Expansion of permutations as products of transpositions

TL;DR

This paper investigates the enumeration of permutations expressed as products of a fixed number of transpositions, utilizing algebraic and combinatorial techniques to derive explicit formulas and symmetry properties.

Contribution

It introduces a novel approach to count permutations as products of transpositions using geometric sequences and explores related symmetry properties in graph matrices and Young diagrams.

Findings

Derived explicit formulas for permutation decompositions

Proved symmetry properties of bipartite graph matrices

Connected combinatorial enumeration with algebraic structures

Abstract

We compute the number of ways a given permutation can be written as a product of exactly $k$ transpositions. We express this number as a linear combination of explicit geometric sequences, with coefficients which can be computed in many particular cases. Along the way we prove several symmetry properties for matrices associated with bipartite graphs, as well as some general (likely known) properties of Young diagrams. The methods involve linear algebra, enumeration of border strip tableau, and a differential operator on symmetric polynomials.