Margaret Readdy, Rutgers Experimental Mathematics Seminar, March 12, 2015
See part 2 at vimeo.com/122984740
Abstract: A q-analogue is a method to enumerate a set of objects by keeping track of one or more of its mathematical properties. After setting q=1, one returns to the naive enumeration.
After reviewing some classical q-analogues, we will discuss the new idea of a negative q-analogue. This concept is motivated by Fu, Reiner, Stanton and Thiem's recent work on the negative q-binomial. We show the classical q-Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1+q. We extend this enumerative result via a decomposition of the Stirling poset of the first kind, as well as a homological version of Stembridge's q = -1 phenomenon.
We then describe a parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind. Time permitting, we give a bijective combinatorial argument a la Viennot showing the (q,t)-Stirling numbers of the first and second kind are orthogonal.
This is joint work with Yue Cai.