THE THEORY OF COMPOSITIONALS

By introducing the notion of compositionals we obtain a combinatorial interpretation of plethysm of formal power series in infinitely many v ariables. The following related problems are studied: Polya's theorem on the plethysm of cycle indices, plethystic inverse, the inverse of a sequence of delta series, the plethystic analog of the partition latt ice, the reduced incidence algebra of the plethystic lattice, and the plethystic Hopf algebra. We also introduce plethystic trees, enriched plethystic trees and plethystic Schroder trees. An involution for plet hystic Schroder trees is devised, which leads to a combinatorial expan sion for the plethystic inverse of a series f(x1, x2, ...) containing the factor x1.