The Tchebyshev Transforms of the First and Second Kind

Abstract

An in-depth study of the Tchebyshev transforms of the first and second kind of a poset is taken. The Tchebyshev transform of the first kind is shown to preserve desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg and Readdy omega map of oriented matroids. The Tchebyshev transform of the second kind U is a Hopf algebra endomorphism on the space of quasisymmetric functions which, when restricted to Eulerian posets, coincides with Stembridge's peak enumerator. The complete spectrum of U is determined, generalizing work of Billera, Hsiao and van Willigenburg. The type B quasisymmetric function of a poset is introduced and, like Ehrenborg's classical quasisymmetric function of a poset, it is a comodule morphism with respect to the quasisymmetric functions QSym. Finally, similarities among the omega map, Ehrenborg's r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps which occur naturally in the setting of combinatorial Hopf algebras.

Tchebyshev triangulations of stable simplicial complexes

“…These new complexes, associated to the original have the property that their face enumerating polynomial F (x) := j 0 f j −1 ((x − 1)/2) j arises from the F -polynomial of the original complex by the linear map that sends the monomial x n into the Tchebyshev polynomials T n (x) (first kind) or U n−1 (x) (second kind) respectively. This construction generalizes the Tchebyshev transform for graded posets introduced and studied by the present author [5,7], and by Ehrenborg and Readdy [2]. The connection to this earlier theory is explained along the way and in Section 5.…”

“…Many interesting results on the Tchebyshev transform may be found in the papers cited above. These include showing that the Tchebyshev transform preserves grading [7], the Eulerian property [7], the positivity of the cd-index [2], and shellability [2]. The result motivating our present paper was shown in [5, Theorem 1.10], although not stated in this generality.…”

“…In analogy to the definition for posets in [2], we define the Tchebyshev triangulation of the second kind as follows. …”

“…Given any poset Q, we may define another poset T (Q) whose elements are the open intervals (x, y) ⊂ Q, and the partial order is given by (x 1 , y 1 ) (x 2 , y 2 ) if either x 1 = x 2 and y 1 < y 2 , or y 1 x 2 holds. Using the operation we may define the Tchebyshev transform of a graded poset (this notation and terminology was introduced by Ehrenborg and Readdy [2]), as follows. Given a poset P with minimum element 0 and maximum element 1, we introduce a new minimum element −1 < 0 and a new maximum element 2.…”

Cyclotomic factors of the descent set polynomial

“…Theorem 4.6 can now be stated as follows. ρ(c) , (8.1) where the sum is over all chains c = {0 = x 0 < x 1 < · · · < x m =1} in the poset P ; see [12]. A different way to write Eq.…”