Counting Standard Young Tableaux With Restricted Runs

TL;DR

This paper explores counting Young tableaux with restricted run lengths, extending classical enumeration results, and presents conjectures and open problems related to these combinatorial objects.

Contribution

It introduces new conjectures on counting Young tableaux with forbidden run lengths, expanding understanding beyond classical formulas.

Findings

Classical formulas for rectangular Young tableaux are reaffirmed.

New conjectures are proposed for tableaux with restricted run lengths.

Open problems are presented for future research in combinatorics.

Abstract

The number of Young Tableaux whose shape is a k by n rectangle is famously (nk)! 0! ... (k-1)!/((n+k-1)!(n+k-2)!... n!) implying that for each specific k, that sequence satisfies a linear recurrence equation with polynomial coefficients of the first order. But what about counting Young tableaux where certain "run lengths" are forbidden? Then things seem to get much more complicated. We conclude with four conjectures and pledge donations to the OEIS in honor of the first provers.