Abstract
AbstractWe present a unified framework of combinatorial descriptions, and the analogous asymptotic growth of the coefficients of two general families of functions related to integer partitions. In particular, we resolve several conjectures and verify several claims that are posted on the On-Line Encyclopedia of Integer Sequences. We perform the asymptotic analysis by systematically applying the Mellin transform, residue analysis, and the saddle point method. The combinatorial descriptions of these families of generalized partition functions involve colorings of Young tableaux, along with their “divisor diagrams”, denoted with sets of colors whose sizes are controlled by divisor functions.
Funder
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México
National Science Foundation
Foundation for Food and Agriculture Research
National Institute of Food and Agriculture
Society of Actuaries
Cummins Incorporated
Gro Master
Lilly Endowment
Sandia National Laboratories
Publisher
Springer Science and Business Media LLC
Reference44 articles.
1. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
2. Euler, L.: Introductio in analysin infinitorum. vol. 1. Marcum-Michaelem Bousquet; 1748. (Introduction to the Analysis of the Infinite, Book 1), Translated by J. D. Blanton. Springer (1988)
3. Hardy, G.H., Ramanujan, S.: Asymptotic formulae for the distribution of integers of various types. Proc. Lond. Math. Soc. Ser. 2(16), 112–132 (1917)
4. Hardy, G.H., Ramanujan, S.: Asymptotic formulæ in combinatory analysis. Proc. Lond. Math. Soc. Ser. 2(17), 75–115 (1918)
5. Berndt, B.C., Robles, N., Zaharescu, A., Zeindler, D.: Partitions with multiplicities associated with divisor functions. J Math Anal Appl. 533(1), Paper No. 127987, 32 (2024). https://doi.org/10.1016/j.jmaa.2023.127987