2,455 research outputs found
Coloring Complexes and Combinatorial Hopf Monoids
We generalize the notion of coloring complex of a graph to linearized
combinatorial Hopf monoids. These are a generalization of the notion of
coloring complex of a graph. We determine when a combinatorial Hopf monoid has
such a construction, and discover some inequalities that are satisfied by the
quasisymmetric function invariants associated to the combinatorial Hopf monoid.
We show that the collection of all such coloring complexes forms a
combinatorial Hopf monoid, which is the terminal object in the category of
combinatorial Hopf monoids with convex characters. We also study several
examples of combinatorial Hopf monoids.Comment: 37 pages, 5 figure
Constructing combinatorial operads from monoids
We introduce a functorial construction which, from a monoid, produces a
set-operad. We obtain new (symmetric or not) operads as suboperads or quotients
of the operad obtained from the additive monoid. These involve various familiar
combinatorial objects: parking functions, packed words, planar rooted trees,
generalized Dyck paths, Schr\"oder trees, Motzkin paths, integer compositions,
directed animals, etc. We also retrieve some known operads: the magmatic
operad, the commutative associative operad, and the diassociative operad.Comment: 12 page
Fomin-Greene monoids and Pieri operations
We explore monoids generated by operators on certain infinite partial orders.
Our starting point is the work of Fomin and Greene on monoids satisfying the
relations and
if Given such a monoid, the non-commutative
functions in the variables are shown to commute. Symmetric functions in
these operators often encode interesting structure constants. Our aim is to
introduce similar results for more general monoids not satisfying the relations
of Fomin and Greene. This paper is an extension of a talk by the second author
at the workshop on algebraic monoids, group embeddings and algebraic
combinatorics at The Fields Institute in 2012.Comment: 33 pages, this is a paper expanding on a talk given at Fields
Institute in 201
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