Combinatorics Seminar: I. Stern's diatomic array and beyond; II. A conjecture on the weak order of the symmetric group

Wednesday, November 28, 2018 at 2:00pm to 3:00pm

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Kaprielian Hall (KAP), 414
3620 South Vermont Avenue, Los Angeles, CA 90089

Richard Stanley, Emeritus Professor of Mathematics, MIT

This talk will have two unrelated parts.
I. Stern’s diatomic array is a certain array of numbers somewhat analogous to Pascal’s triangle. We consider a slight variation whose nth row is the sequence of coefficients of the polynomial \prod_{i=0}^{n-1} (1 + x^{2i} + x^{2i+1}). After reviewing the basic properties of this array and the closely related diatomic sequence of Stern, we consider the problem of computing the sum of the rth powers of the elements of the array. We then consider some generalizations, some of which have especially elegant properties.
II. A finite graded poset P is said to have the Sperner property if the largest level of P is an antichain of P of maximum size. (It is always an antichain.) It is unknown whether the weak (Bruhat) order on the symmetric group S_n has the Sperner property. We define some matrices whose entries are specializations of Schubert polynomials and give an explicit conjectured value for their determinant. If this conjecture is true, then the weak order on Sn does indeed have the Sperner property.

 

Event Type

Lecture / Talk / Workshop

Campus

University Park Campus

Department
Mathematics
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