3,890 research outputs found
Cycles and sorting index for matchings and restricted permutations
We prove that the Mahonian-Stirling pairs of permutation statistics (\sor,
\cyc) and (\inv, \mathrm{rlmin}) are equidistributed on the set of
permutations that correspond to arrangements of non-atacking rooks on a
Ferrers board with rows and columns. The proofs are combinatorial and
use bijections between matchings and Dyck paths and a new statistic, sorting
index for matchings, that we define. We also prove a refinement of this
equidistribution result which describes the minimal elements in the permutation
cycles and the right-to-left minimum letters. Moreover, we define a sorting
index for bicolored matchings and use it to show analogous equidistribution
results for restricted permutations of type and .Comment: 23 page
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
A note on the -coefficients of the "tree Eulerian polynomial"
We consider the generating polynomial of the number of rooted trees on the
set counted by the number of descending edges (a parent with
a greater label than a child). This polynomial is an extension of the descent
generating polynomial of the set of permutations of a totally ordered -set,
known as the Eulerian polynomial. We show how this extension shares some of the
properties of the classical one. B. Drake proved that this polynomial factors
completely over the integers. From his product formula it can be concluded that
this polynomial has positive coefficients in the -basis and we show
that a formula for these coefficients can also be derived. We discuss various
combinatorial interpretations of these positive coefficients in terms of
leaf-labeled binary trees and in terms of the Stirling permutations introduced
by Gessel and Stanley. These interpretations are derived from previous results
of the author and Wachs related to the poset of weighted partitions and the
free multibracketed Lie algebra.Comment: 13 pages, 6 figures, Interpretations derived from results in
arXiv:1309.5527 and arXiv:1408.541
Restricted -Stirling Numbers and their Combinatorial Applications
We study set partitions with distinguished elements and block sizes found
in an arbitrary index set . The enumeration of these -partitions
leads to the introduction of -Stirling numbers, an extremely
wide-ranging generalization of the classical Stirling numbers and the
-Stirling numbers. We also introduce the associated -Bell and
-factorial numbers. We study fundamental aspects of these numbers,
including recurrence relations and determinantal expressions. For with some
extra structure, we show that the inverse of the -Stirling matrix
encodes the M\"obius functions of two families of posets. Through several
examples, we demonstrate that for some the matrices and their inverses
involve the enumeration sequences of several combinatorial objects. Further, we
highlight how the -Stirling numbers naturally arise in the enumeration
of cliques and acyclic orientations of special graphs, underlining their
ubiquity and importance. Finally, we introduce related generalizations
of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on
generalized combinatorial sequences
Skyscraper Numbers
We introduce numbers depending on three parameters which we call skyscraper
numbers. We discuss properties of these numbers and their relationship with
Stirling numbers of the first kind, and we also introduce a skyscraper
sequence.Comment: 7 pages, 1 figur
- …