Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux

TL;DR

This paper explores the enumeration of three-dimensional analogs of standard Young tableaux, filling a gap in known sequences and highlighting computational and theoretical challenges in counting these complex combinatorial objects.

Contribution

It introduces the concept of Solid Standard Young Tableaux, providing initial enumeration and discussing the computational and theoretical difficulties involved.

Findings

Identified the sequence for 3D tableaux as missing from OEIS

Provided initial counts for solid standard Young tableaux

Highlighted computational and theoretical challenges in enumeration

Abstract

In how many ways can you place n chocolate pieces all of different sizes in an n by n chocolate box, in such a way that when you go from left to right and from top to bottom, there are no gaps AND the sizes increase along each row and each column? The answer is the well-known OEIS Sequence Number 85. To our amazement, the analogous sequence for a three-dimensional chocolate box was not there. Here we fill this gap, and more importantly, offer some computational and theoretical challenges about enumerating families of Solid Standard Young Tableaux.