# Reciprocals of exponential polynomials and permutation enumeration

@article{Gessel2019ReciprocalsOE, title={Reciprocals of exponential polynomials and permutation enumeration}, author={Ira M. Gessel}, journal={Australas. J Comb.}, year={2019}, volume={74}, pages={364-370} }

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… Expand

#### One Citation

A lifting of the Goulden-Jackson cluster method to the Malvenuto-Reutenauer algebra

- Mathematics
- 2021

The Goulden–Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted by Elizalde and Noy for counting permutations by occurrences of… Expand

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