Finite Cardinalities of Mis\`ere Quotients
Simon Rubinstein-Salzedo, Stephen Zhou
TL;DR
This paper investigates the possible sizes of finite misère quotient structures, revealing that partisan misère quotients can have any finite size except three, unlike impartial ones which are always even.
Contribution
It proves that partisan misère quotients can have any finite size other than three, answering a longstanding open question.
Findings
Partisan misère quotients can have any finite cardinality except three.
Impartial misère quotients must have even cardinality.
The paper resolves a question posed by Allen.
Abstract
We find that partisan mis\`ere quotients can have any finite cardinality other than 3, answering a question of Allen. This contrasts with impartial mis\`ere quotients, which must have even cardinality.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Graph Labeling and Dimension Problems