科研之友
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摘要
For ka parts per thousand yen2 and ka parts per thousand yenia parts per thousand yen1, let B (k,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 A (k,i) (n) denote the number of partitions with parts not congruent to 0, +/- i modulo 2k. Bressoud's theorem states that A (k,i) (n)=B (k,i) (n). Corteel, Lovejoy, and Mallet found an overpartition analogue of Bressoud's theorem for i=1, that is, for overpartitions not containing non-overlined part 1. We obtain an overpartition analogue of Bressoud's theorem in the general case. For ka parts per thousand yen2 and ka parts per thousand yenia parts per thousand yen1, let D (k,i) (n) denote the number of overpartitions of n such that 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 non-overlined part l+1 together appear exactly k-1 times then the sum of the parts l and non-overlined parts l+1 has the same parity as the number of overlined parts that are less than l+1 plus i-1. Let C (k,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 C (k,i) (n)=D (k,i) (n). Note that this relation can also be considered as a Rogers-Ramanujan-Gordon type theorem for overpartitions.
