4,023 research outputs found
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
Multiple Gamma Function and Its Application to Computation of Series
The multiple gamma function , defined by a recurrence-functional
equation as a generalization of the Euler gamma function, was originally
introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the
pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma
function has been revived. This paper discusses some theoretical aspects of the
function and their applications to summation of series and infinite
products.Comment: 20 pages, Latex, uses kluwer.cls, will appear in The Ramanujan
Journa
The -Bell numbers
The notion of generalized Bell numbers has appeared in several works but
there is no systematic treatise on this topic. In this paper we fill this gap.
We discuss the most important combinatorial, algebraic and analytic properties
of these numbers which generalize the similar properties of the Bell numbers.
Most of these results seem to be new. It turns out that in a paper of Whitehead
these numbers appeared in a very different context. In addition, we introduce
the so-called -Bell polynomials
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