Generalized Stirling Numbers, Convolution Formulae and p, q-Analogues

Médicis, Anne de ; Leroux, Pierre

Canadian journal of mathematics, Tome 47 (1995) no. 3, pp. 474-499

Résumé

In this paper, we study two generalizations of the Stirling numbers of the first and second kinds, inspired from their combinatorial interpretation in terms of 0-1 tableaux. They are the A-Stirling numbers and the partial Stirling numbers. In particular, we give a q and a p, q-analogue of convolution formulae for Stirling numbers of the second kind, due to Chen and Verde-Star, and we extend these formulae to Stirling numbers of the first kind. Included in this study are the a, d-progressive Stirlingnumbers, corresponding to 0-1 tableaux with column lengths from an arithmetic progression {a + id}i≥0.

Détail

DOI : 10.4153/CJM-1995-027-x

Mots-clés : 05A15, 11B65, 11B73

Bibliographie

[Bu] Butler, L.M., The q-log concavity of q-binomial coefficients, J. Combin. Theory Ser. A 54(1990), 53–62. Google Scholar

[Cal] Carlitz, L., On abelian fields, Trans. Amer. Math. Soc. 35(1933), 122–136. Google Scholar

[Ca2] Carlitz, L., q-Bernoulli numbers and polynomials, Duke Math. J. 15(1948), 987–1000. Google Scholar

[Ch] Chen, W.Y., Context-free grammars, differential operators and formal power series. In: Actes du Colloque sur les séries formelles et la combinatoire algébrique, (eds. Delest, M., Jacob, G. and Leroux, P.), Publ. LaBRI, Bordeaux, 1991. 145–160. Google Scholar

[Co] Comtet, L., Nombres de Stirling généraux et fonctions symétriques, C. R. Acad. Sci. Paris 275(1972), 747–750. Google Scholar

[dMLe] de Médicis, A. and Leroux, P., A unified combinatorial approach for q-(and p,q-)Stirling numbers, J. Statist. Plann. Inference 34(1993), 89–105. Google Scholar

[Do] Dowling, T.A., A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14(1973), 61–85. Google Scholar

[Ge] Gessel, I., A q-analogue of the exponential formula, Discrete Math. 40(1982), 69–80. Google Scholar

[Go] Gould, H.W., The q-Stirling Numbers of the First and Second Kinds, Duke Math. J. 28( 1961 ), 281–289. Google Scholar

[Ha] Habsieger, L., Inégalités entre fonctions symétriques élémentaires: applications à des problèmes de log concavité, Discrete Math. 115(1993), 167–174. Google Scholar

[Ko] Koutras, M., Non-central Stirling numbers and some applications, Discrete Math. 42(1982), 73–89. Google Scholar

[Le] Leroux, P., Reduced Matrices and q-Log Concavity Properties ofq-Stirling Numbers, J. Combin. Theory Ser. A 54(1990), 64–84. Google Scholar

[Mi] Milne, S.C., Restricted Growth Functions and Incidence Relations of the Lattice of Partitions of an n-Set, Adv. Math. 26(1977), 290–305. Google Scholar

[ReWa] Remmel, J.B. and Wachs, M., p, q-analogues of the Stirling numbers, private communication, 1986. Google Scholar

[Ri] Riordan, , Combinatorial Identities, Wiley, 1968. Google Scholar

[RuVo] Rucinski, A. and Voigt, B., A local limit theorem for generalized Stirling numbers, Rev. Roumaine Math. Pures Appl. (2) 35(1990), 161–172. Google Scholar

[Sa] Sagan, B.E., Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants, Amer. Math. Soc. Transi. 329(1992), 795–811. Google Scholar

[Stl] Stanley, R.P., Supersolvable lattices, Algebra Universalis 2(1972), 197–217. Google Scholar

[St2] Stanley, R.P., Finite lattices and Jordan-Holder sets, Algebra Universalis 4(1974), 361–371. Google Scholar

[VS] Verde-Star, L., Interpolation and Combinatorial Functions, Stud. Appl. Math. 79(1988), 65–92. Google Scholar

[Vo] Voigt, B., A common generalization of binomial coefficients, Stirling numbers and Gaussian coefficients. In: Proc. 11th Winter School on Abstract Analysis, Železná Ruda, 1983. Rend. Circ. Mat. Palermo (2) Suppl., 1984. 339–359. Google Scholar