## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### A positivity conjecture related first positive rank and crank moments for overpartitions

Xinhua Xiong

#### Abstract

Recently, Andrews, Chan, Kim and Osburn 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 $m \geq 3$, \begin{equation*} \frac{1}{(q)_{∞}} (h(q) - m h(q^{m})) \end{equation*} has positive power series coefficients for all powers of $q$. Byungchan Kim, Eunmi Kim and Jeehyeon Seo 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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 9 (2016), 117-120.

Dates
First available in Project Euclid: 2 November 2016

https://projecteuclid.org/euclid.pja/1478052017

Digital Object Identifier
doi:10.3792/pjaa.92.117

Mathematical Reviews number (MathSciNet)
MR3567597

Zentralblatt MATH identifier
06705717

#### Citation

Xiong, Xinhua. A positivity conjecture related first positive rank and crank moments for overpartitions. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 9, 117--120. doi:10.3792/pjaa.92.117. https://projecteuclid.org/euclid.pja/1478052017

#### References

• G. E. Andrews, S. H. Chan, B. Kim and R. Osburn, The first positive rank and crank moments for overpartitions, Ann. Comb. 20 (2016), no. 2, 193–207.
• K. Bringmann and K. Mahlburg, Asymptotic inequalities for positive crank and rank moments, Trans. Amer. Math. Soc. 366 (2014), no. 2, 1073–1094.
• S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1623–1635 (electronic).
• B. Kim, E. Kim and J. Seo, On the number of even and odd strings along the overpartitions of $n$, Arch. Math. (Basel) 102 (2014), no. 4, 357–368.
• E. M. Wright, Asymptotic partition formulae: (II) weighted partitions, Proc. London Math. Soc. S2-36 (1934), no. 1, 117–141.