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Foundations of the chromatic polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
For integers n and k with 0≤k≤n the Stirling number of the second kind, S(n,k), counts the number of ways to partition a set of n objects into k nonempty subsets. For any 1≤k≤n, the function S(n,k) satisfies the recursion S(n,k)=S(n−1,k−1)+kS(n−1,k).
Combinatorics
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
[Recurrence formula for Stirling number of the second kind] For non-negative integers n and k we have: n+1k=knk+nk-1 $$ \left\{ n+1\atop k\right\} = k\left\{ n\atop k\right\} +\left\{ n\atop k-1\right\} $$
Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Stirling subset numbers are also called Stirling numbers of the second kind, and are denoted by S(n,k) $ S(n, k) $ .
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.
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).