135 research outputs found
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
A generalization of the "probléme des rencontres"
In this paper, we study a generalization of the classical \emph{probl\'eme des rencontres} (\emph{problem of coincidences}),
consisting in the enumeration of all permutations \pi \in \SS_n with fixed points,
and, in particular, in the enumeration of all permutations \pi \in \SS_n with no fixed points (derangements).
Specifically, we study this problem for the permutations of the
symbols , , \ldots, , , , \ldots, ,
where for every .
In this way, we obtain a generalization of the derangement numbers,
the rencontres numbers and the rencontres polynomials.
For these numbers and polynomials, we obtain the exponential generating series, some recurrences and representations,
and several combinatorial identities. Moreover, we obtain the expectation and the variance
of the number of fixed points in a random permutation of the considered kind.
Finally, we obtain some asymptotic formulas
for the generalized rencontres numbers and the generalized derangement numbers
Closed-form formulas, determinantal expressions, recursive relations, power series, and special values of several functions used in Clark--Ismail's two conjectures
In the paper, by virtue of the famous formula of Fa\`a di Bruno, with the aid
of several identities of partial Bell polynomials, by means of a formula for
derivatives of the ratio of two differentiable functions, and with availability
of other techniques, the authors establish closed-form formulas in terms of the
Bernoulli numbers and the second kind Stirling numbers, present determinantal
expressions, derive recursive relations, obtain power series, and compute
special values of the function , its
derivatives, and related ones used in Clark--Ismail's two conjectures. By these
results, the authors also discover a formula for the determinant of a
Hessenberg matrix and derive logarithmic convexity of a sequence related to the
function and its derivatives.Comment: 23 page
\u3cem\u3eq\u3c/em\u3e-Stirling Identities Revisited
We give combinatorial proofs of q-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz\u27s identity, a new proof of the q-Frobenius identity of Garsia and Remmel and of Ehrenborg\u27s Hankel q-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version
Theorems, Problems and Conjectures
These notes are designed to offer some (perhaps new) codicils to related
work, a list of problems and conjectures seeking (preferably) combinatorial
proofs. The main items are Eulerian polynomials and hook/contents of Young
diagram, mostly on the latter. The new additions include items on Frobenius
theorem and multi-core partitions; most recently, some problems on (what we
call) colored overpartitions. Formulas analogues to or in the spirit of works
by Han, Nekrasov-Okounkov and Stanley are distributed throughout. Concluding
remarks are provided at the end in hopes of directing the interested
researcher, properly.Comment: 14 page
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