Related papers: Extended Bell and Stirling numbers from hypergeome…
We introduce a new generalization of Stirling numbers of the second kind and analyze their properties, including generating functions, integral representations, and recurrence relations. These numbers are used to approximate Riemann zeta…
In 2008, Spivey found a recurrence relation for the Bell numbers. We consider the probabilistic r-Bell polynomials associated with which are a probabilistic extension of the r-Bell polynomials. Here Y is a random variable whose moment…
It is known that the ordered Bell numbers count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$. In this paper, we introduce the deranged Bell numbers that count the total number of deranged partitions of $[n]$. We first study…
We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of…
We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…
We study symmetric function analogues of the higher order Bell numbers. Their construction involves iterated plethystic exponential towers mimicking the single variable exponential generating functions for the higher order Bell numbers. We…
The aim of this paper is to study the $\lambda$-Stirling numbers of both kinds which are $\lambda$-analogues of Stirling numbers of both kinds. Those numbers have nice combinatorial interpretations when $\lambda$ are positive integers. If…
This work deals with a new generalization of $r$-Stirling numbers using $l$-tuple of permutations and partitions called $(l,r)$-Stirling numbers of both kinds. We study various properties of these numbers using combinatorial interpretations…
In this paper, we show that the r-Stirling numbers of both kinds, the r-Whitney numbers of both kinds, the r-Lah numbers and the r-Whitney-Lah numbers form particular cases of family of polynomials forming a generalization of the partial…
A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.
In this paper, we study classes of subexcedant functions enumerated by the Bell numbers and present bijections on set partitions. We present a set of permutations whose transposition arrays are the restricted growth functions, thus defining…
The Stirling number of the second kind $n\brace k$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As an extension of this, we consider $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls…
In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.
The aim of this paper is to study some basic properties of the r$-extended Lah-Bell numbers and polynomials associated with r-Lah numbers and to show the connection between the r-extended Lah-Bell polynomials and the rising factorial…
Following Spivey's pivotal discovery of a recurrence relation for Bell numbers, significant research has emerged concerning various generalizations of Bell numbers and polynomials. For example, Kim and Kim established a Spivey-type…
We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…
This short note addresses Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same…
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…
In this paper, we study some combinations of the degenerate and incomplete Stirling numbers of the second kind. We use a combinatorial approach and provide some asymptotic results.
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…