2012/03/20 by William Y. C. Chen, Chen, William Y. C., Doris D. M. Sang +3
Mathematics · #05A17 #11P84 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1203.4302
openalex publication_date 2012/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For k≥ i≥ 1, let Bk,i(n) denote the number of partitions of n such that part 1 appears at most i-1 times, two consecutive integers l and l+1 appear at most k-1 times and if l and l+1 appear exactly k-1 times then the total sum of the parts l and l+1 is congruent to i-1 modulo 2. Let Ak,i(n) denote the number of partitions with parts not congruent to i, 2k-i and 2k modulo 2k. Bressoud's theorem states that Ak,i(n)=Bk,i(n). Corteel, Lovejoy, and Mallet found an overpartition analogue of Bressoud's theorem for i=1, that is, for partitions not containing nonoverlined part 1. We obtain an overpartition analogue of Bressoud's theorem in the general case. For k≥ i≥ 1, let Dk,i(n) denote the number of overpartitions of n such that the nonoverlined part 1 appears at most i-1 times, for any integer l, l and nonoverlined l+1 appear at most k-1 times and if the parts l and the nonoverlined part l+1 appear exactly k-1 times then the total sum of the parts l and nonoverlined part l+1 is congruent to the number of overlined parts that are less than l+1 plus i-1 modulo 2. Let Ck,i(n) denote the number of overpartitions with the nonoverlined parts not congruent to ± i and 2k-1 modulo 2k-1. We show that Ck,i(n)=Dk,i(n). This relation can also be considered as a Rogers-Ramanujan-Gordon type theorem for overpartitions.