80 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
Eulerian digraphs and Dyck words, a bijection
The main goal of this work is to establish a bijection between Dyck words and
a family of Eulerian digraphs. We do so by providing two algorithms
implementing such bijection in both directions. The connection between Dyck
words and Eulerian digraphs exploits a novel combinatorial structure: a binary
matrix, we call Dyck matrix, representing the cycles of an Eulerian digraph
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers
Ordering identities in the Weyl-Heisenberg algebra generated by single-mode
boson operators are investigated. A boson string composed of creation and
annihilation operators can be expanded as a linear combination of other such
strings, the simplest example being a normal ordering. The case when each
string contains only one annihilation operator is already combinatorially
nontrivial. Two kinds of expansion are derived: (i) that of a power of a string
in lower powers of another string , and (ii) that of a power
of in twisted versions of the same power of . The expansion
coefficients are shown to be, respectively, the generalized Stirling numbers of
Hsu and Shiue, and certain generalized Eulerian numbers. Many examples are
given. These combinatorial numbers are binomial transforms of each other, and
their theory is developed, emphasizing schemes for computing them: summation
formulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminating
hypergeometric series, and closed-form expressions. The results on the first
type of expansion subsume a number of previous results on the normal ordering
of boson strings.Comment: 36 pages (preprint format
Normally ordered forms of powers of differential operators and their combinatorics
We investigate the combinatorics of the general formulas for the
powers of the operator h∂k, where h is a central element of a ring
and ∂ is a differential operator. This generalizes previous work on
the powers of operators h∂. New formulas for the generalized Stirling
numbers are obtained.Ministerio de EconomÃa y competitividad MTM2016-75024-PJunta de AndalucÃa P12-FQM-2696Junta de AndalucÃa FQM–33
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