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In Many Circles. Permutations as Products of Cycles.
Published in Miklós Bóna, Combinatorics of Permutations, 2022
The symmetric group is a quintessential ingredient of group theory. It is well known, for instance, that every finite group of n elements is a subgroup of Sn. Extensive research of the symmetric group is therefore certainly justified. In this chapter, we will concentrate on the enumerative combinatorics of the symmetric group, that is, we are going to count permutations according to statistics that are relevant to this second way of looking at them.
Trees
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
The final two sections include a brief excursion into enumerative combinatorics. In §3.7, Cayley’s formula for the number of labeled n-vertex trees is derived using Prüfer Encoding. In §3.8, a recurrence relation for the number of different binary trees is established and then solved to obtain a closed formula.
Generalized degenerate Stirling numbers arising from degenerate Boson normal ordering
Published in Applied Mathematics in Science and Engineering, 2023
Taekyun Kim, Dae San Kim, Hye Kyung Kim
The Stirling number of the second kind is the number of ways to partition a set of n objects into k nonempty subsets (see (3)). The Stirling numbers of the second kind arise in various different contexts and have numerous applications, for example to enumerative combinatorics and quantum mechanics. In recent years, intensive explorations have been done for degenerate versions of many special numbers and polynomials, which was initiated by Carlitz in his work on degenerate Bernoulli and degenerate Euler polynomials. They have been studied by using such tools as combinatorial methods, generating functions, p-adic analysis, umbral calculus techniques, probability theory, mathematical physics, operator theory, special functions, analytic number theory and differential equations. The degenerate Stirling numbers of the second kind appear naturally when we replace the power by the generalized falling factorial polynomial in the defining equation of (see (3) and (6)). It turns out that they appear very frequently when we study degenerate versions of many special polynomials and numbers.
Degenerate 2D bivariate Appell polynomials: properties and applications
Published in Applied Mathematics in Science and Engineering, 2023
Shahid Ahmad Wani, Arundhati Warke, Javid Gani Dar
Polynomial sequences are of interest in enumerative combinatorics, algebraic combinatorics, and applied mathematics. The Laguerre polynomials, Chebyshev polynomials, Legendre polynomials, and Jacobi polynomials are a few polynomial sequences that appear as solutions to particular ordinary differential equations in physics and approximation theory. The most significant polynomial sequences is a class of Appell polynomial sequences [1]. Many applications of the Appell polynomial sequence may be found in theoretical physics, approximation theory, mathematics, and related fields of mathematics. The set of all Appell sequences is closed as a result of umbral polynomial sequence composition. This process turns the collection of all Appell sequences into an abelian group.
Normal ordering of degenerate integral powers of number operator and its applications
Published in Applied Mathematics in Science and Engineering, 2022
Taekyun Kim, Dae San Kim, Hye Kyung Kim
The Stirling number of the second is the number of ways to partition a set of n objects into k nonempty subsets. The Stirling numbers of the second kind have been extensively studied and repeatedly and independently rediscovered during their long history. The Stirling numbers of the second kind appear in many different contexts and have numerous applications, for example to enumerative combinatorics and quantum mechanics. They are given either by (5) or by (7). The study of degenerate versions of some special numbers and polynomials began with Carlitz's paper in [1], where the degenerate Bernoulli and Euler numbers were investigated. It is remarkable that in recent years quite a few degenerate versions of special numbers and polynomials have been explored with diverse tools and yielded many interesting results (see [2–4] and the references therein). It turns out that the degenerate Stirling numbers of the second play an important role in this exploration for degenerate versions of many special numbers and polynomials. The normal ordering of an integral power of the number operator in terms of boson operators a and can be written in the form In addition, the normal ordering of the degenerate kth power of the number operator , namely , in terms of boson operators a and can be written in the form where the generalized falling factorials are given by (3) and the degenerate Stirling numbers by (4) and (6).