Sprout Symmetric Functions: Part 1

A \emph{sprout sequence} is a sequence $\frakr=(R_0=1,R_1,R_2,\dots)$ of symmetric functions in the variables $\bmx=(x_1,x_2,\dots)$ over a field $K$ generated from a power series $F(t)=1+a_1t+a_2t^2+\cdots$ by the rule $\sum_{n\geq 0}R_nt^n = \prod_{i\geq 1} F(x_it)$. The power series $F(t)$ is called the \emph{seed} of $\frakr$. This concept originated in the work of Littlewood and Richardson (though not with the name ``sprout sequence''), and numerous examples of sprout sequences have appeared in the literature. They are related to chromatic Tutte polynomials of complete graphs and complete hypergraphs, binomial posets, upper homogeneous (upho) posets, topological genera, etc. We first develop the basic theory of sprout sequences and then look at the special case $F(t)=\sec(\sqrt{t})$. We give five characterizations of sprout sequences and consider the expansion of sprout symmetric functions in terms of well-known symmetric function bases. The Schur positivity, elementary symmetric function positivity, and complete homogeneous symmetric function positivity of $R_n$ for all $n$ are completely characterized using the Edrei-Thoma theorem from the theory of total positivity. The seed $F(t)=\sec(\sqrt{t})$ is especially interesting. The expansion of $R_n$ in the power sum or monomial basis is related to alternating permutations. The Schur function expansion is related to standard Young skew tableaux. The expansion in terms of the complete symmetric functions has nonnegative integer coefficients, but we don't know a combinatorial interpretation. Finally we give a formula for $R_n$ as a sum of chromatic symmetric functions of interval orders.