Explicit expressions for the extremal excedance set statistics

Authors:
Eric Clark;Richard Ehrenborg
Affiliations:
Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA;Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA
Venue:
European Journal of Combinatorics
Year:
2010

Citing 4
Cited 0

Enumerative combinatorics

Enumerative combinatorics
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Proof of a conjecture of Ehrenborg and Steingrímsson on excedance statistic

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Abstract

The excedance set of a permutation @p=@p"1@p"2...@p"k is the set of indices i for which @p"ii. We give explicit formulas for the number of permutations whose excedance set is the initial segment {1,2,...,m} and also of the form {1,2,...,m,m+2}. We provide two proofs. The first is an explicit combinatorial argument using rook placements. The second uses the chromatic polynomial and two variable exponential generating functions. We then recast these explicit formulas as LDU-decompositions of associated matrices and show that these matrices are totally non-negative.