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

2014/06/20 by Chris Jennings-Shaffer, Jennings-Shaffer, Chris
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #math.NT

paper · pdf · doi:10.48550/arxiv.1406.5458

7 pages

arxiv created 2014/06/20 · arxiv updated 2014/06/23

Abstract

By considering the M₂-rank of an overpartition as well as a residual crank, we give another combinatorial refinement of the congruences spt₂(3n)≡ spt₂(3n+1)≡ 0\pmod3. Here spt₂(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 M₂-rank of an overpartition. This congruence, along with a modulo 5 congruence, has previously been refined using the rank of an overpartition.

Related

• Extending Congruences for the number of smallest parts in overpartitions with smallest part even
• A conjecture of Baruah and Begum on the smallest parts function of restricted overpartitions
• Arithmetic Properties of Overpartition Triples
• The spt-crank for overpartitions
• Arithmetic Properties modulo powers of 2 and 3 for Overpartition k-Tuples with Odd Parts