1,026 research outputs found
The -colored composition poset
We generalize Bj\"{o}rner and Stanley's poset of compositions to -colored
compositions. Their work draws many analogies between their (1-colored)
composition poset and Young's lattice of partitions, including links to
(quasi-)symmetric functions and representation theory. Here we show that many
of these analogies hold for any number of colors. While many of the proofs for
Bj\"{o}rner and Stanley's poset were simplified by showing isomorphism with the
subword order, we remark that with 2 or more colors, our posets are not
isomorphic to a subword order.Comment: 12 pages, 1 figur
Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras
We develop a theory of multigraded (i.e., -graded) combinatorial Hopf
algebras modeled on the theory of graded combinatorial Hopf algebras developed
by Aguiar, Bergeron, and Sottile [Compos. Math. 142 (2006), 1--30]. In
particular we introduce the notion of canonical -odd and -even
subalgebras associated with any multigraded combinatorial Hopf algebra,
extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our
results are specific categorical results for higher level quasisymmetric
functions, several basis change formulas, and a generalization of the
descents-to-peaks map.Comment: 49 pages. To appear in the Journal of Algebraic Combinatoric
Pre-lie algebras and incidence categories of colored rooted trees
The incidence category \C_{\F} of a family \F of colored posets closed under disjoint unions and the operation of taking convex sub-posets was introduced by the author in \cite{Sz}, where the Ringel-Hall algebra \H_{\F} of \C_{\F} was also defined. We show that if the Hasse diagrams underlying \F are rooted trees, then the subspace \n_{\F} of primitive elements of \H_{\F} carries a pre-Lie structure, defined over ℤ, and with positive structure constants. We give several examples of \n_{\F}, including the nilpotent subalgebras of n, Ln, and several others
Intersection Graph of a Module
Let be a left -module where is a (not necessarily commutative)
ring with unit. The intersection graph \cG(V) of proper -submodules of
is an undirected graph without loops and multiple edges defined as follows: the
vertex set is the set of all proper -submodules of and there is an edge
between two distinct vertices and if and only if We
study these graphs to relate the combinatorial properties of \cG(V) to the
algebraic properties of the -module We study connectedness, domination,
finiteness, coloring, and planarity for \cG (V). For instance, we find the
domination number of \cG (V). We also find the chromatic number of \cG(V)
in some cases. Furthermore, we study cycles in \cG(V), and complete subgraphs
in \cG (V) determining the structure of for which \cG(V) is planar
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