Partitions without repeated odds, q-Catalan numbers, and the Shapiro convolution (continued)

Combinatorics/Partitions Seminar

Meeting Details

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Speaker: Dr. George Andrews, Department of Mathematics, The Pennsylvania State University

Abstract: Twenty years ago a short paper in the JCT(A) related the Catalan numbers C_n = Binomial(2n,n)/(n+1) to partitions without repeated odd parts. Also in that paper, the familiar Catalan recurrence C_(n+1) = C_0 C_n + C_1 C_(n-1) + ... + C_n C_0 was shown to be an instance of the Chu-Vandermonde convolution. In my talk on this topic to this seminar in January, I covered these background developments. In January 2009, T. Koshy published a 422 page book titled Catalan Numbers. On page 123, we find Shapiro's convolution 4^n C_n = C_0 C_(2n) + C_2 C_(2n-2) + ... + C_(2n) C_0 Richard Stanley in his Catalan Addendum poses as a research problem to find a bijective proof of Shapiro's convolution. This talk will build on the previous talk and will include a sketch of a combinatorial proof of a q-analog of Shapiro's convolution and the surprising implication for q-series summations.

Room Reservation Information

Room Number: 106 McAllister

Date: 05/05/2009

Time: 11:15am - 12:05pm