A generalized recurrence for Bell polynomials: An alternate approach to Spivey and Gould–Quaintance formulas

Belbachir, Hacène; Mihoubi, Miloud

doi:10.1016/j.ejc.2008.11.012

Cited by 20 publications

(17 citation statements)

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“…Furthermore, they extended these results to ordinary single variable Bell polynomials. Independently from Gould and Quaintance, Belbachir and Mihoubi [1] also obtained the same identity (15) in [11] and their proof follows a different approach. More recently, by using Faà di Bruno's formula [9,12] for higher order derivatives of composite functions, Xu and Cen [22] obtained some recurrence sequences including the Bell polynomials.…”

Section: Introductionmentioning

confidence: 85%

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Extensions of Spivey's Bell Number Formula

2012

Electron. J. Combin.

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Section: Introductionmentioning

confidence: 85%

“…This new formula includes both (1) and (2) as special cases and Spivey's proof was combinatorial. Gould and Quaintance [11] provided a generating function proof of Spivey's result and used Spivey's formula to obtain a new formula for B n .…”

Section: Introductionmentioning

confidence: 93%

Extensions of Spivey's Bell Number Formula

2012

Electron. J. Combin.

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“…and gives it a simple combinatorial proof. This recurrence has been generalized by Belbachir and Mihoubi [1], Gould and Quaintance [9]. We also have a similar formula for A n+k,k .…”

Section: Proof By Lemma 21 One Hasmentioning

confidence: 72%

The largest singletons of set partitions

Sun

2011

European Journal of Combinatorics

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Recently, Deutsch and Elizalde studied the largest and the smallest fixed points of permutations. Motivated by their work, we consider the analogous problems in set partitions. Let A n,k denote the number of partitions of {1, 2, . . . , n + 1} with the largest singleton {k + 1} for 0 ≤ k ≤ n. In this paper, several explicit formulas for A n,k , involving a Dobinski-type analog, are obtained by algebraic and combinatorial methods, many combinatorial identities involving A n,k and Bell numbers are presented by operator methods, and congruence properties of A n,k are also investigated. It will been showed that the sequences (A n+k,k ) n≥0 and (A n+k,k ) k≥0 (mod p) are periodic for any prime p, and contain a string of p − 1 consecutive zeroes. Moreover their minimum periods are conjectured to be N p = p p −1 p−1 for any prime p.

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“…Suppose that the cells are ( ), ( ) and [ ]. The partitions of the set [3] such that each cell contains at most 2 elements are listed below.…”

Section: Mixed Restricted Partition Numbersmentioning

confidence: 99%

Mixed restricted Stirling numbers

et al. 2019

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In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.2000 Mathematics Subject Classification. 05A18, 11B73. Key words and phrases. Multiplicative partition function; Stirling numbers of the second kind; mixed partitions of a set.

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A generalized recurrence for Bell polynomials: An alternate approach to Spivey and Gould–Quaintance formulas

Cited by 20 publications

References 1 publication

Extensions of Spivey's Bell Number Formula

Extensions of Spivey's Bell Number Formula

The largest singletons of set partitions

Mixed restricted Stirling numbers

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