Sprout symmetric functions

Let F(t) = 1 + a_1 t + a_2 t^2 + … be a formal power series over a field K of characteristic 0. Define homogeneous symmetric functions R_n of degree n in the variables x = (x_1, x_2, …) by

F(x_1 t)F(x_2 t) … = \sum_{n\geq 0} R_n t^n.

We call the sequence R = (R_0=1, R_1, R_2, …) a *sprout sequence* with *seed* F(t). First we review some basic properties of sprout sequences and give some examples. When K is the field of real numbers, the problem of deciding whether each R_n is Schur positive (i.e., a nonnegative linear combination of Schur functions) is related to the Edrei-Thoma theorem from the theory of total positivity.

We then give an example of a sprout sequence A arising from work of Amdeberhan-Ono-Singh on expressing a certain theta function of Ramanujan in terms of Eisenstein series. This sprout sequence has many interesting combinatorial properties connected with alternating permutations. We discuss several generalizations and numerous related open problems.

This talk assumes a basic knowledge of symmetric functions. The research was done in collaboration with Tewodros Amdeberhan and John Shareshian.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/