Generalized Binomial Coefficients for Molecular Species

Let ξ be a complex variable. We associate a polynomial in ξ, denoted (MN)ξ, to any two molecular species M=M(X) and N=N(X) by means of a binomial-type expansion of the formM(ξ+X)=∑N MNξ N(X).In the special case M(X)=Xm, the species of linear orders of length m, the above formula reduces to the classical binomial expansion(ξ+X)m=∑n mn ξm−nXn.When ξ=1, a M(1+X)-structure can be interpreted as a partially labelled M-structure and (MN)1 is a nonnegative integer, denoted (MN) for simplicity. We develop some basic properties of these “generalized binomial coefficients” and apply them to study solutions, Φ, of combinatorial equations of the form M(Φ)=Ψ in the context of C-species, M being molecular and Ψ being a given C-species. This generalizes the study of symmetric square roots (where M=E2, the species of 2-element sets) initiated by P. Bouchard, Y. Chiricota, and G. Labelle in (1995, Discrete Math.139, 49–56).