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arXiv · 2607.20132

Generalised flip order on the faces of nestohedra

Pierre-Louis Curien·Bérénice Delcroix-Oger·Jovana Obradović
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Abstract

Classical shuffle products on permutations and binary planar rooted trees (i.e., on the vertices of permutohedra and associahedra) admit descriptions in terms of intervals in the weak Bruhat order and the Tamari order, respectively. Palacios and Ronco extended these products as well as their interval description to surjections and planar rooted trees (i.e. on all faces of permutohedra and associahedra). In this article, we present a broad generalisation of this phenomenon. We show that the shuffle product on faces of certain families of nestohedra admits an interval description with respect to the generalised flip order, a partial order defined on the faces of nestohedra through elementary splitting and fusion operations on the tree-like combinatorial objects encoding them. The generalised flip order extends the flip order of Barnard and McConville from vertices to all faces of nesthedra. We further compare it with the facial weak order of Dermenjian-Hohlweg-Pilaud and the generalised Tamari order of Ronco, and we provide its characterisation in terms of (generalised) inversions.

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Pierre-Louis Curien, Bérénice Delcroix-Oger, Jovana Obradović. 2026-07-22. Generalised flip order on the faces of nestohedra. https://arxiv.org/abs/2607.20132

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