Reciprocals of exponential polynomials and permutation enumeration
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.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.09290
 Bibcode:
 2018arXiv180709290G
 Keywords:

 Mathematics  Combinatorics;
 05A15
 EPrint:
 Australasian Journal of Combinatorics 74 (2) (2019) 364370