Abstract
Following the lead of Stanley and Gessel, we consider a linear map 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 (or of P -partitions of the poset).We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusion- exclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs.
Highlights
Given an acyclic directed graph G = (V, EG), it is natural to consider the following multivariate generating functionΓG(x1, x2, · · · ) = xf (v) (1)f :V →N v∈V f non-decreasing where N is the set of positive integers and non-decreasing means that (i, j) ∈ EG implies f (i) ≤ f (j)
A similar definition can be considered for a poset P = (V,
1365–8050 c 2016 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France. This is a classical object in the algebraic combinatorics literature: using the terminology of the seminal book of Stanley [10], the non-decreasing functions on posets correspond to P -partitions when P has a natural labelling
Summary
The main result of the present paper is a combinatorial description of the kernel of the map Γ from the graph algebra to quasi-symmetric functions (Theorem 2) In the long version of this paper [4], we consider restrictions of the linear maps Γ and Γnc to bipartite graphs and describe the kernel of these restrictions via cyclic inclusion-exclusion. We refer the reader to the long version [4, Section 6] for details This operation of cyclic inclusion-exclusion has been fruitful in a quite different context in [2], where the purpose was to study some rational functions introduced by Greene [7]. This gives an efficient way to compute these rational functions and a powerful tool to study them; see [2]
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