Please use this identifier to cite or link to this item: http://hdl.handle.net/10497/22490
Title: 
Authors: 
Keywords: 
Partitions
Stanley’s partition function
Ramanujan’s congruences
Issue Date: 
2020
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. Advance online publication. 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).
Description: 
This is the final draft, after peer-review, of a manuscript published in The Ramanujan Journal. The published version is available online at https://doi.org/10.1007/s11139-020-00315-5
URI: 
ISSN: 
1382-4090 (print)
1572-9303 (online)
Other Identifiers: 
10.1007/s11139-020-00315-5
Website: 
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)
Appears in Collections:Journal Articles

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