Orbit lengths for promotion on 2-row and near-hook tableaux
Laura Pierson
TL;DR
This work analyzes promotion orbits for standard Young tableaux beyond rectangular shapes by introducing stable sequences $T[n]$ that extend fixed bottom shapes with a growing top row. It proves that for generic 2-row shapes, orbit lengths divide monic polynomials in $n$, with degree tied to the number of distinct bottom-row runs, and it identifies a CSP in special 2-row subsets linked to a modified major index. The authors develop a track-based combinatorial framework to describe general 2-row orbit lengths, yielding a recursive polynomial structure $P_d(n,\vec{\ell},\vec{r})$ governing orbit sizes and providing constructive bijections between tableaux and track configurations. They extend the analysis to near-hook shapes, obtaining linear and quadratic divisors in $n$ depending on run and singleton structures. Overall, the paper establishes a polynomial, CSP-connected framework for promotion in non-rectangular shapes and connects CSP to generalized major-index statistics, with potential extensions to quasipolynomials and representation-theoretic interpretations.
Abstract
Promotion has been well-studied for rectangular standard Young tableaux, in which case the orbit lengths divide the total number of boxes and are described by a cyclic sieving phenomenon (CSP), but little is known about the orbit lengths for tableaux of general shape. We approach this problem by building a stable sequence of tableaux where we fix the bottom portion and add extra boxes to the first row to get $n$ total boxes, with $n$ varying. We show that for 2-row tableaux with a fixed bottom row, the orbit lengths are divisors of certain monic polynomials in $n$, with degree generally equal to the number of distinct lengths of runs of consecutive numbers in the bottom row. For the subsets of 2-row tableaux where all runs have the same length, we show that the orbit lengths are characterized by a CSP polynomial that is a slightly modified version of the major index generating function, like in the rectangle case. We also show that for any stable sequence of tableaux, the orbit lengths are linear in $n$ as long as all non-first-row entries differ from each other by at least 2, which asymptotically happens for almost all tableaux in the limit as $n\to\infty.$ We also calculate the orbit lengths for near-hook tableaux, which are divisors of certain linear or quadratic polynomials in $n$.