Options
A variation of the Andrews–Stanley partition function and two interesting q-series identities
Citation
Lin, B. L. S., Peng, L., & Toh, P. C. (2020). A variation of the Andrews–Stanley partition function and two interesting q-series identities. The Ramanujan Journal, 56, 297–307. https://doi.org/10.1007/s11139-020-00315-5
Abstract
Stanley introduced a partition statistic srank(π)=O(π)−O(π′), where O(π) denote the number of odd parts of the partition π, and π′ is the conjugate of π. Let pi(n) denote the number of partitions of n with srank ≡i(mod4). Andrews proved the following refinement of Ramanujan’s partition congruence modulo 5:
p0(5n+4)≡p2(5n+4)≡0(mod5).
In this paper, we consider an analogous partition statistic
lrank(π)=O(π)+O(π′).
Let p+i(n) denote the number of partitions of n with lrank ≡i(mod4). We will establish the generating functions of p+0(n) and p+2(n) and show that they satisfy similar properties to pi(n). We also utilize a pair of interesting q-series identities to obtain a direct proof of the congruences
p+0(5n+4)≡p+2(5n+4)≡0(mod5).
Date Issued
2020
Publisher
Springer
Journal
The Ramanujan Journal
Grant ID
National Natural Science Foundation of China (Grant no. 11871246)
Natural Science Foundation of Fujian Province of China (Grant no. 2019J01328)
Program for New Century Excellent Talents in Fujian Province University (Grant no. B17160)