Séminaire Lotharingien de Combinatoire, 89B.55 (2023), 12 pp.
Alejandro H. Morales, Greta Panova and GaYee Park
Minimal Skew Semistandard Young Tableaux and the Hillman-Grassl Correspondence
Abstract.
Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula (NHLF) as a positive sum over excited diagrams of products of hook-lengths. Subsequently, Morales, Pak, and Panova gave several proofs and generalized it to two q-analogues. They also showed, partly algebraically, that the Hillman-Grassl map restricted to skew shapes must be a bijection. We study the problem of circumventing the algebraic part and proving the bijection completely combinatorially. For a skew shape, we define a new set of semi-standard Young tableaux, called the \emph{minimal SSYT}, that are equinumerous with excited diagrams via a new description of the Hillman-Grassl bijection and a version of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula (OOF) for counting SYTs of skew shape. Our construction immediately implies that the summands in the NHLF are less than the summands in the OOF.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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