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A bijection between symmetric plane partitions and quasi transpose complementary plane partitions

Takuya Inoue

TL;DR

The paper resolves the explicit bijection between $SPP(n;M)$ and $QTCPP(n;M)$ by linking Proctor’s parallel equinumerosities for $SPP$, $e$-SPP, stair-PP, and p-stair-PP to a signed-bijection framework. The core approach uses 1:2^{|S|} correspondences, non-intersecting lattice-path configurations, and the Lindström–Gessel–Viennot lemma, organized through the index sets $I_m$ and $J_m$ to pair staircase and even SPP decompositions. The main result is an explicit sijection, hence an explicit bijection between $SPP(n;M)$ and $QTCPP(n;M)$, resolving a 35-year open problem and enriching the combinatorial toolkit with signed-bijections and path-transform techniques. The framework also suggests connections to Pfaffian evaluations and provides a blueprint for extending bijections to other symmetry classes and parity-restricted partitions.

Abstract

We resolve the explicit bijection problem between symmetric plane partitions (SPPs) and quasi transpose complementary plane partitions (QTCPPs), introduced by Schreier-Aigner, who proved their equinumerosity. First, we relate this problem to Proctor's parallel equinumerosities for SPPs, even SPPs, staircase plane partitions, and parity staircase plane partitions, by constructing several bijections. As a result, we reduce the task to constructing a compatible bijection between even SPPs and staircase plane partitions. We then provide non-intersecting lattice path configurations for these objects, apply the LGV lemma, and transform the resulting path configurations. This process leads us to new combinatorial objects, $I_m$ and $J_m$, and the task is further reduced to constructing a compatible sijection (signed bijection) between $I_m$ and $J_m$, which is carried out in the final part of this paper. Our construction also answers the 35-year-old open problem posed by Proctor: constructing an explicit bijection between even SPPs and staircase plane partitions.

A bijection between symmetric plane partitions and quasi transpose complementary plane partitions

TL;DR

The paper resolves the explicit bijection between and by linking Proctor’s parallel equinumerosities for , -SPP, stair-PP, and p-stair-PP to a signed-bijection framework. The core approach uses 1:2^{|S|} correspondences, non-intersecting lattice-path configurations, and the Lindström–Gessel–Viennot lemma, organized through the index sets and to pair staircase and even SPP decompositions. The main result is an explicit sijection, hence an explicit bijection between and , resolving a 35-year open problem and enriching the combinatorial toolkit with signed-bijections and path-transform techniques. The framework also suggests connections to Pfaffian evaluations and provides a blueprint for extending bijections to other symmetry classes and parity-restricted partitions.

Abstract

We resolve the explicit bijection problem between symmetric plane partitions (SPPs) and quasi transpose complementary plane partitions (QTCPPs), introduced by Schreier-Aigner, who proved their equinumerosity. First, we relate this problem to Proctor's parallel equinumerosities for SPPs, even SPPs, staircase plane partitions, and parity staircase plane partitions, by constructing several bijections. As a result, we reduce the task to constructing a compatible bijection between even SPPs and staircase plane partitions. We then provide non-intersecting lattice path configurations for these objects, apply the LGV lemma, and transform the resulting path configurations. This process leads us to new combinatorial objects, and , and the task is further reduced to constructing a compatible sijection (signed bijection) between and , which is carried out in the final part of this paper. Our construction also answers the 35-year-old open problem posed by Proctor: constructing an explicit bijection between even SPPs and staircase plane partitions.

Paper Structure

This paper contains 28 sections, 12 theorems, 168 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. The SPP-QTCPP bijection problem
  4. Four classes of plane partitions considered by Proctor
  5. $1 : 2^{\# S}$ correspondences
  6. A bijection between p-stair-PPs and QTCPPs
  7. $1 : 2^{\# S}$ correspondences between QTCPPs and stair-PPs
  8. $1 : 2^{\# S}$ correspondences between SPPs and even SPPs
  9. Preliminaries
  10. Signed Sets
  11. Sijections
  12. Composition of sijections
  13. Cartesian products of sijections
  14. Disjoint unions of sijections
  15. Compatibility
  16. ...and 13 more sections

Key Result

Theorem 2.5

Let $n \in \mathbb{Z}_{>0}$ and $M \in \mathbb{Z}_{\geq 0}$. Then, we have

Figures (5)

  • Figure 1: An outline of our construction. The red arrows indicate the bijections and relations established in this paper, which together form the desired bijection $\operatorname{SPP}\left({n};{M}\right) \to \operatorname{QTCPP}\left({n};{M}\right)$.
  • Figure 2: An illustration of a sijection.
  • Figure 3: A composition of sijections.
  • Figure 4: An example of the use of the double LGV lemma technique.
  • Figure 5: Cutting, translating and reflecting lattice paths as explained in Section 5.

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9: Proctor 1990
  • proof
  • Example 3.2
  • ...and 38 more