5,223 research outputs found
Combinatorially interpreting generalized Stirling numbers
Let be a word in alphabet with 's and 's.
Interpreting "" as multiplication by , and "" as differentiation with
respect to , the identity , valid
for any smooth function , defines a sequence , the terms of
which we refer to as the {\em Stirling numbers (of the second kind)} of .
The nomenclature comes from the fact that when , we have , the ordinary Stirling number of the second kind.
Explicit expressions for, and identities satisfied by, the have been
obtained by numerous authors, and combinatorial interpretations have been
presented. Here we provide a new combinatorial interpretation that retains the
spirit of the familiar interpretation of as a count of
partitions. Specifically, we associate to each a quasi-threshold graph
, and we show that enumerates partitions of the vertex set of
into classes that do not span an edge of . We also discuss some
relatives of, and consequences of, our interpretation, including -analogs
and bijections between families of labelled forests and sets of restricted
partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00
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
Some formulas for the restricted r-Lah numbers
The r-Lah numbers, which we denote here by `(r)(n, k), enumerate partitions
of an (n+r)-element set into k+r contents-ordered blocks in which the
smallest r elements belong to distinct blocks. In this paper, we consider a
restricted version `(r)
m (n, k) of the r-Lah numbers in which block cardinalities
are at most m. We establish several combinatorial identities for `(r)
m (n, k) and
obtain as limiting cases for large m analogous formulas for `(r)(n, k). Some
of these formulas correspond to previously established results for `(r)(n, k),
while others are apparently new also in the r-Lah case. Some generating function
formulas are derived as a consequence and we conclude by considering a
polynomial generalization of `(r)
m (n, k) which arises as a joint distribution for
two statistics defined on restricted r-Lah distributions.
Keywords: restricted Lah numbers, polynomial generalization, r-Lah numbers,
combinatorial identities
MSC: 11B73, 05A19, 05A1
Crystal bases and q-identities
The relation of crystal bases with -identities is discussed, and some new
results on crystals and -identities associated with the affine Lie algebra
are presented.Comment: 25 pages, style file axodraw.sty require
Polynomial Triangles Revisited
A polynomial triangle is an array whose inputs are the coefficients in
integral powers of a polynomial. Although polynomial coefficients have appeared
in several works, there is no systematic treatise on this topic. In this paper
we plan to fill this gap. We describe some aspects of these arrays, which
generalize similar properties of the binomial coefficients. Some combinatorial
models enumerated by polynomial coefficients, including lattice paths model,
spin chain model and scores in a drawing game, are introduced. Several known
binomial identities are then extended. In addition, we calculate recursively
generating functions of column sequences. Interesting corollaries follow from
these recurrence relations such as new formulae for the Fibonacci numbers and
Hermite polynomials in terms of trinomial coefficients. Finally, properties of
the entropy density function that characterizes polynomial coefficients in the
thermodynamical limit are studied in details.Comment: 24 pages with 1 figure eps include
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