q-Combinatorics: A new view

Yue Cai, University of Kentucky
Fine Hall 214

Please note special day (Friday).  The idea of q-analogues can be traced back to Euler in the 1700's who was studying q-series, especially specializations of theta functions. Recall a q-analogue is a method to numerate a set of objects by keeping track of its mathematical structure. For example, a combinatorial interpretation of the q-analogue of the Gaussian polynomial due to MacMahon in 1916 is given by summing over all 0-1 bit strings consisting of n-k zeroes and k ones the statistic q to the inversion number. Setting q=1 returns to the familiar binomial coefficient. In this talk we show the classical q-Stirling numbers of the second kind can be expressed more compactly 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 a new poset whose rank generating function is the q-Stirling number S_q[n,k] which we call the Stirling poset of the second kind. This poset supports an algebraic complex and a basis for integer homology is determined. This is another instance of Hersh, Shareshian and Stanton's homological version of the Stembridge q = -1 phenomenon. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done beginning with de Médicis and Leroux's rook placement formulation. Time permitting, we will indicate a bijective argument à la Viennot showing the (q,t)-Stirling numbers of the first and second kind are orthogonal.  This is joint work with Margaret Readdy.