Square-sum identity for shifted partition coordinates (${\sum }_{{\ell }_i\ge 0,{\sum }_i{\ell }_i=m(m+1)/2}\Delta (\ell {)}^2/{\prod }_i({\ell }_i!{)}^2=1$)

Repeated coordinates contribute zero. Sorting a distinct tuple and subtracting the staircase gives a partition of $m$, and its $m!$ orderings have identical squared summands. The beta-set hook-product identity turns each summand into $(\dim S^{\lambda }/m!{)}^2$. Summing and using the Artin–Wedderburn theorem for $\mathbb{C}{S}_m$ gives $m!{\sum }_{\lambda ⊢m}(\dim S^{\lambda }{)}^2/(m!{)}^2=1$. This is a normalized group algebra dimension identity.