A Natural Bijection between Permutations and a Family of Descending Plane Partitions

TL;DR

This paper establishes a natural, direct bijection between descending plane partitions without special parts and permutations, clarifying their structural relationship without relying on lattice path references.

Contribution

It introduces a new bijection that simplifies understanding the connection between descending plane partitions and permutations, avoiding complex path-based methods.

Findings

Provides a direct bijection avoiding lattice paths

Offers an interpretation for defining conditions of descending plane partitions

Relates the number of inversions to descending plane partitions through a simple expression

Abstract

We construct a direct natural bijection between descending plane partitions without any special part and permutations. The directness is in the sense that the bijection avoids any reference to nonintersecting lattice paths. The advantage of the bijection is that it provides an interpretation for the seemingly long list of conditions needed to define descending plane partitions. Unfortunately, the bijection does not relate the number of parts of the descending plane partition with the number of inversions of the permutation as one might have expected from the conjecture of Mills, Robbins and Rumsey, although there is a simple expression for the number of inversions of a permutation in terms of the corresponding descending plane partition.