منابع مشابه
M2-rank Differences for Overpartitions
Abstract. This is the third and final installment in our series of papers applying the method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences. The study of rank differences was initiated by Atkin and Swinnerton-Dyer in their proof of Dyson’s conjectures concerning Ramanujan’s congruences for the partition function. Since then, other types of rank differences for statistics ...
متن کاملRank Differences for Overpartitions
The rank of a partition is the largest part minus the number of parts. This statistic was introduced by Dyson [14], who observed empirically that it provided a combinatorial explanation for Ramanujan’s congruences p(5n + 4) ≡ 0 (mod 5) and p(7n + 5) ≡ 0 (mod 7). Here p(n) denotes the usual partition function. Specifically, Dyson conjectured that if N(s,m, n) denotes the number of partitions of ...
متن کاملRank and Crank Moments for Overpartitions
We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for som...
متن کاملOverpartitions with Restricted Odd Differences
We use q-difference equations to compute a two-variable q-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpar...
متن کاملDyson’s Rank, Overpartitions, and Weak Maass Forms
In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decomposit...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2010
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa144-2-8