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Peer-reviewedCC BY 4.0The Electronic Journal of Combinatorics

Proof of a Conjecture of Matherne, Morales, and Selover on Encodings of Unit Interval Orders

  • Félix Gélinas
  • Adrien Segovia
  • Hugh Thomas
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Abstract

There are two bijections from unit interval orders on n elements to Dyck paths from (0,0) to (n,n). One is to consider the pairs of incomparable elements as forming the set of boxes between a Dyck path and the diagonal. Another is to find a particular part listing (in the sense of Guay-Paquet, Rowland and Morales) which yields an isomorphic poset, and to interpret the part listing as the area sequence of a Dyck path. Matherne, Morales, and Selover conjectured that, for any unit interval order, these two Dyck paths are related by Haglund's well-known zeta bijection. In this paper we prove their conjecture.

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Cite

BibTeXRIS
BibTeX
@article{gelinas2026proof,
  title = {{Proof of a Conjecture of Matherne, Morales, and Selover on Encodings of Unit Interval Orders}},
  author = {Félix Gélinas and Adrien Segovia and Hugh Thomas},
  journal = {The Electronic Journal of Combinatorics},
  year = {2026},
  month = nov,
  doi = {10.37236/11825},
  url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p31},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Proof of a Conjecture of Matherne, Morales, and Selover on Encodings of Unit Interval Orders
AU  - Félix Gélinas
AU  - Adrien Segovia
AU  - Hugh Thomas
PY  - 2026
DA  - 2026/11/06
JO  - The Electronic Journal of Combinatorics
DO  - 10.37236/11825
UR  - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p31
AB  - There are two bijections from unit interval orders on n elements to Dyck paths from (0,0) to (n,n). One is to consider the pairs of incomparable elements as forming the set of boxes between a Dyck path and the diagonal. Another is to find a particular part listing (in the sense of Guay-Paquet, Rowland and Morales) which yields an isomorphic poset, and to interpret the part listing as the area sequence of a Dyck path. Matherne, Morales, and Selover conjectured that, for any unit interval order, these two Dyck paths are related by Haglund's well-known zeta bijection. In this paper we prove their conjecture.
SN  - 1077-8926
ER  -