Directed Graphs and the Jacobi-Trudi Identity

Abstract

Let |aij|n×n denote the n × n determinant with (i, j)-entry aij, and hk = hk(x1, …, xn) denote the kth-homogeneous symmetric function of x1, …, xn defined bywhere the summation is over all m1, …, mn ≧ 0 such that m1 + … + mn = k. We adopt the convention that hk = 0 for k < 0. For integers α1 ≧ α2 … ≧ αn ≧ 0, the Jacobi-Trudi identity (see [6], [7]) states thatIn this paper we give a combinatorial proof of an equivalent identity, Theorem 1.1, obtained by moving the denominator on the RHS to the numerator on the LHS.