arXiv Analytics

Sign in

arXiv:1406.5458 [math.NT]AbstractReferencesReviewsResources

Another proof of two modulo 3 congruences and another SPT crank for the number of smallest parts in overpartitions with even smallest part

Chris Jennings-Shaffer

Published 2014-06-20Version 1

By considering the $M_2$-rank of an overpartition as well as a residual crank, we give another combinatorial refinement of the congruences $\overline{\mbox{spt}}_2(3n)\equiv \overline{\mbox{spt}}_2(3n+1)\equiv 0\pmod{3}$. Here $\overline{\mbox{spt}}_2(n)$ is the total number of occurrences of the smallest parts among the overpartitions of $n$ where the smallest part is even and not overlined. Our proof depends on Bailey's Lemma and the rank difference formulas of Lovejoy and Osburn for the $M_2$-rank of an overpartition. This congruence, along with a modulo $5$ congruence, has previously been refined using the rank of an overpartition.

Comments: 7 pages
Categories: math.NT
Related articles: Most relevant | Search more
arXiv:1109.2340 [math.NT] (Published 2011-09-11, updated 2011-10-19)
An Extension of a Congruence by Kohnen
arXiv:1604.00445 [math.NT] (Published 2016-04-02)
A congruence involving the quotients of Euler and its applications (III)
arXiv:0812.2841 [math.NT] (Published 2008-12-15, updated 2009-02-19)
On a congruence only holding for primes II