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

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.

10 pages, title "modestified", unnecessary definitions removed, remarks shortened

Cited by in corpus (7)

• On the weighted enumeration of alternating sign matrices and descending plane partitions
• Multiply-refined enumeration of alternating sign matrices
• Alternating paths of fully packed loops and inversion number
• A direct bijection between descending plane partitions with no special parts and permutation matrices
• A simple bijection between permutation matrices and descending plane partitions without special parts
• Weight-preserving bijections between integer partitions and a class of alternating sign trapezoids
• A bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruplet of statistics considered by Behrend, Di Francesco and Zinn--Justin