2013/10/31 by William Y. C. Chen, Chen, William Y. C., Kathy Q. Ji +3
Mathematics · #05A17 #05A30 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1310.8556
openalex publication_date 2013/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By introducing k-marked Durfee symbols, Andrews found a combinatorial interpretation of 2k-th symmetrized moment η2k(n) of ranks of partitions of n in terms of (k+1)-marked Durfee symbols of n. In this paper, we consider the k-th symmetrized positive moment ηk(n) of ranks of partitions of n which is defined as the truncated sum over positive ranks of partitions of n. As combintorial interpretations of η2k(n) and η2k-1(n), we show that for fixed k and i with 1≤ i≤ k+1, η2k-1(n) equals the number of (k+1)-marked Durfee symbols of n with the i-th rank being zero and η2k(n) equals the number of (k+1)-marked Durfee symbols of n with the i-th rank being positive. The interpretations of η2k-1(n) and η2k(n) also imply the interpretation of η2k(n) given by Andrews since η2k(n) equals η2k-1(n) plus twice of η2k(n). Moreover, we obtain the generating functions of η2k(n) and η2k-1(n).