Partitions with congruence conditions
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- by M. M. Robertson PDF
- Proc. Amer. Math. Soc. 57 (1976), 45-49 Request permission
Abstract:
Let $A = \cup _{i = 1}^q\{ a(i) + \nu M:\nu = 0,1,2, \ldots \}$, where $q,M$ and the $a(i)$ are positive integers such that $a(1) < a(2) < \cdots < a(q) \leqslant M$. Asymptotic formulae are obtained for $p(n,k,A),{p^ \ast }(n,k,A)$ the number of partitions of $n$ into $k$ parts, $k$ unequal parts respectively, where all the parts belong to $A$.References
- Paul Erdös and Joseph Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8 (1941), 335–345. MR 4841
- Emil Grosswald, Some theorems concerning partitions, Trans. Amer. Math. Soc. 89 (1958), 113–128. MR 97371, DOI 10.1090/S0002-9947-1958-0097371-5
- Peter Hagis Jr., A problem on partitions with a prime modulus $p\geq 3$, Trans. Amer. Math. Soc. 102 (1962), 30–62. MR 146166, DOI 10.1090/S0002-9947-1962-0146166-3 G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. (2) 17 (1918), 75-115.
- Shô Iseki, A partition function with some congruence condition, Amer. J. Math. 81 (1959), 939–961. MR 108473, DOI 10.2307/2372997
- Joseph Lehner, A partition function connected with the modulus five, Duke Math. J. 8 (1941), 631–655. MR 5523
- John Livingood, A partition function with the prime modulus $P>3$, Amer. J. Math. 67 (1945), 194–208. MR 12101, DOI 10.2307/2371722
- Ivan Niven, On a certain partition function, Amer. J. Math. 62 (1940), 353–364. MR 1235, DOI 10.2307/2371459 H. Rademacher, On the partition function $p(n)$, Proc. London Math. Soc. (2) 43 (1937), 241-254.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 57 (1976), 45-49
- MSC: Primary 10J20; Secondary 10A45
- DOI: https://doi.org/10.1090/S0002-9939-1976-0404184-7
- MathSciNet review: 0404184