The enumeration of fully commutative affine permutations

TL;DR

This paper derives a generating function for fully commutative affine permutations, revealing periodicity in their length distribution and providing structural insights, extending previous formulas by Stembridge and others.

Contribution

It introduces new generating functions for affine permutations and uncovers their periodic length coefficients, advancing understanding of their combinatorial structure.

Findings

Length generating functions have periodic coefficients with period dividing the rank.

Extended formulas for enumerating fully commutative affine permutations.

Structural results elucidate the nature of these permutations.

Abstract

We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations.