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p, q-Stirling numbers and set partition statistics

1991
*
Journal of combinatorial theory. Series A
*

ACKNOWLEDGMENTS We thank Professor Dennis Stanton for calling our attention to rs

doi:10.1016/0097-3165(91)90020-h
fatcat:zhdxzdcli5ezthzxni6tkv4rui
*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 ...##
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P-partitions and q-stirling numbers

1994
*
Journal of combinatorial theory. Series A
*

New

doi:10.1016/0097-3165(94)90090-6
fatcat:iue7puoft5cx3psn6zzuyxn47e
*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. ...##
###
Study of the p,q-deformed Touchard polynomials
[article]

2019
*
arXiv
*
pre-print

A two-parameter deformation of the Touchard polynomials, based on the NEXT

arXiv:1904.07674v1
fatcat:a4d4wyd5bvgodmtwt4va4sxkne
*q*-exponential function of Tsallis, defines two*statistics*on*set**partitions*. ...*and**q*. ... Pinsky for comments*and*useful discussions. ...##
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A maj Statistic for Set Partitions

1991
*
European journal of combinatorics (Print)
*

Finally, we generalize these constructions to an analog of Rawling's rmaj

doi:10.1016/s0195-6698(13)80009-1
fatcat:vx353fvf4bh3npuy7rhqzqvxfi
*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. ...##
###
Partition statistics on permutations

1992
*
Discrete Mathematics
*

In [7] four

doi:10.1016/0012-365x(92)90366-n
fatcat:c7n6ygv655cklggjm5ybf4ggbq
*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*. ...##
###
q-Stirling numbers: A new view

2017
*
Advances in Applied Mathematics
*

We show the classical

doi:10.1016/j.aam.2016.11.007
fatcat:ssocityyobbfbdme4vxt2ewvhm
*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. ...##
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Average Values of some Z-Parameters in a Random Set Partition

2011
*
Electronic Journal of Combinatorics
*

We find exact

doi:10.37236/715
fatcat:6curq6mwj5fgbdipi4rq5lfymu
*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 ...##
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Average values of some Z-parameters in a random set partition
[article]

2013
*
arXiv
*
pre-print

We find exact

arXiv:1304.4813v1
fatcat:qma3fz7s2ze5ddjvqtzqjcqovm
*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 ...##
###
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*. ...##
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Reduced matrices and q-log-concavity properties of q-Stirling numbers

1990
*
Journal of combinatorial theory. Series A
*

We prove the

doi:10.1016/0097-3165(90)90006-i
fatcat:wn2tgnkvx5hzfjzk2hbmukhgra
*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). ...##
###
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*. ...##
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Determinants involving q-Stirling numbers

2003
*
Advances in Applied Mathematics
*

Let S [i, j ] denote the

doi:10.1016/s0196-8858(03)00029-0
fatcat:wlmz5eihtrg4ppcgwcccplvrby
*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? ...##
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Statistics on ordered partitions of sets
[article]

2019
*
arXiv
*
pre-print

The distribution of these

arXiv:math/0605670v6
fatcat:pu5sjqywbrhktgalm4pjoh5jom
*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. ...##
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Generalized Stirling numbers, convolution formulae and $p,q$-analogues

1995
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

In particular, we give a

doi:10.4153/cjm-1995-027-x
fatcat:pyhkbxipx5fddjexzyl6wnnh5u
*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*. ...
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