Groups Geometry and Dynamics· 2026Q1
Maximal elements in the Hurwitz graph
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- 2026year
Short summary
Maximal elements in the Hurwitz graph are in bijection with non-crossing partitions, revealing a connection to the permutahedron and enabling diameter computation.
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Key points
- Maximal elements of the Hurwitz graph are in bijection with non-crossing partitions.
- This bijection maps the graph structure to that of the permutahedron (restricted to non-crossing partitions).
- The diameter of the graph of maximal elements can be computed using this structural correspondence.
- The paper also considers a related graph involving cycle factorizations and commutation relations.
AI-generated from the title and abstract; the full text is not read.
Abstract
The Hurwitz graph has as its vertices the maximal chains in the lattice of non-crossing partitions, with an edge between two chains which differ at exactly one place. Adin and Roichman introduced an order on this graph by choosing a base element and showed that the maximal elements are counted by the Catalan numbers. We show that this set of maximal elements is in natural bijection with non-crossing partitions so that the graph structure corresponds to that of the permutahedron, restricted to non-crossing partitions. We use this to compute the diameter of the graph of maximal elements. We also briefly consider the graph obtained by identifying maximal chains with minimal factorizations of a full cycle into transpositions in the symmetric groups and quotienting by commutation relations.
The authors' abstract, as published at the source. Groups Geometry and Dynamics, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics