Mathematics | |
On the Distribution of the spt-Crank | |
关键词: partitions; partition crank; partition rank; spt-crank; unimodal; | |
DOI : 10.3390/math1030076 | |
来源: DOAJ |
【 摘 要 】
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 conjecture is true and allows us to asymptotically establish the conjecture. Additionally, we give an asymptotic study for the distribution of the spt-crank statistic. Finally, we give some speculations about a definition for the spt-crank in terms of “marked” partitions. A “marked” partition is an unrestricted integer partition where each part is marked with a multiplicity number. It remains an interesting and apparently challenging problem to interpret the spt-crank in terms of ordinary integer partitions.
【 授权许可】
Unknown