2015/06/30 by Yue Cai, Margaret A. Readdy · 21 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic number #Algebraic structures and combinatorial models #Bijection #Function (biology) #Generating function #Integer (computer science) #Partially ordered set #Rank (graph theory) #Stirling numbers of the second kind #math.CO #msc:05A18 #msc:05A30 #msc:06A07 #msc:11B73 #msc:18G35
paper · pdf · doi:10.1016/j.aam.2016.11.007
published in Advances in Applied Mathematics 86, 50-80 (Elsevier BV)
arxiv created 2016/11/12 · openalex publication_date 2016/12/14 · arxiv updated 2017/05/30 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
We show the classical q-Stirling numbers of the second kind can be expressed 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 Π(n,k) which we call the Stirling poset of the second kind. Its rank generating function is the q-Stirling number Sq[n,k]. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. Letting t = 1+q we give a bijective argument showing the (q,t)-Stirling numbers of the first and second kind are orthogonal.