Abstract
The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of permutations. In this paper, new combinatorial objects "Young books'' are introduced and shown to have a connection with the Selberg integral. This connection gives an enumeration formula for Young books. It is shown that special cases of Young books become standard Young tableaux of various shapes: shifted staircases, squares, certain skew shapes, and certain truncated shapes. As a consequence, enumeration formulas for standard Young tableaux of these shapes are obtained. L’intégrale de Selberg est une partie intégrante importante abord évaluée par Selberg en 1944. Stanley a trouvé une interprétation combinatoire de la Selberg aide en permutations. Dans ce papier, de nouveaux objets combinatoires “livres de Young” sont introduits et présentés à avoir un lien avec l’intégrale de Selberg. Cette connexion donne une formule d'énumération pour les livres de Young. Il est démontré que des cas spéciaux de livres de Young deviennent tableaux standards de Young de formes diverses: escaliers décalés, places, certaines formes gauches et certaines formes tronquées. En conséquence, l’énumération des formules pour tableaux standards de Young de ces formes sont obtenues.
Highlights
The Selberg integral is the following integral first evaluated by Selberg [Sel44] in 1944: Sn(α, β, γ) = · · · xiα−1(1 − xi)β−1|xi − xj |2γ dx1 · · · dxn (1) 0 i=1 1≤i 0, Re(β) > 0, and Re(γ) > − min{1/n, Re(α)/(n − 1), Re(β)/(n − 1)}
We refer the reader to Forrester and Warnaar’s exposition [FW08] for the history and importance of the Selberg integral
We show that there is a simple relation between the number of Selberg books and the number of Young books by finding generating functions for both objects
Summary
The Selberg integral is the following integral first evaluated by Selberg [Sel44] in 1944: 1n. We show that there is a simple relation between the number of Selberg books and the number of Young books by finding generating functions for both objects. Using this relation and the Selberg integral formula, we obtain some enumeration results on standard Young tableaux of certain shapes. We show that there is a simple relation between their cardinalities by finding generating functions for them Using this relation and (1) we obtain a formula for the number of Young books.
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