Catalan structures arising from pattern-avoiding Stoimenow matchings and other Fishburn objects

Summary · qwen2.5:32b

The paper solves a problem posed by Bevan et al., identifying subsets of Stoimenow matchings counted by Catalan numbers through five solutions involving pattern-avoiding matchings; it also proves that matchings avoiding four infinite families of generalized forbidden patterns are equinumerous, linking these structures to other Fishburn objects like ascent sequences and permutations.

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Exploring the connection between Vassiliev's knot invariants and Catalan numbers in pattern-avoiding Stoimenow matchings.

Excerpt

arXiv:2509.09115v2 Announce Type: replace Abstract: In connection with Vassiliev's knot invariants, Stoimenow introduced in 1998 a class of matchings, also known as regular linearized chord diagrams. These matchings are linked to various combinatorial structures, all of which are associated with the Fishburn numbers. In this paper, we address a problem posed by Bevan et al.\ in 2025 concerning the identification of subsets of Stoimenow matchings that are counted by the Catalan numbers. We present five solutions in terms of pattern-avoiding matchings. We also consider four infinite families of patterns that generalize four of the five forbidden patterns appearing in the solution to the problem we solved and prove that the matchings avoiding them are equinumerous. Finally, we establish numerous results on distributions and joint equidistribution of statistics over these Catalan-counted subsets of Fishburn structures, namely Stoimenow matchings, $(2+2)$-free posets, ascent sequences, and Fishburn permutations, notably expressing some of them in terms of Narayana numbers and others in terms of ballot numbers.