Pieri rule

In the induction ring of ordinary symmetric-group characters, $[\lambda ][r]$ is the sum, with multiplicity one, of $[\nu ]$ for which $\nu /\lambda$ is a horizontal strip of $r$ cells. Here $[r]$ is the trivial representation of $S_r$. In the Littlewood–Richardson rule, content $(r)$ means every skew entry is $1$; strict column increase is possible exactly when there is at most one new cell in every column. This proves the rule.