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 g n = A 2 n ( 2 ) , A 2 n − 1 ( 2 ) , B n ( 1 ) , D n ( 1 ) and D n + 1 ( 2 ) 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 ) κ i where  κ i = 0 , 1  or 2, and κ i  varies periodically. Moreover, we generalize 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 U q ( g n ) .