Some results on LCTR, an impartial game on partitions

TL;DR

This paper analyzes the impartial game LCTR on partitions using the Sprague-Grundy Theorem, establishing bounds on Grundy values, computing them for certain families, and providing an efficient algorithm for their determination.

Contribution

It introduces LCTR, applies the Sprague-Grundy Theorem to it, and develops an $O(n)$ dynamic programming method to compute Grundy values efficiently.

Findings

Sprague-Grundy value of any partition is at most 2

Determined Grundy values for several infinite families

Developed an $O(n)$ algorithm for computing Grundy values

Abstract

We apply the Sprague-Grundy Theorem to LCTR, a new impartial game on partitions in which players take turns removing either the Left Column or the Top Row of the corresponding Young diagram. We establish that the Sprague-Grundy value of any partition is at most $2$, and determine Sprague-Grundy values for several infinite families of partitions. Finally, we devise a dynamic programming approach which, for a given partition $\lambda$ of $n$, determines the corresponding Sprague-Grundy value in $O(n)$ time.