A simple combinatorial interpretation of certain generalized Bell and Stirling numbers

Codara, Pietro; DʼAntona, Ottavio M.; Hell, Pavol

doi:10.1016/j.disc.2013.11.010

Cited by 9 publications

(11 citation statements)

References 6 publications

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“…In the case w = (xD) n , G w turns out to be the empty graph on n vertices and we recover the usual combinatorial interpretation of n k . In the case w = (x s D s ) n , G w is the disjoint union of n copies of the complete graph on s vertices, and we recover a recent result of Codara et al [14,Proposition 2.2]. To the best of our knowledge, ours is the first combinatorial interpretation of S w (k) for arbitrary w as a count of (restricted) partitions.…”

Section: A New Combinatorial Interpretation Of S W (K)supporting

confidence: 78%

“…In 1973 Navon [22] provided a lovely combinatorial interpretation of S w (k) for all w, associating a Ferrers board to w and realizing S w (k) as the number of ways of placing non-attacking rooks on the board (see Section 4 for more details); Varvak thoroughly explored this interpretation, and obtained q-analogs for it, in [28]. Very recently Codara et al [14] gave a combinatorial interpretation in terms of graph coloring in the case w = (x s D s ) n ; our Theorem 2.3, which was developed independently, generalizes this interpretation to arbitrary w.…”

Section: Introductionmentioning

confidence: 99%

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Combinatorially interpreting generalized Stirling numbers

Engbers

Galvin

Hilyard

2015

European Journal of Combinatorics

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Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any smooth function $f(x)$, defines a sequence $(S_w(k))_k$, the terms of which we refer to as the {\em Stirling numbers (of the second kind)} of $w$. The nomenclature comes from the fact that when $w=(xD)^n$, we have $S_w(k)={n \brace k}$, the ordinary Stirling number of the second kind. Explicit expressions for, and identities satisfied by, the $S_w(k)$ have been obtained by numerous authors, and combinatorial interpretations have been presented. Here we provide a new combinatorial interpretation that retains the spirit of the familiar interpretation of ${n \brace k}$ as a count of partitions. Specifically, we associate to each $w$ a quasi-threshold graph $G_w$, and we show that $S_w(k)$ enumerates partitions of the vertex set of $G_w$ into classes that do not span an edge of $G_w$. We also discuss some relatives of, and consequences of, our interpretation, including $q$-analogs and bijections between families of labelled forests and sets of restricted partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00

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Section: A New Combinatorial Interpretation Of S W (K)supporting

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Combinatorially interpreting generalized Stirling numbers

Engbers

Galvin

Hilyard

2015

European Journal of Combinatorics

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“…Note that the general formulas for S r,s (n, k) given in [24] can be used to compute our numbers S G , B G (defined in Section 2). Interpretation to S r,r (n, k) with colorings of complete graphs introduced in [7] is very close to the meaning of S λ (n, k) (λ = (r, r, . .…”

Section: The λ-Stirling Numbersmentioning

confidence: 58%

Walks, Partitions, and Normal Ordering

Dzhumadil’daev

Yeliussizov

2015

Electron. J. Combin.

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“…The investigation of the normal ordered expression of ((a † ) r a s ) n naturally gives rise to generalized Stirling numbers S r,s (n, k) and Bell polynomials [6,7]. The interpretations of some special cases are well known and related to combinatorial numbers [11,8]. For instance, for r = 2 and s = 1, the number S 2,1 (n, k) = n−1 k−1 n!…”

Section: Introductionmentioning

confidence: 99%

A combinatorial Hopf algebra for the boson normal ordering problem

Bousbaa

Chouria

Luque

2018

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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In the aim to understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak et al. defined combinatorial objects allowing to interpret the number of Sr,s(k) appearing in thewhere α is assumed to be non-negative. These objects are used to define a combinatorial Hopf algebra which specializes to the enveloping algebra of the Heisenberg Lie algebra. Here, we propose a new variant of this construction which admits a realization with variables. This means that we construct our algebra from a free algebra C⟨A⟩ using quotient and shifted product. The combinatorial objects (B-diagrams) are slightly different from those proposed by Blasiak et al., but give also a combinatorial interpretation of the generalized Stirling numbers together with a combinatorial Hopf algebra related to Heisenberg Lie algebra. The main difference comes from the fact that the B-diagrams have the same number of inputs and outputs. After studying the combinatorics and the enumeration of B-diagrams, we propose two constructions of algebras called Fusion algebra F, defined using formal variable and another algebra B constructed directly from the B-diagrams. We show the connection between these two algebras and that B can be endowed with a Hopf structure. We recognize two already known combinatorial Hopf subalgebras of B : WSym the algebra of word symmetric functions indexed by set partitions and BWSym the algebra of biword symmetric functions indexed by set partitions into lists.

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A simple combinatorial interpretation of certain generalized Bell and Stirling numbers

Cited by 9 publications

References 6 publications

Combinatorially interpreting generalized Stirling numbers

Combinatorially interpreting generalized Stirling numbers

Walks, Partitions, and Normal Ordering

A combinatorial Hopf algebra for the boson normal ordering problem

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