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A positivity conjecture related first positive rank and crank moments for overpartitions. (English) Zbl 1415.11136

Summary: Recently, G. Andrews et al. [Ann. Comb. 20, No. 2, 193–207 (2016; Zbl 1405.11136)] introduced a \(q\)-series \(h(q)\) for the study of the first positive rank and crank moments for overpartitions. They conjectured that for all integers \(m \geq 3\), \[ \frac{1}{(q)_{\infty}} (h(q) - m h(q^{m})) \] has positive power series coefficients for all powers of \(q\). B. Kim et al. [Arch. Math. 102, No. 4, 357–368 (2014; Zbl 1305.11086)] provided a combinatorial interpretation and proved it is asymptotically true. In this note, we show this conjecture is true if \(m\) is any positive power of 2, and we show that in order to prove this conjecture, it is only to prove it for all primes \(m\). Moreover we give a stronger conjecture. Our method is completely different from that of Kim et al. [loc. cit.].

MSC:

11P82 Analytic theory of partitions
05A17 Combinatorial aspects of partitions of integers
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