A simple bijection between permutation matrices and descending plane partitions without special parts

TL;DR

This paper introduces a straightforward bijection linking permutation matrices and descending plane partitions without special parts, utilizing inversion words and lattice path representations, and discusses enumeration via Andrew's determinant.

Contribution

It provides a simple, explicit bijection between permutation matrices and a specific class of descending plane partitions, clarifying their combinatorial relationship.

Findings

Established a bijection using inversion words and lattice paths

Connected enumeration of descending plane partitions to Andrew's determinant

Clarified the combinatorial structure linking permutations and plane partitions

Abstract

We present a simple bijection between permutation matrices and descending plane partitions without special parts. This bijection is already mentioned in work of P. Lalonde (without giving the details); it involves the inversion words of permutations and the (well-known) representation of descending plane partitions as families of non--intersecting lattice paths. (Taking a short detour, we will also exhibit how the (well--known) enumeration of descending plane partitions follows easily from the evaluation of Andrew's determinant.)