On refined enumerations of plane partitions of a given shape with bounded entries
Takuya Inoue
TL;DR
This work addresses refined enumerations of plane partitions of a fixed shape $\\lambda$ with entries in $\\{0,\\ldots,m\\}$ by showing that the generating function by the number of rows containing $0$ coincides with that by the number of rows containing $m$: $\\sum_{P \\in \\mathrm{PP}(\\lambda; m)} x^{\\#\\text{rows with }0} = \\sum_{P \\in \\mathrm{PP}(\\lambda; m)} x^{\\#\\text{rows with }m}$. The authors deploy a framework of signed bijections (sijections) and compatibility, map plane partitions to families of non-intersecting paths, and derive a determinant formula via the LGV lemma, followed by a purely bijective proof using LGV involutions. They further generalize the technique to a Double LGV framework and apply it to give an elementary combinatorial proof of Schur function symmetry by permuting path directions, highlighting the broader applicability of compatibility in refined combinatorial proofs. Overall, the paper provides a robust method for refined bijections and connects refined plane-partition enumerations to classical symmetric-function identities. The results illuminate how signed-bijection compatibility can yield both determinant-based and bijective proofs, with potential impact on a range of refined combinatorial problems and symmetry phenomena.
Abstract
In this paper, we consider plane partitions $\text{PP}(λ; m)$ of a given shape $λ$, with entries at most $m$. We prove that the distributions of two statistics on $\text{PP}(λ; m)$ coincide: one is the number of rows containing $0$ and the other is the number of rows containing $m$. We also provide a bijective proof.