Reciprocals of exponential polynomials and permutation enumeration

TL;DR

This paper explores the relationship between reciprocals of exponential polynomials and permutation enumeration, revealing new combinatorial interpretations and extending results to noncommutative symmetric functions.

Contribution

It introduces a novel connection between reciprocals of exponential polynomials and permutation run-length restrictions, extending to noncommutative symmetric functions.

Findings

Reciprocal of partial sums relates to permutations with specific run-length conditions.

Generalizes to polynomials with reciprocals as generating functions for restricted permutations.

Extends results to noncommutative symmetric functions counting words with run restrictions.

Abstract

We show that the reciprocal of a partial sum with 2m terms of the alternating exponential series is the exponential generating function for permutations in which every increasing run has length congruent to 0 or 1 modulo 2m. More generally we study polynomials whose reciprocals are exponential generating functions for permutations whose run lengths are restricted to certain congruence classes, and extend these results to noncommutative symmetric functions that count words with the same restrictions on run lengths.