Random Young Tableaux and Combinatorial Identities

TL;DR

This paper derives new combinatorial identities related to infinite Young tableaux, connecting probability distributions with hypergeometric series and harmonic analysis on the symmetric group.

Contribution

It introduces novel multivariate combinatorial identities by analyzing probability distributions on infinite Young tableaux, extending previous work in harmonic analysis.

Findings

Derived identities as multivariate analogs of hypergeometric summation formulas

Calculated probabilities of entries in random Young tableaux

Established connections between combinatorics and harmonic analysis

Abstract

We derive new combinatorial identities which may be viewed as multivariate analogs of summation formulas for hypergeometric series. As in the previous paper [Re], we start with probability distributions on the space of the infinite Young tableaux. Then we calculate the probability that the entry of a random tableau at a given box equals n=1,2,.... Summing these probabilities over n and equating the result to 1 we get a nontrivial identity. Our choice for the initial distributions is motivated by the recent work on harmonic analysis on the infinite symmetric group and related topics.