Matrices of formal power series associated to binomial posets

We introduce an operation that assigns to each binomial poset a partially ordered set for which the number of saturated chains in any interval is a function of two parameters. We develop a corresponding theory of generating functions involving noncommutative formal power series modulo the closure of a principal ideal, which may be faithfully represented by the limit of an infinite sequence of lower triangular matrix representations. The framework allows us to construct matrices of formal power series whose inverse may be easily calculated using the relation between the Möbius and zeta functions. Introduction In a recent paper [5] I introduced a sequence of Eulerian partially ordered sets whose ce-indices provide a noncommutative generalization of the Tchebyshev polynomials. The partially ordered sets were obtained by looking at intervals in a poset obtained from the the fairly “straightforward” lower Eulerian poset 0 < −2, 2 < −3, 3 < · · · , and using an operator that could be applied to any partially ordered set. This operator, which I call Tchebyshev operator, creates a partial order on the non-singleton intervals of its input, by setting (x1, y1) ≤ (x2, y2) when either y1 ≤ x2, or x1 = x2 and y1 ≤ y2. The property of having a rank function is preserved by the Tchebyshev operator. The existence of a unique minimum element is not preserved, but if we “augment” the poset that has a unique minimum element 0̂ by adding a new minimum element −̂1, then the Tchebyshev transform of the augmented poset will have a unique minimum element associated to the interval (−̂1, 0̂). In this paper we study the effect of the augmented Tchebyshev operator on binomial posets. As it is well known, binomial posets provide a framework for studying generating functions. Those functions of the incidence algebra that depend only on the rank of the interval, form a subalgebra, and there is a homomorphism from this subalgebra into the ring of formal power series in one variable. Combinatorial enumeration problems stated in terms of binomial posets may be solved using generating functions and, conversely, identities of formal power series may be explained by exposing the combinatorial background. ∗2000 Mathematics Subject Classification: Primary 05A99; Secondary 06A07, 13F25.