Permutations with given peak set

TL;DR

This paper investigates the enumeration of permutations with a specified set of peaks, deriving a formula involving a polynomial and exponential terms, and explores conjectures related to these counts.

Contribution

The authors derive a general formula for counting permutations with a given peak set for large n and compute specific polynomials for various cases, including those depending on n.

Findings

Number of permutations with fixed peak set follows a polynomial times an exponential for large n.

Explicit polynomials p(n) are computed for various peak sets.

Two conjectures about polynomial coefficients and maximizing peak set counts are discussed.

Abstract

Let Sym_n denote the symmetric group of all permutations pi = a_1...a_n of {1,...,n}. An index i is a peak of pi if a_{i-1} < a_i > a_{i+1} and we let P(pi) be the set of peaks of pi. Given any set S of positive integers we define P(S;n) to be the set pi in Sym_n with P(pi)=S. Our main result is that for all fixed subsets of positive integers S and all sufficiently large n we have #P(S;n)= p(n) 2^{n-#S-1} for some polynomial p(n) depending on S. We explicitly compute p(n) for various S of probabilistic interest, including certain cases where S depends on n. We also discuss two conjectures, one about positivity of the coefficients of the expansion of p(n) in a binomial coefficient basis, and the other about sets S maximizing #P(S;n) when #S is fixed.