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Number of permutations with same peak set for signed permutations

MSRI-UP 2013: Algebraic Combinatorics June 15, 2013 - July 28, 2013

July 26, 2013 (03:00 PM PDT - 03:45 PM PDT)
Speaker(s): Francis Castro (Massachusetts Institute of Technology), Jose Pastrana (University of Puerto Rico), Rita Zevallos (Swarthmore College)
Location: SLMath: Eisenbud Auditorium



Let Bn be the group of all signed permutations of [n]. A signed permutation has a peak in a position i=2,…,n−1 if πi−1<πi>πi+1. Let P(π) be the set of peaks of πP(S,n) be the set of signed permutations πBn such that P(π)=S, and #P(S,n) be the cardinality of P(S,n). We show #P(∅,n)=22n−1 and #P(S,n)=p(n)22n−|S|−1 where p(n) is some polynomial. We also consider the case in which we add a zero at the beginning of the permutation to also allow peaks at position i=1.

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