A direct bijection between descending plane partitions with no special parts and permutation matrices

TL;DR

This paper introduces a direct bijection linking descending plane partitions with no special parts to permutation matrices, aligning parts with permutation inversion numbers and positions of ones.

Contribution

It provides a novel, explicit bijection connecting these combinatorial objects and explores potential extensions to alternating sign matrices.

Findings

Number of parts equals permutation inversion number.

Maximum parts correspond to position of the one in the last column.

Bijection preserves key combinatorial statistics.

Abstract

We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices.