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