Let $s\_{\lambda}(x\_1,\dots,x\_k)$ be the [Schur polynomial](https://en.wikipedia.org/wiki/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.



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> **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](https://oeis.org/A001858)?
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Enumerating monomials in Schur polynomials
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.)


