The Selberg integral and Young books

TL;DR

This paper introduces Young books, new combinatorial objects linked to the Selberg integral, providing enumeration formulas and connecting to standard Young tableaux of various shapes, thus extending combinatorial interpretations of the integral.

Contribution

The paper introduces Young books and establishes their connection to the Selberg integral, deriving new enumeration formulas for standard Young tableaux of specific shapes.

Findings

Young books are combinatorial objects related to the Selberg integral.

Enumeration formulas for Young books are derived.

Connections to standard Young tableaux of various shapes are established.

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, product formulas for the number of standard Young tableaux of these shapes are obtained.