The Group of Generalized Stirling Numbers

Bickel, Thomas

doi:10.1006/aama.2000.0703

Cited by 7 publications

(3 citation statements)

References 13 publications

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“…This form is more useful for us later. Further information about r -Stirling numbers and generalized Stirling numbers -all of these are Stirling numbers in the sense of the definition here -can be found in the papers by Bickel [2], Broder [4,3], Hsu [12] and Meső [19].…”

Section: The Stirling Recursionmentioning

confidence: 99%

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Metric intersection problems in Cayley graphs and the Stirling recursion

Phongpattanacharoen

Siemons

2013

Aequat. Math.

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In Sym(n) with n ≥ 5 let H be a conjugacy class of elements of order 2 and let Γ be the Cayley graph whose vertex set is the group G generated by H (so G = Sym(n) or Alt(n) ) and whose edge set is determined by H. We are interested in the metric structure of this graph. In particular, for g ∈ G let Br(g) be the metric ball in Γ of radius r and centre g. We show that the intersection numbers Φ(Γ; r, g) := | Br(e) ∩ Br(g) | are generalized Stirling functions in n and r. The results are motivated by the study of error graphs in Levenshtein [15,16,17] and related reconstruction problems.Keywords: Intersection numbers in graphs, the k -transposition Cayley graph on Sym(n) , error graphs, reconstruction AMS Classification: 05C12 Distance in graphs, 05C25 Graphs and abstract algebra, 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures Metric Intersections in Cayley graphsLet G be a finite group and let H be a subset of G which generates it. We assume that H = {h −1 : h ∈ H} and that H does not contain the identity element e of G. In this situation H defines an undirected simple Cayley graph Γ = Γ H G on the vertex set G. The usual graph distance on Γ is denoted d : G × G → N ∪ {0}; thus d = d(u, w) for u, w ∈ G is the least number of h i ∈ H so that w = uh 1 ...h d . This defines a metric on Γ. For an integer r ≥ 0 and g ∈ G the setis the metric ball of radius r and centre g. In this paper we are interested in the metric intersection numbers Φ(Γ; r, g) := | B r (Γ, e) ∩ B r (Γ, g) | considered as a function on G for r ≥ 0. Notice in particular, Φ(Γ; r, e) is the cardinality of B r (Γ, e), and this information also provides the diameter of Γ . We determine these intersection numbers when 1

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Section: The Stirling Recursionmentioning

confidence: 99%

“…To give an example let u = (1, 5)(2, 4)(3) and g = (1, 2)(1, 3)(1, 4) =(1,2,3,4) in Sym(5) . This gives the path…”

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confidence: 99%

Metric intersection problems in Cayley graphs and the Stirling recursion

Phongpattanacharoen

Siemons

2013

Aequat. Math.

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“…Alternative approaches can be found in [3,12,19,22]. In general, the reduced equation is used to define combinatorial number families, see de Médicis and Leroux In connection with probabilities their q-analogues are useful in many applications, see Crippa and Simon [7].…”

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confidence: 99%

Birth Processes and Symmetric Polynomials

Bickel

Galli

Simon

2001

Annals of Combinatorics

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Many discrete-time pure birth processes are symmetrical in the following sense: The probability that the process is in a fixed state is independent of the sequence of transitions inducing . This is always the case whenever a transition is a time-dependent or a state-dependent random variable or the product of such independent variables. We use this property in order to derive algebraic descriptions in terms of symmetric polynomials. Besides new solutions, our approach offers a uniform point of view on a large class of often considered distributions.

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Stirling, bell, bernoulli, euler and eulerian numbers

Sándor

Crstici²

2004

Handbook of Number Theory II

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The Group of Generalized Stirling Numbers

Cited by 7 publications

References 13 publications

Metric intersection problems in Cayley graphs and the Stirling recursion

Metric intersection problems in Cayley graphs and the Stirling recursion

Birth Processes and Symmetric Polynomials

Stirling, bell, bernoulli, euler and eulerian numbers

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