Title:  New Perspectives of Quantum Analogues

Abstract:  In this talk we show the classical q-binomial can be expressed more compactly as a pair of statistics on a subset of 01-permutations via major index, an instance of the cyclic sieving phenomenon related to unitary spaces is also given. We then generalize this idea to q-Stirling numbers of the second kind using 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 Sq[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.