Assembling crystals of type A

Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q({\mathfrak{s}\mathfrak{l}}_{n+1})$. For such a crystal $K$ with colors $1,2,\mathrm{.}\mathrm{.}\mathrm{.},n$, we consider its maximal connected subcrystals with colors $1,\mathrm{.}\mathrm{.}\mathrm{.},n-1$ and with colors $2,\mathrm{.}\mathrm{.}\mathrm{.},n$ and characterize the interlacing structure for all pairs of these subcrystals. This is used to give a recursive description of the combinatorial structure of $K$ and develop an efficient procedure of assembling $K$.