Some Congruences for Andrews’ Partition Function EO(n)
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Recently, Andrews introduced partition functions EO(n) and EO(n) where the function EO(n) denotes the number of partitions of n in which every even part is less than each odd part and the function EO(n) denotes the number of partitions enumerated by EO(n) in which only the largest even part appears an odd number of times. In this paper we obtain some congruences modulo 2, 4, 10 and 20 for the partition function EO(n). We give a simple proof of the first Ramanujan-type congruences EO (10n + 8) ≡ 0 (mod 5) given by Andrews.
Abstract
Recently, Andrews introduced partition functions EO(n) and EO(n) where the function EO(n) denotes the number of partitions of n in which every even part is less than each odd part and the function EO(n) denotes the number of partitions enumerated by EO(n) in which only the largest even part appears an odd number of times. In this paper we obtain some congruences modulo 2, 4, 10 and 20 for the partition function EO(n). We give a simple proof of the first Ramanujan-type congruences EO (10n + 8) ≡ 0 (mod 5) given by Andrews.
참고문헌 (7)
- G. E. Andrews, "The theory of partitions" (1998), Cambridge Univ. Press
- G. E. Andrews, "Integer partitions with even parts below odd parts and the mock theta functions", Ann. Comb, 22, pp.433~445 (2018)
- G. E. Andrews, "Partitions associated with the Ramanujan", Watson mock theta functions ω(q), ν(q) and ϕ(q), 2015, pp.1(19), 25~
- G. E. Andrews, "Congruences related to the Ramanujan", Watson mock theta functions ω(q), ν(q), 2017, pp.43, 347~357
- M. D. Hirschhorn, "Another short proof of Ramanujan’s mod 5 partition congruence, and more", Amer. Math. Monthly, 106, pp.580~583 (1999)
- M. D. Hirschhorn, "The power of q, Developments in Mathematics 49" (2017), Springer
- S. Ramanujan, "Some properties of p(n), the number of partitions of n", Proc. Camb. Philos. Soc, 19, pp.207~210 (1919)