2022/06/26 by Zhang, Helen W. J., Zhong, Ying
#05A20 #11P82 #11P83 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2206.12833
Bessenrodt and Ono initially found the strict log-subadditivity of partition function p(n), that is, p(a+b)< p(a)p(b) for a,b>1 and a+b>9. Many other important statistics of partitions are proved to enjoy similar properties. Lovejoy introduced the overpartition rank as an analog of Dyson's rank for partitions from the q-series perspective. Let N(a,c,n) denote the number of overpartitions with rank congruent to a modulo c. Ciolan computed the asymptotic formula of N(a,c,n) and showed that N(a, c, n) > N(b, c, n) for c≥7 and n large enough. In this paper, we derive an upper bound and a lower bound of N(a,c,n) for each c≥3 by using the asymptotics of Ciolan. Consequently, we establish the strict log-subadditivity of N(a,c,n) analogous to the partition function p(n).