, a three-leg RPP of shape $(\lambda, \mu, \nu)$ is an object in which we remove boxes from cylindrical legs of shape $\lambda$, $\mu$, and $\nu$, including their two- and three-way intersections.

Ex: the minimal-weight RPP of shape $(\lambda, \mu, \nu)$ for $\lambda = (3, 2)$, $\mu = (2, 1, 1)$, and $\nu = (4, 2, 1)$, colored by region.

We denote the set of orange boxes as $\operatorname{I}$, the set of yellow (i.e. two-way intersection) boxes as $\operatorname{II}$, and the set of purple (i.e. three-way intersection) boxes as $\operatorname{III}$.

RPP Regions

We write $\operatorname{I} = \operatorname{I}_1 \sqcup \operatorname{I}_2 \sqcup \operatorname{I}_3$, where $\operatorname{I}_1$, $\operatorname{I}_2$, and $\operatorname{I}_3$ are the sets of boxes in the legs corresponding to $\lambda$, $\mu$, and $\nu$, respectively. Similarly, $\operatorname{II} = \operatorname{II}_{\overline{1}} \sqcup \operatorname{II}_{\overline{2}} \sqcup \operatorname{II}_{\overline{3}}$, where $\operatorname{II}_{\overline{1}}$, $\operatorname{II}_{\overline{2}}$, and $\operatorname{II}_{\overline{3}}$ are the sets of two-way intersection boxes not from the legs corresponding to $\lambda$, $\mu$, and $\nu$, respectively.

We then define a three-leg RPP as a pair $\rho = (A, B)$, where $A \subseteq \operatorname{I} \cup \operatorname{III}$ and $B \subseteq \operatorname{II} \cup \operatorname{III}$ and both satisfy the plane partition inequalities.

The Labeling Condition

The weight of $\rho = (A, B)$ is the total number of boxes removed from both $A$ and $B$; note that since region $\operatorname{III}$ is present in both, its boxes can be present with multiplicity 2. This notion of a labeled $AB$ configuration was introduced by Jenne, Webb, and Young, who were also the first to prove the three-leg PT–DT correspondence.

Three-Leg Challenges

Extending the bijectivization methods of the one- and two-leg cases to the three-leg one is immediately more difficult: three-leg RPPs do not naturally admit a vertex operator expression at all.

In the hopes of producing one via a toggle-like operation, however, we investigated the entry posets of specific boxes of three-leg RPPs.

Ex: a three-leg RPP and the posets of valid values $(a,b)$ for its outer corners.

Three-Leg Results

Thm (G.): All outer corners' entry posets have this right-angled shape.

However, the height of the left part of the angle may not equal the width of the infinite vertical part, meaning there is not always a grading-preserving bijection between these posets and products of intervals.

This means a toggle operation is likely impossible to define, since no analogous involution may exist.