Associated Lah numbers and r-Stirling numbers

Belbachir, Hacène; Bousbaa, Imad Eddine

doi:10.48550/arxiv.1404.5573

Cited by 4 publications

(9 citation statements)

References 10 publications

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“…k even, and v 1 not a left-most child of a vertex w with d(w) = s d (2). If v 1 is not deleted then these properties hold for v in F exactly as in T .…”

Section: Proofsmentioning

confidence: 98%

“…We now show items 2 and 3 in the case that A(T ) is produced from T by an uncontraction at vertex v j in step 1(b) of the algorithm. Suppose (2). It follows that A(T ) has all down-degrees in R(d), and we have item 2.…”

Section: Proofsmentioning

confidence: 99%

“…Make v ′ j be the left-most child of v j and let the other d children of v j retain the linear ordering they had before. Now d(v j ) = s d (2). Leave the orderings on the children of all other vertices v = v j , v ′ j unchanged.…”

Section: Proofsmentioning

confidence: 99%

“…}, and obtained recurrence relations and generating functions for them. Belbachir and Bousbaa [2] studied r-associated Lah numbers, L(n, k) R also with R = {r, r + 1, r + 2, . .…”

Section: Introductionmentioning

confidence: 99%

“…In [4] Choi, Long, Ng and Smith note that n k −1 [2] is a Bessel number [13, A100861] and has many combinatorial interpretations. For example, (−1) n−k n k −1 [2] counts the number of size n − k matchings of the complete graph K 2n−1−k [6]. They asked if n k −1…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Restricted Stirling and Lah number matrices and their inverses

Engbers¹,

Galvin²,

Smyth³

2019

Journal of Combinatorial Theory, Series A

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Given R ⊆ N let n k R , n k R , and L(n, k) R count the number of ways of partitioning the set [n] := {1, 2, . . . , n} into k non-empty subsets, cycles and lists, respectively, with each block having cardinality in R. We refer to these as the R-restricted Stirling numbers of the second and first kind and the R-restricted Lah numbers, respectively. Note that the classical Stirling numbers of the second kind and first kind, and Lah numbers are n k = n k N , n k = n k N and L(n, k) = L(n, k) N , respectively.n,k≥1 and [L(n, k) R ] −1 n,k≥1 exist and have integer entries if and only if 1 ∈ R. We express each entry of each of these matrices as the difference between the cardinalities of two explicitly defined families of labeled forests. In particular the entries of [ n k [r] ] −1 n,k≥1 have combinatorial interpretations, affirmatively answering a question of Choi, Long, Ng and Smith from 2006.If we have 1, 2 ∈ R and if for all n ∈ R with n odd and n ≥ 3, we have n ± 1 ∈ R, we additionally show that each entry of [ n k R ] −1 n,k≥1 , [ n k R ] −1 n,k≥1 and [L(n, k) R ] −1 n,k≥1 is up to an explicit sign the cardinality of a single explicitly defined family of labeled forests. With R as before we also do the same for restriction sets of the form R(d) = {d(r−1)+1 : r ∈ R} for all d ≥ 1. Our results also provide combinatorial interpretations of the kth Whitney numbers of the first and second kinds of Π 1,d n , the poset of partitions of [n] that have each part size congruent to 1 mod d.

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“…k even, and v 1 not a left-most child of a vertex w with d(w) = s d (2). If v 1 is not deleted then these properties hold for v in F exactly as in T .…”

Section: Proofsmentioning

confidence: 98%

Section: Proofsmentioning

confidence: 99%

Section: Proofsmentioning

confidence: 99%

“…}, and obtained recurrence relations and generating functions for them. Belbachir and Bousbaa [2] studied r-associated Lah numbers, L(n, k) R also with R = {r, r + 1, r + 2, . .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Restricted Stirling and Lah number matrices and their inverses

Engbers¹,

Galvin²,

Smyth³

2019

Journal of Combinatorial Theory, Series A

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Some applications of S-restricted set partitions

Bényi

Ramírez

2018

Period Math Hung

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Generalized r-Lah numbers

Shattuck¹

2016

Proc Math Sci

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In this paper, we consider a two-parameter polynomial generalization, denoted by G a,b (n, k; r), of the r-Lah numbers which reduces to these recently introduced numbers when a = b = 1. We present several identities for G a,b (n, k; r) that generalize earlier identities given for the r-Lah and r-Stirling numbers. We also provide combinatorial proofs of some identities involving the r-Lah numbers which were established previously using algebraic methods. Generalizing these arguments yields orthogonality-type relations that are satisfied by G a,b (n, k; r).

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Associated Lah numbers and r-Stirling numbers

Cited by 4 publications

References 10 publications

Restricted Stirling and Lah number matrices and their inverses

Restricted Stirling and Lah number matrices and their inverses

Some applications of S-restricted set partitions

Generalized r-Lah numbers

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