Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-20T10:29:03.350Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 November 2018

Anne de Médicis  and
Pierre Leroux
Anne de Médicis
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota U.S.A.
Pierre Leroux
Affiliation:
LAC, IM Université du Québec à Montréal, Montréal, Québec
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Résumé

Résumé

Dans cet article, nous étudions deux généralisations des nombres de Stirling de première et deuxième espèces, inspirées par leur interprétation combinatoire en termes de tableaux 0-1. Il s'agit des 𝔄-nombres de Stirlinget des nombres de Stirlingpartiels.Nous donnons en particulier des qet p, q-analogues de formules de convolution des nombres de Stirling de deuxième espèce, dues à Chen et Verde-Star, et nous étendons ces formules aux nombres de Stirling de première espèce. Les nombres de Stirlinga, d-progressifs,correspondant aux tableaux 0-1 dont les longueurs des colonnes font partie d'une progression arithmétique ﹛a + id﹜i≥0, sont également inclus dans cette étude.

Keywords

05A1511B6511B73
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 3 , 01 June 1995 , pp. 474 - 499
Copyright
Copyright © Canadian Mathematical Society 1995

References

[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