Inequivalent Factorizations of Permutations

TL;DR

This paper investigates the enumeration of inequivalent permutation factorizations under a specific equivalence relation, providing closed-form generating series for certain cases and revealing connections to acyclic digraphs and monotone factorizations.

Contribution

It introduces a new correspondence between inequivalent factorizations and acyclic alternating digraphs, enabling explicit enumeration for specific classes of permutations.

Findings

Closed-form generating series for permutations with up to three cycles.

Enumeration of permutations with four cycles when factors are transpositions.

Established connections to monotone factorizations and other permutation classes.

Abstract

Two factorizations of a permutation into products of cycles are equivalent if one can be obtained from the other by repeatedly interchanging adjacent disjoint factors. This paper studies the enumeration of equivalence classes under this relation. We obtain closed form expressions for generating series of inequivalent minimal transitive factorizations of permutations having up to three cycles, and also of permutations with four cycles when factors are restricted to be transpositions. Our derivations rely on a new correspondence between inequivalent factorizations and acyclic alternating digraphs. Strong similarities between the enumerative results derived here and analogous ones for "ordinary" factorizations suggest that a unified theory remains to be discovered. We also establish connections between inequivalent factorizations and other well-studied classes of permutation factorizations. In particular, a relationship with monotone factorizations allows us to recover an exact counting formula for the latter due to Goulden, Guay-Paquet and Novak.