An Overpartition Analogue of Bressoud's Theorem of Rogers-Ramanujan-Gordon Type
William Y.C. Chen, Doris D. M. Sang and Diane Y. H. Shi
Abstract: For k≥ 2 and 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 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 non-overlined part 1. We obtain an overpartition analogue of Bressoud's theorem in the general case. For k≥ 2 and k≥i≥1, let Dk,i(n) denote the number of overpartitions of n such that the non-overlined part 1 appears at most i-1 times, for any integer l, l and non-overlined l+1 appear at most k-1 times and if the parts l and the non-overlined part l+1 together appear exactly k-1 times then the sum of the parts l and non-overlined part l+1 has the same parity as the number of overlined parts that are less than l + 1 plus i - 1. Let Ck, i(n) denote the number of overpartitions of n with the non-overlined parts not congruent to ±i and 2k - 1 modulo 2k - 1. We show that Ck, i(n) = Dk, i(n). Note that this relation can also be considered as a Rogers-Ramanujan-Gordon type theorem for overpartitions.
AMS Classification: 05A17, 11P84
Keywords: the Rogers-Ramanujan-Gordon theorem, overpartition, Bressoud's theorem