Cyclotomic factors of the descent set polynomial

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.Comment: 21 pages, revised the proof of the opening result and cleaned up notatio

Asymptotics of the Euler number of bipartite graphs

We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph $G$ with the path $P_m$ in terms of self adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product $P_2 \Box P_m$, we numerically solve the eigenvalue problem.Comment: 13 pages, 6 figure, submitted to JCT

Euler flag enumeration of Whitney stratified spaces

The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi-graded poset. This is a poset endowed with an order-preserving rank function and a weighted zeta function. This allows us to generalize the classical notion of Eulerianness, and obtain a cd-index in the quasi-graded poset arena. We also extend the semi-suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components.Comment: 41 pages, 3 figures. Final versio

Classification of the factorial functions of Eulerian binomial and Sheffer posets

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with $n!$, the factorial function of the infinite Boolean algebra, or $2^{n-1}$, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function $B(n) = n!$ has Sheffer factorial function D(n) identical to that of the infinite Boolean algebra, the infinite Boolean algebra with two new coatoms inserted, or the infinite cubical poset. Moreover, we are able to classify the Sheffer factorial functions of Eulerian Sheffer posets with binomial factorial function $B(n) = 2^{n-1}$ as the doubling of an upside down tree with ranks 1 and 2 modified. When we impose the further condition that a given Eulerian binomial or Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite Boolean algebra $B_X$ or the infinite cubical lattice $C_X^{< \infty}$. We also include several poset constructions that have the same factorial functions as the infinite cubical poset, demonstrating that classifying Eulerian Sheffer posets is a difficult problem.Comment: 23 pages. Minor revisions throughout. Most noticeable is title change. To appear in JCT

The Tchebyshev transforms of the first and second kind

We give an in-depth study of the Tchebyshev transforms of the first and second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform (of the first kind) preserves desirable combinatorial properties, including Eulerianess (due to Hetyei) and EL-shellability. It is also a linear transformation on flag vectors. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg and Readdy omega map of oriented matroids. One consequence is that nonnegativity of the cd-index is maintained. The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on the space of quasisymmetric functions QSym. It coincides with Stembridge's peak enumerator for Eulerian posets, but differs for general posets. The complete spectrum is determined, generalizing work of Billera, Hsiao and van Willigenburg. The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's classical quasisymmetric function of a poset, this map is a comodule morphism with respect to the quasisymmetric functions QSym. Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps. One such occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and Sottile's result on the terminal object in the category of combinatorial Hopf algebras. In contrast, the chain map of the first kind is both an algebra map and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page