Young tableau reconstruction via minors

TL;DR

This paper investigates the tableau reconstruction problem using minors, proving a sharp lower bound for reconstructing standard Young tableaux from their 2-minors and exploring bounds for multisets of minors.

Contribution

The paper extends previous work by solving the tableau reconstruction problem for 2-minors and providing initial bounds for multisets of minors.

Findings

Reconstruction from 2-minors is possible for tableaux of size at least 8.

Established a lower bound for multiset minors reconstruction for arbitrary k.

Solved the problem for k=2, complementing the known solution for k=1.

Abstract

The tableau reconstruction problem, posed by Monks (2009), asks the following. Starting with a standard Young tableau $T$, a 1-minor of $T$ is a tableau obtained by first deleting any cell of $T$, and then performing jeu de taquin slides to fill the resulting gap. This can be iterated to arrive at the set of $k$-minors of $T$. The problem is this: given $k$, what are the values of $n$ such that every tableau of size $n$ can be reconstructed from its set of $k$-minors? For $k=1$, the problem was recently solved by Cain and Lehtonen. In this paper, we solve the problem for $k=2$, proving the sharp lower bound $n \geq 8$. In the case of multisets of $k$-minors, we also give a lower bound for arbitrary $k$, as a first step toward a sharp bound in the general multiset case.