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

## Combinatorics/Partitions Seminar

## Meeting Details

For more information about this meeting, contact Calendar Administrator

**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