Partial transpose of permutation matrices

TL;DR

This paper investigates the combinatorial properties of the partial transpose operation on permutation matrices, providing enumeration results for matrices that are invariant or remain permutation after partial transpose, including symmetric cases.

Contribution

It introduces a combinatorial framework for analyzing partial transpose of permutation matrices and solves enumeration problems for invariant and permutation-preserving cases.

Findings

Counted permutation matrices equal to their partial transpose

Counted permutation matrices whose partial transpose remains a permutation

Extended results to symmetric permutation matrices

Abstract

The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of partial transpose from a combinatorial point of view. In this perspective, we solve some basic enumeration problems concerning the partial transpose of permutation matrices. More specifically, we count the number of permutations matrices which are equal to their partial transpose and the number of permutation matrices whose partial transpose is still a permutation. We solve these problems also when restricted to symmetric permutation matrices only.