Asymptotic relations for partitions

Richmond, L. B.

doi:10.1090/S0002-9947-1976-0412137-2

Asymptotic relations for partitions
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by L. B. Richmond

Trans. Amer. Math. Soc. 219 (1976), 379-385

DOI: https://doi.org/10.1090/S0002-9947-1976-0412137-2

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Abstract:

Asymptotic relations are obtained for the number ${p_A}(n)$ of partitions of the integer n into summands from a set A of integers. The set A is subject to certain conditions; however the only arithmetic condition is that A have property ${P_k}$ of Bateman and Erdös. A conjecture of Bateman and Erdös concerning the kth differences of ${p_A}(n)$ may be verified using these asymptotic relations.

References

P. T. Bateman and P. Erdös, Monotonicity of partition functions, Mathematika 3 (1956), 1–14. MR 80121, DOI 10.1112/S002557930000084X
L. B. Richmond, Asymptotic relations for partitions, J. Number Theory 7 (1975), no. 4, 389–405. MR 382210, DOI 10.1016/0022-314X(75)90043-8
L. Bruce Richmond, The moments of partitions. II, Acta Arith. 28 (1975/76), no. 3, 229–243. MR 401687, DOI 10.4064/aa-28-3-229-243
K. F. Roth and G. Szekeres, Some asymptotic formulae in the theory of partitions, Quart. J. Math. Oxford Ser. (2) 5 (1954), 241–259. MR 67913, DOI 10.1093/qmath/5.1.241

Asymptotic results for partitions

Conjecture of Bateman and Erdös