منابع مشابه
On 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 کاملCombinatorial Interpretations of Congruences for the Spt-function
Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. In 1988, the second author gave new combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions. In 2008, the first author found Ramanujantype congruences for the spt-function mod 5, 7 and 13. We give new combinatorial inte...
متن کاملRank and Crank Analogs for some Colored Partitions
We establish some rank and crank analogs for partitions into distinct colors and give combinatorial interpretations for colored partitions such as partitions defined by Toh, Zhang and Wang congruences modulo 5, 7.
متن کاملPartitions 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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal für die reine und angewandte Mathematik (Crelles Journal)
سال: 2016
ISSN: 0075-4102,1435-5345
DOI: 10.1515/crelle-2014-0035