A Combinatorial Proof of a Refinement of the Andrews—Olsson Partition Identity

The dual of Göllnitz’s (big) partition theorem

“…It was noticed by Andrews and Olsson [18], that Theorem A1 could be reformulated in a more convenient form, and this was exploited by Bessenrodt [20] to provide a nice combinatorial proof by using N -modular Ferrers graphs. Such an approach did not work for Theorem A2.…”

A unification of two refinements of Euler’s partition theorem

“…To provide a purely combinatorial proof of Andrews-Olsson's theorem [5], Bessenrodt [8] constructs an explicit bijection on the sets of partitions in Andrews-Olsson's theorem, which we call Bessenrodt's insertion algorithm. The original insertion algorithm does not imply the bijection in Theorem 2.4, but we find that the generalized insertion algorithm given by Bessenrodt [10] in 1995 can be used to establish the bijection required by Theorem 2.4.…”

“…Then we have the following theorem which will be needed to prove Theorem 2.4. We outline the first approach by constructing a bijection Φ between C 1 (A 2N ; n, 2N) and C 2 (A 2N ; n, 2N) based on a variant of Bessenrodt's insertion algorithm [8].…”

“…It can be shown that γ ∈ C 2 (A 2N ; n, 2N) for β 1 ≤ 2N · l(α). For the details of the proof, see [8,20].…”

q-Deformed Fock spaces and modular representations of spin symmetric groups

“…, p − 1} and N = p in Theorem 2 of [2]. A combinatorial proof of a refinement of the Andrews-Olsson partition identity has been given by Bessenrodt [6].…”