On the number of special permutation-invariant orbits and terms (Q1392667)

Let \(G\) be a permutation group on the set \(\{1, \ldots, n\}\). If \(R\) is a commutative ring, then \(G\) acts on \(T:=R[X_1, \ldots, X_d]\). If \(t \in T\) is a monomial, let \(\text{orb}_G(t)\) denote its trace (in \(T^G\)). A monomial \(t\) is called special if the collection of exponents of the variables occurring in \(t\) has no breaks. The purpose of this paper is to estimate the number of possibilities for \(\text{orb}_G(t)\) with \(t\) special. The upper bound obtained is \(2^{n-1}n!/| G| \) (attained for \(G=S_n\)) and the lower bound is the least integer which is at least as large as \({\beta_n/| G| }\) where \(\beta_n\) is the \(n\)-th derivative of the function \((2-e^x)^{-1}\) evaluated at \(0\). The lower bound is attained for the trivial group.