The number of permutations with a given number of sequences

TL;DR

This paper derives explicit formulas for the number of permutations with a fixed number of sequences, building on prior work that expressed these counts as sums of polynomials.

Contribution

It explicitly determines the polynomials representing P(n,s) and provides formulas for P(n,s) and its generating function u_s(x).

Findings

Explicit formulas for P(n,s) derived

Polynomials representing permutation counts are explicitly characterized

Generating functions for fixed s are obtained

Abstract

P(n,s) denotes the number of permutations of 1,2,...n that have exactly s sequences. Canfield and Wilf [math.CO/0609704] recently showed that P(n,s) can be written as a sum of s polynomials in n. We determine these polynomials explicitly and also obtain explicit expressions for P(n,s) and its fixed-s generating function u_s(x).