The Deranged Bell Numbers
-
Hacène Belbachir
and
László Németh
ABSTRACT
Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set [n] = {1, 2, … n}, this paper introduces the deranged partition as a free-fixed-block permutation of its blocks, then defines the deranged Bell numbers that count the total number of the deranged partitions of [n].
At first, we study the classical properties of these numbers (generating function, explicit formula, convolutions, etc.), we then present an asymptotic behavior of the deranged Bell numbers. Finally, we briefly review some results regarding the r-extension of these numbers by considering that the r first elements must be in distinct blocks.
Funding statement: For H. Belbachir and Y. Djemmada, the paper was partially supported by the DGRSDT grant C0656701.
Acknowledgement
We would like to thank the anonymous reviewers for their suggestions and comments which improved the quality of the present paper.
REFERENCES
[1] AIGNER, M.: Combinatorial Theory, Springer Science & Business Media, 2012.Search in Google Scholar
[2] APOSTOL, T. M.: Calculus: One-Variable Calculus, with an Introduction to Linear Algebra, Vol. 1, John Wiley & Sons, 2007.Search in Google Scholar
[3] BONA, M.: Combinatorics of Permutations, 2nd ed., Chapman and Hall/CRC, 2012.Search in Google Scholar
[4] BRODER, A. Z.: The r-Stirling numbers, Discrete Math. 49(3) (1984), 241–259.10.1016/0012-365X(84)90161-4Search in Google Scholar
[5] CAICEDO, J. B.—MOLL, V. H.—RAMÍREZ, J. L.—VILLAMIZAR, D.: Extensions of set partitions and permutations, Electron. J. Combin. 26(2) (2019), Art. No. P2.20.10.37236/8223Search in Google Scholar
[6] CHARALAMBIDES, C. A.: Enumerative Combinatorics, CRC Press, 2018.10.1201/9781315273112Search in Google Scholar
[7] COMTET, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Springer Science & Business Media, 2012.Search in Google Scholar
[8] DOBINSKI, G.: Summirung der Reihe
[9] EPSTEIN, L. F.: A function related to the series for ee x , Stud. Appl. Math. 18(1-4) (1939), 153–173.10.1002/sapm1939181153Search in Google Scholar
[10] EULER, L.: Institutiones Calculi Differentialis, Teubner, 1755.Search in Google Scholar
[11] FLAJOLET, P.—SEDGEWICK, R.: Analytic Combinatorics, Cambridge University Press, 2009.10.1017/CBO9780511801655Search in Google Scholar
[12] FONSECA, C. M. D.: On a closed form for derangement numbers: an elementary proof, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2020), Art. No. 146.10.1007/s13398-020-00879-3Search in Google Scholar
[13] GROSS, O. A.: Preferential arrangements, Amer. Math. Monthly 69(1) (1962), 4–8.10.1080/00029890.1962.11989826Search in Google Scholar
[14] LOEHR, N.: Combinatorics, CRC Press, 2017.Search in Google Scholar
[15] MANSOUR, T.—SCHORK, M.: Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press, 2015.10.1201/b18869Search in Google Scholar
[16] MEZÖ, I.: The r-Bell numbers, J. Integer Seq. 14(1) (2011), 1–14.Search in Google Scholar
[17] PITMAN, J.: Some probabilistic aspects of set partitions, Amer. Math. Monthly 104(3) (1997), 201–209.10.1080/00029890.1997.11990624Search in Google Scholar
[18] QI, F.—ZHAO, J. L.—GUO, B. N.: Closed forms for derangement numbers in terms of the Hessenberg determinants, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112(4) (2018), 933–944.10.1007/s13398-017-0401-zSearch in Google Scholar
[19] QUAINTANCE, J.—GOULD, H. W.: Combinatorial Identities for Stirling Numbers: the Unpublished Notes of HW Gould, World Scientific, 2015.10.1142/9821Search in Google Scholar
[20] ROTA, G.-C.: The number of partitions of a set, Amer. Math. Monthly 71(5) (1964), 498–504.10.1080/00029890.1964.11992270Search in Google Scholar
[21] TANNY, S. M.: On some numbers related to the Bell numbers, Canad. Math. Bull. 17(5) (1975), 733–738.10.4153/CMB-1974-132-8Search in Google Scholar
[22] WANG, C.—MISKA, P.—MEZO, I.: The r-derangement numbers, Discrete Math. 340(7) (2017), 1681–1692.10.1016/j.disc.2016.10.012Search in Google Scholar
[23] WHITWORTH, W. A.: Choice and Chance, D. Bell and Company, 1870.Search in Google Scholar
[24] WILF, H. S.: Generatingfunctionology, CRC press, 2005.10.1201/b10576Search in Google Scholar
© 2023 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- A Note on Special Subsets of the Rudin-Frolík Order for Regulars
- The 2-Class Group of Certain Families of Imaginary Triquadratic Fields
- The Deranged Bell Numbers
- On Index and Monogenity of Certain Number Fields Defined by Trinomials
- The k-Generalized Lucas Numbers Close to a Power of 2
- Shifted Power of a Polynomial with Integral Roots
- Further Insights into the Mysteries of the Values of Zeta Functions at Integers
- Memoryless Properties on Time Scales
- A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi
- Besov and Triebel-Lizorkin Capacity in Metric Spaces
- Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part
- An Elliptic Type Inclusion Problem on the Heisenberg Lie Group
- Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator
- A New Series Space Derived by Absolute Generalized Nörlund Means
- Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors
- The Uniform Effros Property and Local Homogeneity
- Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions
- The Unit-Gompertz Quantile Regression Model for the Bounded Responses
- An Extended Gamma-Lindley Model and Inference for the Prediction of Covid-19 in Tunisia
- Modeling Bivariate Data Using Linear Exponential and Weibull Distributions as Marginals
Articles in the same Issue
- A Note on Special Subsets of the Rudin-Frolík Order for Regulars
- The 2-Class Group of Certain Families of Imaginary Triquadratic Fields
- The Deranged Bell Numbers
- On Index and Monogenity of Certain Number Fields Defined by Trinomials
- The k-Generalized Lucas Numbers Close to a Power of 2
- Shifted Power of a Polynomial with Integral Roots
- Further Insights into the Mysteries of the Values of Zeta Functions at Integers
- Memoryless Properties on Time Scales
- A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi
- Besov and Triebel-Lizorkin Capacity in Metric Spaces
- Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part
- An Elliptic Type Inclusion Problem on the Heisenberg Lie Group
- Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator
- A New Series Space Derived by Absolute Generalized Nörlund Means
- Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors
- The Uniform Effros Property and Local Homogeneity
- Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions
- The Unit-Gompertz Quantile Regression Model for the Bounded Responses
- An Extended Gamma-Lindley Model and Inference for the Prediction of Covid-19 in Tunisia
- Modeling Bivariate Data Using Linear Exponential and Weibull Distributions as Marginals
