Let B k,i (n) be the number of partitions of n with certain difference condition and let A k,i (n) be the number of partitions of n with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that B k,i (n) = A k,i (n). Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases i = 1 and i = k. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let D k,i (n) be the number of overpartitions of n satisfying certain difference condition and C k,i (n) be the number of overpartitions of n whose non-overlined parts satisfy certain congruence condition. We show that C k,i (n) = D k,i (n). By using a function introduced by Andrews, we obtain a recurrence relation that implies that the generating function of D k,i (n) equals the generating function of C k,i (n). By introducing the Gordon marking of an overpartition, we find a generating function formula for D k,i (n) that can be considered an overpartition analogue of an identity of Andrews for ordinary partitions.

For k ≥ i ≥ 1, 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 total 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 i, 2k − i and 2k 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 partitions not containing nonoverlined part 1. We obtain an overpartition analogue of Bressoud's theorem in the general case. For k ≥ i ≥ 1, let D k,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 C k,i (n) denote the number of overpartitions with the nonoverlined parts not congruent to ±i and 2k − 1 modulo 2k − 1. We show that C k,i (n) = D k,i (n). This relation can also be considered as a Rogers-Ramanujan-Gordon type theorem for overpartitions.

We show that the number of anti-lecture hall compositions of n with the first entry not exceeding k − 2 equals the number of overpartitions of n with non-overlined parts not congruent to 0, ±1 modulo k. This identity can be considered as a finite version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers-Ramanujan type identities for overpartitions which are analogous to the Rogers-Ramanujan type identities due to Andrews. When k is odd, we give another proof by using the bijections of Corteel and Savage for the anti-lecture hall theorem and the generalized Rogers-Ramanujan identity also due to Andrews.

In 1961, Gordon found a combinatorial generalization of the RogersRamanujan identities, which has been called the Rogers-Ramanujan-Gordon theorem. In 1974, Andrews derived an identity which can be considered as the generating function counterpart of the Rogers-Ramanujan-Gordon theorem, and it has been called the Andrews-Gordon identity. The Andrews-Gordon identity is an analytic generalization of the Rogers-Ramanujan identities with odd moduli. In 1979, Bressoud obtained a Rogers-Ramanujan-Gordon type theorem and the corresponding Andrews-Gordon type identity with even moduli. In 2003, Lovejoy proved two overpartition analogues of two special cases of the Rogers-Ramanujan-Gordon theorem. In 2013, Chen, Sang and Shi found the overpartition analogue of the Rogers-Ramanujan-Gordon theorem in general cases and the corresponding Andrews-Gordon type identity with even moduli. In 2008, Corteel, Lovejoy and Mallet found an overpartition analogue of a special case of Bressoud's theorem of the Rogers-Ramanujan-Gordon type. In 2012, Chen, Sang and Shi obtained the overpartition analogue of Bressoud's theorem in the general case. In this paper, we obtain an Andrews-Gordon type identity corresponding to this overpartition theorem with odd moduli using the Gordon marking representation of an overpartition.

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