تعداد نشریات | 43 |
تعداد شمارهها | 1,650 |
تعداد مقالات | 13,402 |
تعداد مشاهده مقاله | 30,203,849 |
تعداد دریافت فایل اصل مقاله | 12,074,494 |
Refinements of the Bell and Stirling numbers | ||
Transactions on Combinatorics | ||
مقاله 4، دوره 7، شماره 4، اسفند 2018، صفحه 25-42 اصل مقاله (269.45 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22108/toc.2018.110171.1560 | ||
نویسنده | ||
Tanay Wakhare* | ||
University of Maryland | ||
چکیده | ||
We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial, restricted Bell, and $r$-derangement numbers (and probably more!). By combining methods from analytic combinatorics, umbral calculus, and probability theory, we derive several recurrence relations and closed form expressions for these numbers. By specializing our results to the classical case, we recover explicit formulae for the Bell and Stirling numbers as sums over compositions. | ||
کلیدواژهها | ||
Bell numbers؛ Stirling numbers | ||
مراجع | ||
[1] T. Amdeb erhan and V. Moll, Involutions and their progenies, J. Comb. 6 (2015) 483{508. [2] P. Blasiak and F. Fla jolet, Combinatorial mo dels of creation-annihilation, Sm. Lothar. Combin. , 65 (2010/12) pp. 78. [3] M. Bona and I. Mez}o, Real zero es and partitions without singleton blo cks, European. J. Combin. , 51 (2016) 500{510. [4] L. Comtet, Advanced combinatorics , The art of nite and innite expansions, D. Reidel Publishing Co., Dordrecht, 1974. [5] V. De Angelis and D. Marcello, Wilf 's conjecture, Amer. Math. Monthly , 123 (2016) 557{573. [6] J. Engb ers, D. Galvin and C. Smyth, Restricted Stirling and Lah numb ers and their inverses, preprint (2016), https://arxiv.org/abs/1610.05803 . [7] P. Fla jolet and R. Sedgewick, Analytic combinatorics , Cambridge University Press, Cambridge, 2009. [8] I. M. Gessel, Applications of the classical umbral calculus, Algebra Universalis , 49 (2003) 397{434. [9] T. Komatsu, K. Liptai and I. Mezo, Incomplete p oly-Bernoulli numb ers asso ciated with incomplete Stirling num- b ers, Publ. Math. Debrecen , 88 (2016) 357{368. [10] I. G. Macdonald, Symmetric functions and orthogonal polynomials , University Lecture Series, 12 American Math-ematical So ciety, Providence, RI, 1998. [11] T. Mansour, Combinatorics of set partitions , Discrete Mathematics and its Applications, CRC Press, Bo ca Raton, FL, 2013 [12] F. L. Miksa, L. Moset and M. Wyman, Restricted partitions of nite sets, Canad. Math. Bul l. , 1 (1958) 87{96. [13] V. H. Moll, J. L. Ramirez and D. Villamizar, Combinatorial and arithmetical prop erties of the restricted and asso ciated Bell and factorial numb ers, preprint (2017), https://arxiv.org/abs/1706.00165 . [14] P. Mongelli, Combinatorial interpretations of particular evaluations of complete and elementary symmetric func- tions, Electron. J. Combin. , 19 (2012) pp. 23. [15] J. P. Nolan, Stable distributions - models for heavy tailed data , In progress, Chapter 1 online at http://fs2. american.edu/jpnolan/www/stable/stable.html , Birkhauser, Boston MA, 2018. [16] S. Roman, The umbral calculus , Academic Press, Inc., New York, 1984 [17] M. Z. Spivey, A generalized recurrence for Bell numb ers, J. Integer Seq. , 11 (2008) pp. 3. [18] C. Vignat, A probabilistic approach to some results by Nieto and Truax, J. Math. Phys. , 51 (2010) pp. 9. [19] C. Vignat and O. Lvque, Pro of of a conjecture by Gazeau et al. using the Gould-Hopp er p olynomials, J. Math. Phys., 54 (2013) pp. 8. [20] C. Vignat and T. Wakhare, Wo on's tree and sums over comp ositions, J. Integer Seq. , 21 (2018) Article 18.3.4. | ||
آمار تعداد مشاهده مقاله: 344 تعداد دریافت فایل اصل مقاله: 616 |