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p, q-Stirling numbers and set partition statistics

Michelle Wachs, Dennis White
1991 Journal of combinatorial theory. Series A  
ACKNOWLEDGMENTS We thank Professor Dennis Stanton for calling our attention to rs and the problem associated with proving (2) using rs. We also thank Mr.  ...  Sagan [Sal has discovered yet another statistic on set partitions with the q-Stirling number as its distribution generating function.  ...  INTRODUCTION The q-Stirling numbers of the second kind, described by Gould [Go] , arise as generating functions for the distribution of certain statistics on set partitions, much as the q-analogue of  ... 
doi:10.1016/0097-3165(91)90020-h fatcat:zhdxzdcli5ezthzxni6tkv4rui

P-partitions and q-stirling numbers

SeungKyung Park
1994 Journal of combinatorial theory. Series A  
New q-analogs of Stifling numbers of the first and the second kind are derived from a poset on [2k] using Stanley's P-partition theory [Mem. Amer. Math. Soc. 119 (1972)].  ...  We also generalize to the poset on the set [rk].  ...  p, q-Stirling numbers of the second kind.  ... 
doi:10.1016/0097-3165(94)90090-6 fatcat:iue7puoft5cx3psn6zzuyxn47e

Study of the p,q-deformed Touchard polynomials [article]

Orli Herscovici
2019 arXiv   pre-print
A two-parameter deformation of the Touchard polynomials, based on the NEXT q-exponential function of Tsallis, defines two statistics on set partitions.  ...  and q.  ...  Pinsky for comments and useful discussions.  ... 
arXiv:1904.07674v1 fatcat:a4d4wyd5bvgodmtwt4va4sxkne

A maj Statistic for Set Partitions

Bruce E. Sagan
1991 European journal of combinatorics (Print)  
Finally, we generalize these constructions to an analog of Rawling's rmaj and to two new kinds of p, q-Stirling numbers.  ...  The corresponding weight generating function yields the q-Stirling numbers of the second kind of Carlitz and Gould.  ...  We hope to present these results on partition Eulerian numbers and the q-Stirling numbers of the first kind in the future.  ... 
doi:10.1016/s0195-6698(13)80009-1 fatcat:vx353fvf4bh3npuy7rhqzqvxfi

Partition statistics on permutations

Paul H. Edelman, Rodica Simion, Dennis White
1992 Discrete Mathematics  
In [7] four statistics on set partitions were described, each having the q-Stirling numbers [2] as their distribution generating function.  ...  This statistic is closely related to a well-known statistic on set partitions.  ...  Also, the number of partitions in II(n, k) which give the same partition This construction leads naturally to a description of the coefficients of the q-Stirling number.  ... 
doi:10.1016/0012-365x(92)90366-n fatcat:c7n6ygv655cklggjm5ybf4ggbq

q-Stirling numbers: A new view

Yue Cai, Margaret A. Readdy
2017 Advances in Applied Mathematics  
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.  ...  Letting t = 1+q we give a bijective argument showing the (q,t)-Stirling numbers of the first and second kind are orthogonal.  ...  for q-analogues.  ... 
doi:10.1016/j.aam.2016.11.007 fatcat:ssocityyobbfbdme4vxt2ewvhm

Average Values of some Z-Parameters in a Random Set Partition

Anisse Kasraoui
2011 Electronic Journal of Combinatorics  
We find exact and asymptotic formulas for the average values of several statistics on set partitions: of Carlitz's $q$-Stirling distributions, of the numbers of crossings in linear and circular representations  ...  of set partitions, of the numbers of overlappings and embracings, and of the numbers of occurrences of a 2-pattern.  ...  The author would like to thank Christian Krattenthaler and Einar Steingrímsson for their encouragement during the preparation of this paper, and the anonymous referee for his helpful suggestions concerning  ... 
doi:10.37236/715 fatcat:6curq6mwj5fgbdipi4rq5lfymu

Average values of some Z-parameters in a random set partition [article]

Anisse Kasraoui
2013 arXiv   pre-print
We find exact and asymptotic formulas for the average values of several statistics on set partitions: of Carlitz's q-Stirling distributions, of the numbers of crossings in linear and circular representations  ...  of set partitions, of the numbers of overlappings and embracings, and of the numbers of occurrences of a 2-pattern.  ...  The author would like to thank Christian Krattenthaler and Einar Steingrímsson for their encouragement during the preparation of this paper, and the anonymous referee for his helpful suggestions concerning  ... 
arXiv:1304.4813v1 fatcat:qma3fz7s2ze5ddjvqtzqjcqovm

Page 5111 of Mathematical Reviews Vol. , Issue 95i [page]

1995 Mathematical Reviews  
The generating functions for P-partitions of Stirling posets with bounded part sizes are shown to give rise to q-analogues of Stirling numbers of first and second kind.  ...  Several properties of these q-analogues are derived. These g-analogues turn out to be basically the q = p” special case of the p, g-Stirling numbers of the first kind by A. de Médicis and P.  ... 

Reduced matrices and q-log-concavity properties of q-Stirling numbers

Pierre Leroux
1990 Journal of combinatorial theory. Series A  
We prove the q-log-concavity of the q-Stirling numbers of the second kind, which was recently conjectured by Lynne Butler, by suitably extending her injective proof of the analogous property of the q-binomial  ...  Other related results, methods, counterexamples, and conjectures are discussed. ci;'  ...  Setting p = 1 in (2.1) means ignoring the statistic a&(~) and yields the q-Stirling numbers of the second kind S,[n, k] defined by (0.8).  ... 
doi:10.1016/0097-3165(90)90006-i fatcat:wn2tgnkvx5hzfjzk2hbmukhgra

Page 1285 of Mathematical Reviews Vol. , Issue 91C [page]

1991 Mathematical Reviews  
The proof is based on an interpretation of the g-Stirling numbers as generating functions for statistics on set partitions described by reduced-row-echelon matrices.  ...  The, paper also discusses some related results, conjectures and counterexamples concerning q-Stirling numbers, unimodality, and connections to the number of subgroups of order p* in a finite abelian p-group  ... 

Page 629 of Mathematical Reviews Vol. , Issue 92b [page]

1992 Mathematical Reviews  
Special cases and a combinatorial identity are given.” 92b:05004 05A18 05A15 11B65 11B73 Wachs, Michelle (1-MIAM); White, Dennis (1-MN) p, q-Stirling numbers and set partition statistics. J. Combin.  ...  for various statistics on set partitions.  ... 

Determinants involving q-Stirling numbers

Richard Ehrenborg
2003 Advances in Applied Mathematics  
Let S [i, j ] denote the q-Stirling numbers of the second kind. We show that the determinant of the We give two proofs of this result, one bijective and one based upon factoring the matrix.  ...  Lastly, a two variable version of a theorem of Sylvester and an application are presented.  ...  One generalization of the q-Stirling numbers is the p, q-Stirling numbers [6, 20] . Can any of the results appearing in this paper be extended to them?  ... 
doi:10.1016/s0196-8858(03)00029-0 fatcat:wlmz5eihtrg4ppcgwcccplvrby

Statistics on ordered partitions of sets [article]

Einar Steingrimsson
2019 arXiv   pre-print
The distribution of these statistics is closely related to the q-Stirling numbers of the second kind.  ...  We introduce several statistics on ordered partitions of sets, that is, set partitions where the blocks are permuted arbitrarily.  ...  Acknowledgments My thanks to Jiang Zeng for fruitful discussions in the initial stage of the research presented here, and to the anonymous referee mentioned after Conjecture 12.  ... 
arXiv:math/0605670v6 fatcat:pu5sjqywbrhktgalm4pjoh5jom

Generalized Stirling numbers, convolution formulae and $p,q$-analogues

Anne de Médicis, Pierre Leroux
1995 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
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  ...  They are the A-Stirling numbers and the partial Stirling numbers.  ...  and partial set partitions.  ... 
doi:10.4153/cjm-1995-027-x fatcat:pyhkbxipx5fddjexzyl6wnnh5u
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