Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5777554 | Journal of Combinatorial Theory, Series A | 2017 | 9 Pages |
Abstract
We say that a permutation Ï=Ï1Ï2â¯ÏnâSn has a peak at index i if Ïiâ1<Ïi>Ïi+1. Let P(Ï) denote the set of indices where Ï has a peak. Given a set S of positive integers, we define P(S;n)={ÏâSn:P(Ï)=S}. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large n, |P(S;n)|=pS(n)2nâ|S|â1 where pS(x) is a polynomial depending on S. They proved this by establishing a recursive formula for pS(x) involving an alternating sum, and they conjectured that the coefficients of pS(x) expanded in a binomial coefficient basis centered at maxâ¡(S) are all nonnegative. In this paper we introduce a new recursive formula for |P(S;n)| without alternating sums and we use this recursion to prove that their conjecture is true.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, Mohamed Omar,