Counting permutations by runs

TL;DR

This paper generalizes Gessel's reciprocity formula for counting permutations with restrictions on increasing run lengths, enabling enumeration of permutations with parity restrictions and systematic generation of related permutation statistics.

Contribution

It extends Gessel's theorem to broader restrictions on increasing runs and provides a systematic method for deriving generating functions for permutation statistics.

Findings

Complete enumeration of permutations with parity restrictions on peaks and valleys

Systematic method for generating functions of permutation statistics based on increasing runs

Extension of results to alternating runs in permutations

Abstract

In his Ph.D. thesis, Ira Gessel proved a reciprocity formula for noncommutative symmetric functions which enables one to count words and permutations with restrictions on the lengths of their increasing runs. We generalize Gessel's theorem to allow for a much wider variety of restrictions on increasing run lengths, and use it to complete the enumeration of permutations with parity restrictions on peaks and valleys, and to give a systematic method for obtaining generating functions for permutation statistics that are expressible in terms of increasing runs. Our methods can also be used to obtain analogous results for alternating runs in permutations.