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Some results on LCTR, an impartial game on partitions

Eric Gottlieb, Jelena Ilić and Matjaž Krnc

Vol. 16 (2023), No. 3, 529–546
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 λ of n, determines the corresponding Sprague–Grundy value in O(n) time.

Keywords
combinatorial game theory, Sprague, Grundy, partition
Mathematical Subject Classification
Primary: 05A17, 91A46, 91A68
Milestones
Received: 12 July 2022
Accepted: 16 July 2022
Published: 10 August 2023

Communicated by Arthur T. Benjamin
Authors
Eric Gottlieb
Department of Mathematics and Computer Science
Rhodes College
Memphis, TN
United States
Jelena Ilić
Faculty of Mathematics
Natural Sciences and Information Technologies
University of Primorska
Koper
Slovenia
Matjaž Krnc
Faculty of Mathematics
Natural Sciences and Information Technologies
University of Primorska
Koper
Slovenia