Year: 2026
Type: article
Abstract: Given a directed graph D with transitive closure $${{\,\textrm{tc}\,}}(D)$$ tc ( D ) and path hypergraph $$\mathbb {P}(D)$$ P ( D ) , we study the connections between the (acyclic) reorientation poset of $${{\,\textrm{tc}\,}}(D)$$ tc ( D ) , the (acyclic) sourcing poset of $$\mathbb {P}(D)$$ P ( D ) , and the (acyclic) ornamentation poset of D . Geometrically, the acyclic reorientation poset of $${{\,\textrm{tc}\,}}(D)$$ tc ( D ) (resp. the acyclic sourcing poset of $$\mathbb {P}(D)$$ P ( D ) ) is the transitive closure of the skeleton of the graphical zonotope of $${{\,\textrm{tc}\,}}(D)$$ tc ( D ) (resp. of the hypergraphic polytope of $$\mathbb {P}(D)$$ P ( D ) ) oriented in a linear direction. When D is a rooted (or even unstarred) increasing tree, we show that the acyclic sourcing poset of $$\mathbb {P}(D)$$ P ( D ) is isomorphic to the ornamentation lattice of D , and that they form a lattice quotient of the acyclic reorientation lattice of $${{\,\textrm{tc}\,}}(D)$$ tc ( D ) . As a consequence, we obtain polytopal realizations of the ornamentation lattices of rooted (or even unstarred) increasing trees, answering an open question of C. Defant and A. Sack. When D is an increasing tree, we show that the ornamentation lattice of D is the MacNeille completion of the acyclic sourcing poset of $$\mathbb {P}(D)$$ P ( D ) . Finally, still when D is an increasing tree, we use the ornamentation lattice of D to characterize the subhypergraphs of the path hypergraph $$\mathbb {P}(D)$$
Source: Mathematische Zeitschrift
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Awards: G1175115578, G1495698476, G2064873908, G2262748287, G2462415295, Beyond Permutahedra and Associahedra, G2766336479, G3005324705, G300979063, G34346518, G3480869486, G3546882995, G425668148, G4274939422, G4532494212, G4784319644, G5080475149, G6054710752, G6325762137, G69051668, G6910422418, G7013227978, G7304226585, Beyond Permutahedra and Associahedra: Geometry, Combinatorics, Algebra, and Probability, G8295830637, G84477964
OpenAlex ID: W4417099778
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