Comptes Rendus
Combinatoire
Associated r-Dowling numbers and some relatives
Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 47-55.

In this paper, we introduce a new generalization of Bell numbers, the s-associated r-Dowling numbers by combining two investigational directions. Here, r distinguished elements have to be in distinct blocks, some elements are coloured according to a colouring rule, and the cardinality of certain blocks is bounded from below by s. Along with them, we define some relatives, the s-associated r-Dowling factorials and the s-associated r-Dowling–Lah numbers, when the underlying set is decomposed into cycles or ordered blocks. The study of these numbers is based on their combinatorial meaning, and the exponential generating functions of their sequences derived from the so-called r-compositional formula.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.145
Classification : 05A15, 05A18, 05A19, 11B73

Eszter Gyimesi 1 ; Gábor Nyul 1

1 Institute of Mathematics, University of Debrecen, H–4002 Debrecen P.O.Box 400, Hungary
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2021__359_1_47_0,
     author = {Eszter Gyimesi and G\'abor Nyul},
     title = {Associated $r${-Dowling} numbers and some relatives},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {47--55},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {1},
     year = {2021},
     doi = {10.5802/crmath.145},
     zbl = {1420.11048},
     language = {en},
}
TY  - JOUR
AU  - Eszter Gyimesi
AU  - Gábor Nyul
TI  - Associated $r$-Dowling numbers and some relatives
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 47
EP  - 55
VL  - 359
IS  - 1
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.145
LA  - en
ID  - CRMATH_2021__359_1_47_0
ER  - 
%0 Journal Article
%A Eszter Gyimesi
%A Gábor Nyul
%T Associated $r$-Dowling numbers and some relatives
%J Comptes Rendus. Mathématique
%D 2021
%P 47-55
%V 359
%N 1
%I Académie des sciences, Paris
%R 10.5802/crmath.145
%G en
%F CRMATH_2021__359_1_47_0
Eszter Gyimesi; Gábor Nyul. Associated $r$-Dowling numbers and some relatives. Comptes Rendus. Mathématique, Volume 359 (2021) no. 1, pp. 47-55. doi : 10.5802/crmath.145. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.145/

[1] Beáta Bényi; Miguel Méndez; José L. Ramírez; Tanay Wakhare Restricted r-Stirling numbers and their combinatorial applications, Appl. Math. Comput., Volume 348 (2019), pp. 186-205 | MR | Zbl

[2] Miklós Bóna; István Mező Real zeros and partitions without singleton blocks, Eur. J. Comb., Volume 51 (2016), pp. 500-510 | DOI | MR | Zbl

[3] Jhon B. Caicedo; Victor H. Moll; José L. Ramírez; Diego Villamizar Extensions of set partitions and permutations, Electron. J. Comb., Volume 26 (2019) no. 2, P2.20 | MR | Zbl

[4] Leonard Carlitz Weighted Stirling numbers of the first and second kind I, Fibonacci Q., Volume 18 (1980), pp. 147-162 | MR | Zbl

[5] Gi-Sang Cheon; Ji-Hwan Jung r-Whitney numbers of Dowling lattices, Discrete Math., Volume 312 (2012) no. 15, pp. 2337-2348 | DOI | MR | Zbl

[6] Cristina B. Corcino; Roberto B. Corcino; I. Mező; José L. Ramírez Some polynomials associated with the r-Whitney numbers, Proc. Indian Acad. Sci., Math. Sci., Volume 128 (2018), 27 | MR | Zbl

[7] Roberto B; Corcino; Cristina B. Corcino; Rodelito Aldema Asymptotic normality of the (r,β)-Stirling numbers, Ars Comb., Volume 81 (2006), pp. 81-96 | MR | Zbl

[8] E. A. Enneking; J. C. Ahuja Generalized Bell numbers, Fibonacci Q., Volume 14 (1976), pp. 67-73 | MR | Zbl

[9] Philippe Flajolet; Robert Sedgewick Analytic Combinatorics, Cambridge University Press, 2009 | Zbl

[10] Eszter Gyimesi The r-Dowling–Lah polynomials (to appear in Mediterranean Journal of Mathematics) | Zbl

[11] Eszter Gyimesi; Gábor Nyul A comprehensive study of r-Dowling polynomials, Aequationes Math., Volume 92 (2018) no. 3, pp. 515-527 | DOI | MR | Zbl

[12] Eszter Gyimesi; Gábor Nyul New combinatorial interpretations of r-Whitney and r-Whitney–Lah numbers, Discrete Appl. Math., Volume 255 (2019), pp. 222-233 | DOI | MR | Zbl

[13] Fredric T. Howard Numbers generated by the reciprocal of e x -x-1, Math. Comput., Volume 31 (1977) no. 138, pp. 581-598 | MR | Zbl

[14] Fredric T. Howard Associated Stirling numbers, Fibonacci Q., Volume 18 (1980), pp. 303-315 | MR | Zbl

[15] Fredric T. Howard Weighted associated Stirling numbers, Fibonacci Q., Volume 22 (1984), pp. 156-165 | MR | Zbl

[16] Zsófia Kereskényi-Balogh; Gábor Nyul Stirling numbers of the second kind and Bell numbers for graphs, Australas. J. Comb., Volume 58 (2014), pp. 264-274 | MR | Zbl

[17] István Mező The r-Bell numbers, J. Integer Seq., Volume 14 (2011) no. 1, 11.1.1 | MR | Zbl

[18] István Mező; Gábor Nyul The r-Fubini and r-Eulerian numbers (manuscript)

[19] István Mező; José L. Ramírez; Chen-Ying Wang On generalized derangements and some orthogonal polynomials, Integers, Volume 19 (2019), A6 | MR | Zbl

[20] Victor Hugo Moll; José L. Ramírez; Diego Villamizar Combinatorial and arithmetical properties of the restricted and associated Bell and factorial numbers, J. Comb., Volume 9 (2018) no. 4, pp. 693-720 | MR | Zbl

[21] Gábor Nyul; Gabriella Rácz Sums of r-Lah numbers and r-Lah polynomials, Ars Math. Contemp., Volume 18 (2020) no. 2, pp. 211-222 | DOI | MR | Zbl

[22] G. Pólya; G. Szegő Problems and Theorems in Analysis. Vol. I: Series. Integral calculus. Theory of functions. Translation by D. Aeppli, Grundlehren der mathematischen Wissenschaften, 193, Springer, 1972 | Zbl

[23] Richard P. Stanley Acyclic orientations of graphs, Discrete Math., Volume 5 (1973), pp. 171-178 | DOI | MR | Zbl

[24] Chenying Wang; Piotr Miska; István Mező The r-derangement numbers, Discrete Math., Volume 340 (2017), pp. 1681-1692 | DOI | MR | Zbl

[25] David G. L. Wang On colored set partitions of type B n , Cent. Eur. J. Math., Volume 12 (2014) no. 9, pp. 1372-1381 | MR | Zbl

Cité par Sources :

Commentaires - Politique