Enumerating monomials in Schur polynomials

Question

Let $s_{\lambda}(x_1,\dots,x_k)$ be the Schur polynomial associated to the partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k>0)$.

Among the many things involved with these polynomial, I was exploring the number of (distinct) monomials appears in them. I was rather amused by what I noticed.

QUESTION. If $\lambda^{(k)}$ denotes the staircase partition $(k,k-1,\dots,1)$, is it true that the number of monomials, denoted $a_n=\#s_{\lambda^{(k)}}(x_1,\dots,x_k)$, equals the number of forests on n labeled nodes?

Answer 1

A monomial $x_1^{a_1}\cdots x_{n}^{a_n}$ will appear in the expansion of the Schur polynomial $s_{\lambda}(x)$ if and only if $(a_1,a_2,...,a_n)\le (\lambda_1,\lambda_2\dots, \lambda_n)$ in the dominance (majorization) order. This is equivalent to saying that $(a_1,a_2,...,a_n)$ is a lattice point in the convex hull of all the points obtained by permuting the coordinates $(\lambda_1,\lambda_2\dots, \lambda_n)$.

The lattice points in the convex hull of all the permutations of the vector $(n,n-1,...,1)$ are in bijection with forests on n labeled nodes, as pointed out in the OEIS link. (In fact, if you follow the Stanley reference there you will see a more general statement about score vectors and forests on graphs.)