Associated Lah numbers and r-Stirling numbers

Hacene Belbachir; Imad Eddine Bousbaa

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Associated Lah numbers and r-Stirling numbers

Hacene Belbachir, Imad Eddine Bousbaa

Combinatorics · Submitted 22 Apr 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim For all these sequences we give combinatorial interpretation, generating functions, recurrence relations, convolution identities.

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We introduce the associated Lah numbers. Some recurrence relations and convolution identities are established. An extension of the associated Stirling and Lah numbers to the r-Stirling and r-Lah numbers are also given. For all these sequences we give combinatorial interpretation, generating functions, recurrence relations, convolution identities. In the sequel, we develop a section on nested sums related to binomial coefficient.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We introduce the associated Lah numbers.	contribution 1.00	0.95	0.18	asserted 0.98
s1	Some recurrence relations and convolution identities are established.	contribution 0.54	0.92	0.12	asserted 1.00
s2	An extension of the associated Stirling and Lah numbers to the r-Stirling and r-Lah numbers are also given.	contribution 0.99	0.92	0.16	asserted 1.00
s3	For all these sequences we give combinatorial interpretation, generating functions, recurrence relations, convolution identities.	contribution 0.95	0.94	0.27	asserted 1.00
s4	In the sequel, we develop a section on nested sums related to binomial coefficient.	future work 0.87	0.75	0.05	asserted 0.95

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Original abstract

We introduce the associated Lah numbers. Some recurrence relations and convolution identities are established. An extension of the associated Stirling and Lah numbers to the r-Stirling and r-Lah numbers are also given. For all these sequences we give combinatorial interpretation, generating functions, recurrence relations, convolution identities. In the sequel, we develop a section on nested sums related to binomial coefficient.

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8 arXiv papers cite this in 11 sentences

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Cites it as background 6 papers

When $r=0$, the $r$-Lah number reduces to the Lah number $nk$ (named for the mathematician Ivo Lah [ref] and often denoted by $L(n,k)$), which counts the number of partitions of $[n]$ into $k$ ordered blocks (see, e.g., [ref] ).

Generalized r-Lah numbers · Introduction · 0.95
Belbachir and Bousbaa [ref] studied $r$-associated Lah numbers, $L(n,k)_R$ also with $R = {r, r+1, r+2, …}$.

Restricted Stirling and Lah number matrices and their inverses · Introduction · 0.99
Several combinatorial properties of these sequences can be found in [ref] .

Some Applications of $S$-restricted Set Partitions · $S$-restricted Set Partitions and Stirling Numbers · 0.95

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