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arXiv · 2509.17064

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

Takuya Inoue
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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.

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Takuya Inoue. 2025-09-21. A bijection between symmetric plane partitions and quasi transpose complementary plane partitions. https://arxiv.org/abs/2509.17064

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