Normal ordering identity for up and down operators
= Normal ordering identity for up and down operators
{title2=$(D+U)^\ell=\sum_{i+j+2m=\ell}\frac{\ell!}{2^m i!j!m!}U^iD^j$}
If $DU-UD=I$, commuting every $D$ past every $U$ gives $(D+U)^\ell=\sum_{i+j+2m=\ell}\ell!U^iD^j/(2^m i!j!m!)$. The recurrence follows from $DU^i=U^iD+iU^{i-1}$. Applied to the empty diagram in the <Young lattice>, it counts <oscillating tableaux>.