23,098 research outputs found
Euler-Mahonian Statistics On Ordered Set Partitions (II)
We study statistics on ordered set partitions whose generating functions are
related to -Stirling numbers of the second kind. The main purpose of this
paper is to provide bijective proofs of all the conjectures of \stein
(Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a
kind of path diagrams and explore the rich combinatorial properties of the
latter structure. We also give a partition version of MacMahon's theorem on the
equidistribution of the statistics inversion number and major index on words.Comment: 27 pages,8 figure
Symmetric unimodal expansions of excedances in colored permutations
We consider several generalizations of the classical -positivity of
Eulerian polynomials (and their derangement analogues) using generating
functions and combinatorial theory of continued fractions. For the symmetric
group, we prove an expansion formula for inversions and excedances as well as a
similar expansion for derangements. We also prove the -positivity for
Eulerian polynomials for derangements of type . More general expansion
formulae are also given for Eulerian polynomials for -colored derangements.
Our results answer and generalize several recent open problems in the
literature.Comment: 27 pages, 10 figure
Distribution of crossings, nestings and alignments of two edges in matchings and partitions
We construct an involution on set partitions which keeps track of the numbers
of crossings, nestings and alignments of two edges.
We derive then the symmetric distribution of the numbers of crossings and
nestings in partitions, which generalizes Klazar's recent result in perfect
matchings. By factorizing our involution through bijections between set
partitions and some path diagrams we obtain the continued fraction expansions
of the corresponding ordinary generating functions.Comment: 12 page
A q-analog of the Seidel generation of Genocchi numbers
A new -analog of Genocchi numbers is introduced through a q-analog of
Seidel's triangle associated to Genocchi numbers. It is then shown that these
-Genocchi numbers have interesting combinatorial interpretations in the
classical models for Genocchi numbers such as alternating pistols, alternating
permutations, non intersecting lattice paths and skew Young tableaux.Comment: 17 page
A unifying combinatorial approach to refined little G\"ollnitz and Capparelli's companion identities
Berkovich-Uncu have recently proved a companion of the well-known
Capparelli's identities as well as refinements of Savage-Sills' new little
G\"ollnitz identities. Noticing the connection between their results and
Boulet's earlier four-parameter partition generating functions, we discover a
new class of partitions, called -strict partitions, to generalize their
results. By applying both horizontal and vertical dissections of Ferrers'
diagrams with appropriate labellings, we provide a unified combinatorial
treatment of their results and shed more lights on the intriguing conditions of
their companion to Capparelli's identities.Comment: This is the second revision submitted to JCTA in June, comments are
welcom
Combinatorial interpretations of the Jacobi-Stirling numbers
The Jacobi-Stirling numbers of the first and second kinds were introduced in
2006 in the spectral theory and are polynomial refinements of the
Legendre-Stirling numbers. Andrews and Littlejohn have recently given a
combinatorial interpretation for the second kind of the latter numbers.
Noticing that these numbers are very similar to the classical central factorial
numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers
of both kinds, which provide a unified treatment of the combinatorial theories
for the two previous sequences and also for the Stirling numbers of both kinds.Comment: 15 page
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