Standard skew Young tableau

A standard skew Young tableau fills the $k$ cells of a skew Young diagram bijectively with $1,\dots ,k$, increasing along each row and column. Equivalently it is a linear extension of a partially ordered set formed from the row and column inequalities. Interchanging consecutive labels preserves standardness exactly when their cells are incomparable in that partial order.

Here $n=|\lambda |-|\mu |$, with zero entries for negative factorial arguments. Extract the squarefree monomial $x_1\cdots x_n$ from the skew Jacobi–Trudi identity. A product $h_{r_1}\cdots h_{r_{\ell }}$ contributes $n!/\prod r_i!$ because its distinct labels are assigned to the factors in that many ways. Summing determinant signs proves the formula, since a squarefree semistandard Young tableau is a standard skew Young tableau.