منابع مشابه
Rank 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 کاملOn Spt-crank Type Functions
In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function spt ω (n), which is closely related to Ramanujan’s third order mock theta function ω(q). Garvan and Jennings-Shaffer introduce a crank function which explains congruences for spt ω (n). In this note, we study asymptotic behavior of this crank function and confirm a positivity conjecture of the crank asymptotically. We...
متن کاملOn the Distribution of the spt-Crank
Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence {NS(m,n)}m is unimodal, where NS(m,n) is the number of S-partitions of size n with crank m weight by the spt-crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conj...
متن کاملThe First Positive Rank and Crank Moments for Overpartitions
In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.
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 ...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2014
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa166-2-3