13 research outputs found
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
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
Polytopes, Hopf algebras and Quasi-symmetric functions
In this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product and a
join of polytopes. is a commutative
associative bigraded ring of polynomials, and is a commutative associative
threegraded ring of polynomials. The ring has the structure of a
graded Hopf algebra. It turns out that has a natural Hopf
comodule structure over . Faces operators that send a
polytope to the sum of all its -dimensional faces define on both rings
the Hopf module structures over the universal Leibnitz-Hopf algebra
. This structure gives a ring homomorphism \R\to\Qs\otimes\R,
where is or . Composing this homomorphism with
the characters of , of
, and with the counit we obtain the ring homomorphisms
f\colon\mathcal{P}\to\Qs[\alpha],
f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha], and
\F^*:\mathcal{RP}\to\Qs, where is the Ehrenborg transformation. We
describe the images of these homomorphisms in terms of functional equations,
prove that these images are rings of polynomials over , and find the
relations between the images, the homomorphisms and the Hopf comodule
structures. For each homomorphism , and \F the images
of two polytopes coincide if and only if they have equal flag -vectors.
Therefore algebraic structures on the images give the information about flag
-vectors of polytopes.Comment: 61 page