9,072 research outputs found
The Algebra of Directed Acyclic Graphs
We give an algebraic presentation of directed acyclic graph structure,
introducing a symmetric monoidal equational theory whose free PROP we
characterise as that of finite abstract dags with input/output interfaces. Our
development provides an initial-algebra semantics for dag structure
Cyclic inclusion-exclusion
Following the lead of Stanley and Gessel, we consider a morphism which
associates to an acyclic directed graph (or a poset) a quasi-symmetric
function. The latter is naturally defined as multivariate generating series of
non-decreasing functions on the graph. We describe the kernel of this morphism,
using a simple combinatorial operation that we call cyclic inclusion-exclusion.
Our result also holds for the natural noncommutative analog and for the
commutative and noncommutative restrictions to bipartite graphs. An application
to the theory of Kerov character polynomials is given.Comment: comments welcom
Algebras associated to acyclic directed graphs
We construct and study a class of algebras associated to generalized layered
graphs, i.e. directed graphs with a ranking function on their vertices. Each
finite directed acyclic graph admits countably many structures of a generalized
layered graph. We construct linear bases in such algebras and compute their
Hilbert series. Our interest to generalized layered graphs and algebras
associated to those graphs is motivated by their relations to factorizations of
polynomials over noncommutative rings.Comment: 20 pages, Latex; an expanded and corrected version; to appear in
"Advances of Applied Mathematics
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