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 $\g_n=A^{(2)}_{2n}$, $A^{(2)}_{2n-1}$, $B^{(1)}_{n}$, $D^{(1)}_{n}$ 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: \begin{center} $\prod^{\infty}_{i=1}(1+t^i)^{\kappa_i}$ where $\kappa_i=0$, $1$ or $2$, and $\kappa_i$ varies periodically. \end{center} 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 $B(\Lambda)$ of the level $1$ highest weight modules $V(\Lambda)$ over $U_q(\g_n)$.