Realizable Standard Young Tableaux

TL;DR

This paper studies the combinatorial structure of realizable standard Young tableaux, establishing tight bounds on their asymptotic counts and connecting them to various mathematical and computational contexts.

Contribution

It provides the first tight asymptotic bounds for the number of realizable rectangular tableaux and related allowable sequences, resolving open questions and extending prior results.

Findings

Established tight asymptotic bounds for realizable rectangular tableaux.

Derived asymptotics for realizable allowable sequences and staircase-shaped tableaux.

Connected realizable tableaux to sorting algorithms, quantum computing, and geometric arrangements.

Abstract

Given two vectors $u$ and $v$, their outer sum is given by the matrix $A$ with entries $A_{ij} = u_{i} + v_{j}$. If the entries of $u$ and $v$ are increasing and sufficiently generic, the total ordering of the entries of the matrix is a standard Young tableau of rectangular shape. We call standard Young tableaux arising in this way realizable. The set of realizable tableaux was defined by Mallows and Vanderbei for studying a deconvolution algorithm, but we show they have appeared in many other contexts including sorting algorithms, quantum computing, random sorting networks, reflection arrangements, fiber polytopes, and Goodman and Pollack's theory of allowable sequences. In our work, we prove tight bounds on the asymptotic number of realizable rectangular tableaux. We also derive tight asymptotics for the number of realizable allowable sequences, which are in bijection with realizable staircase-shaped standard Young tableaux with the notion of realizability coming from the theory of sorting networks. As a consequence, we resolve an open question of Angel, Gorin, and Holroyd from 2012 and improve upon a 1986 result of Goodman and Pollack.