186 research outputs found
Stirling permutations on multisets
A permutation of a multiset is called Stirling permutation if
as soon as and In our
paper we study Stirling polynomials that arise in the generating function for
descent statistics on Stirling permutations of any multiset. We develop
generalizations of the classical Stirling numbers and present their
combinatorial interpretations. Particularly, we apply the theory of
-partitions. Using certain specifications we also introduce the Stirling
numbers of odd type and generalizations of the central factorial numbers.Comment: Accepted for publication in the European Journal of Combinatorics. 17
pages, 4 figure
Partial -Positivity for Quasi-Stirling Permutations of Multisets
We prove that the enumerative polynomials of quasi-Stirling permutations of
multisets with respect to the statistics of plateaux, descents and ascents are
partial -positive, thereby confirming a recent conjecture posed by Lin,
Ma and Zhang. This is accomplished by proving the partial -positivity
of the enumerative polynomials of certain ordered labeled trees, which are in
bijection with quasi-Stirling permutations of multisets. As an application, we
provide an alternative proof of the partial -positivity of the
enumerative polynomials on Stirling permutations of multisets.Comment: arXiv admin note: text overlap with arXiv:2106.0434
A short note on the Stanley-Wilf Conjecture for permutations on multisets
The concept of pattern avoidance respectively containment in permutations can
be extended to permutations on multisets in a straightforward way. In this note
we present a direct proof of the already known fact that the well-known
Stanley-Wilf Conjecture, stating that the number of permutations avoiding a
given pattern does not grow faster than exponentially, also holds for
permutations on multisets.Comment: The contents of this paper have been integrated in the more
comprehensive paper "On restricted permutations on regular multisets",
http://arxiv.org/abs/1306.478
On restricted permutations on regular multisets
The extension of pattern avoidance from ordinary permutations to those on
multisets gave birth to several interesting enumerative results. We study
permutations on regular multisets, i.e., multisets in which each element occurs
the same number of times. For this case, we close a gap in the work of Heubach
and Mansour (2006) and complete the study of permutations avoiding a pair of
patterns of length three. In all studied cases, closed enumeration formulae are
given and well-known sequences appear. We conclude this paper by some remarks
on a generalization of the Stanley-Wilf conjecture to permutations on multisets
and words.Comment: 28 page
Poisson and independent process approximation for random combinatorial structures with a given number of components, and near-universal behavior for low rank assemblies
We give a general framework for approximations to combinatorial assemblies,
especially suitable to the situation where the number of components is
specified, in addition to the overall size . This involves a Poisson
process, which, with the appropriate choice of parameter, may be viewed as an
extension of saddlepoint approximation.
We illustrate the use of this by analyzing the component structure when the
rank and size are specified, and the rank, , is small relative to
. There is near-universal behavior, in the sense that apart from cases where
the exponential generating function has radius of convergence zero, for
, when for fixed , the size of the largest
component converges in probabiity to . Further, when for a positive integer , and ,
, with the choice governed by a
Poisson limit distribution for the number of components of size . This
was previously observed, for the case and the special cases of
permutations and set partitions, using Chen-Stein approximations for the
indicators of attacks and alignments, when rooks are placed randomly on a
triangular board. The case is especially delicate, and was not handled
by previous saddlepoint approximations.Comment: 35 page
Path decompositions of digraphs and their applications to Weyl algebra
We consider decompositions of digraphs into edge-disjoint paths and describe
their connection with the -th Weyl algebra of differential operators. This
approach gives a graph-theoretic combinatorial view of the normal ordering
problem and helps to study skew-symmetric polynomials on certain subspaces of
Weyl algebra. For instance, path decompositions can be used to study minimal
polynomial identities on Weyl algebra, similar as Eulerian tours applicable for
Amitsur--Levitzki theorem. We introduce the -Stirling functions which
enumerate decompositions by sources (and sinks) of paths
Mixed coloured permutations
In this paper we introduce mixed coloured permutation, permutations with
certain coloured cycles, and study the enumerative properties of these
combinatorial objects. We derive the generating function, closed forms,
recursions and combinatorial identities for the counting sequence, mixed
Stirling numbers of the first kind. In this comprehensive study we consider
further the conditions on the length of the cycles, -mixed Stirling numbers
and the connection to Bell polynomials
A note on Stirling permutations
In this note we generalize an identity of John Riordan and Robert Donaghey
relating the enumerator for Stirling permutations to the Eulerian polynomials.Comment: Originally written in 1978 but not published until no
Restricted Stirling permutations
In this paper, we study the generating functions for the number of pattern
restricted Stirling permutations with a given number of plateaus, descents and
ascents. Properties of the generating functions, including symmetric properties
and explicit formulas are studied. Combinatorial explanations are given for
some equidistributions.Comment: 14 pages in Taiwanese Journal of Mathematics, 201
A generalization of Stirling numbers
We generalize the Stirling numbers of the first kind to the case
where may be an arbitrary real number. In particular, we study the case in
which is an integer. There, we discover new combinatorial properties held
by the classical Stirling numbers, and analogous properties held by the
Stirling numbers with a negative integer.
On g\'{e}n\'{e}ralise ici les nombres de Stirling du premier ordre
au cas o\`u est un r\'eel quelconque. On s'interesse en particulier au cas
o\`u est entier. Ceci permet de mettre en evidence de nouvelles
propri\'et\'es combinatoires aux quelles obeissent les nombres de Stirling
usuels et des propri\'et\'es analougues auquelles obeissent les nombres de
Stirling o\`u est un entier n\`egatif
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