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Robinson–Schensted correspondence for unit interval orders

Abstract

The Stanley–Stembridge conjecture associates a symmetric function to each natural unit interval order \(\mathcal {P}\). In this paper, we define relations à la Knuth on the symmetric group for each \(\mathcal {P}\) and conjecture that the associated \(\mathcal {P}\)-Knuth equivalence classes are Schur-positive, refining theorems of Gasharov, Brosnan-Chow, Guay-Paquet, and Shareshian-Wachs. The resulting equivalence graphs fit into the framework of D graphs studied by Assaf. Furthermore, we conjecture that the Schur expansion is given by column-readings of \(\mathcal {P}\)-tableaux that occur in the equivalence class. We prove these conjectures for \(\mathcal {P}\) avoiding two specific suborders by introducing \(\mathcal {P}\)-analog of Robinson–Schensted insertion, giving an answer to a long standing question of Chow.

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Correspondence to Pavlo Pylyavskyy.

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Kim, D., Pylyavskyy, P. Robinson–Schensted correspondence for unit interval orders. Sel. Math. New Ser. 27, 97 (2021). https://doi.org/10.1007/s00029-021-00708-4

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Mathematics Subject Classification

  • 05E10
  • 20C32