687 research outputs found
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
Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari
order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths
or (m+1)-ary trees. On another hand, the Tamari order is related to the product
in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new
combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is
described by the m-Tamari lattices.
In the same way as planar binary trees can be interpreted as sylvester
classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what
we call m-permutations. These objects are no longer in bijection with
decreasing (m+1)-ary trees, and a finer congruence, called metasylvester,
allows us to build Hopf algebras based on these decreasing trees. At the
opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of
graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions
and quasi-symmetric functions in a natural way. Finally, the algebras of packed
words and parking functions also admit such m-analogues, and we present their
subalgebras and quotients induced by the various congruences.Comment: 51 page
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