The Andrews-Olsson identity and Bessenrodt insertion algorithm on Young walls

We extend the Andrews-Olsson identity to two-colored partitions. Regarding the sets of proper Young walls of quantum affine algebras gn = A (2) 2n , A (2) 2n−1, B (1) n , D (1) n and D (2) n+1 as the sets of two-colored partitions, the extended Andrews-Olsson identity implies that the generating functions of the sets of reduced Young walls have very simple formulae: ∏ ∞ i=1(1 + t )i where κi = 0, 1 or 2, and κi varies periodically. Moreover, we generalize the Bessenrodt’s algorithms to prove the extended Andrews-Olsson identity in an alternative way. From these algorithms, we can give crystal structures on certain subsets of pair of strict partitions which are isomorphic to the crystal bases B(Λ) of the level 1 highest weight modules V (Λ) over Uq(gn). Introduction A weakly decreasing sequence of nonnegative integers λ = (λ1, λ2, . . .) is called a partition of m, denoted by λ ⊢ m, if m = ∑ i λi. A partition λ is called a strict partition if all parts are strictly decreasing and an odd partition if all parts are odd. Let P[m] (respectively, S [m] and O[m]) be the set of all (respectively, strict and odd) partitions of m. Denote by P (respectively, S and O) the set of all (respectively, strict and odd) partitions. Theorem (Euler’s partition theorem) For all m ∈ Z≥0, (0.1) |S [m]| = |O[m]|. Euler proved the identity (0.1) by showing the equality of the corresponding generating functions: ∞ ∑