The crank of partitions mod $8,\;9$ and $10$
نویسندگان
چکیده
منابع مشابه
Partitions weighted by the parity of the crank
Abstract. A partition statistic ‘crank’ gives combinatorial interpretations for Ramanujan’s famous partition congruences. In this paper, we establish an asymptotic formula, Ramanujan type congruences, and q-series identities that the number of partitions with even crank Me(n) minus the number of partitions with odd crank Mo(n) satisfies. For example, we show that Me(5n + 4)− Mo(5n + 4) ≡ 0 (mod...
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Let N(r,m, n) (resp. M(r,m, n)) denote the number of partitions of n whose ranks (resp. cranks) are congruent to r modulo m. Atkin and Swinnerton-Dyer gave the relations between the numbers N(r,m,mn+k) when m = 5, 7 and 0 ≤ r, k < m. Garvan gave the relations between the numbers M(r,m,mn+k) when m = 5, 7, and 11, 0 ≤ r, k < m. Here, we show that the methods of Atkin and Swinnerton-Dyer can be e...
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Forms of Nim have been played since antiquity and a complete theory was published as early as 1902 (see [3]). Martin Gardner described the game in one of his earliest columns [7] and returned to it many times over the years ([8–16]). Central to the analysis of Nim is Nim-addition. The Nim-sum is calculated by writing the terms in base 2 and adding the columns mod 2, with no carries. A Nim posit...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1990
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-1990-1012520-8