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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arXiv · 2212.12171

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, which form the set of boxes between some Dyck path and the diagonal. Another is to find a particular part listing (in the sense of Guay-Paquet) 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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Félix Gélinas, Adrien Segovia, Hugh Thomas. 2022-12-23. Proof of a conjecture of Matherne, Morales, and Selover on encodings of unit interval orders. https://arxiv.org/abs/2212.12171

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