Discrete CATS Seminar
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UNIVERSITY OF KENTUCKY **

DISCRETE CATS SEMINAR

DISCRETE MATH AND COMBINATORICS: ALGEBRAIC & TOPOLOGICAL SEMINAR

113 PATTERSON OFFICE TOWER

FALL 2007

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"Cyclotomic Factors of the Descent Set Polynomial"

Margaret Readdy

University of Kentucky

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Monday, November 19, 2007

4:00 pm, 113 Patterson Office Tower

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Abstract:

One usually encodes the number of permutations beta(S) in the
symmetric group having descent set S via the Eulerian polynomial,
where the number of such permutations is the coefficient of t^|S|. We
instead introduce the descent set polynomial where the statistic
beta(S) is the exponent of t. Descent set polynomials exhibit
interesting factorization patterns. We explore the question of when
particular cyclotomic factors divide these polynomials. As an
instance we deduce that the proportion of odd entries in the descent
set statistics for the symmetric group on n elements only depends on
the number of 1's in the binary expansion of n. Similar properties
hold for the signed descent set statistics.

This is joint work with Denis Chebikin (MIT), Richard Ehrenborg
(U Kentucky) and Pavlo Pylyavskyy (U Michigan).