A126110 Number of misere quotients of order 2n.

1, 0, 1, 1, 1, 6, 9, 50, 211

Offset: 1

Author

Jonathan Vos Post, Mar 05 2007

Keywords

• nonn
• hard

Comments

Siegel's abstract: "A bipartite monoid is a commutative monoid Q together with an identified subset P subset of Q. In this paper we study a class of bipartite monoids, known as misere quotients, that are naturally associated to impartial combinatorial games. We introduce a structure theory for misere quotients with |P| = 2 and give a complete classification of all such quotients up to isomorphism. One consequence is that if |P| = 2 and Q is finite, then |Q| = 2^n+2 or 2^n+4. We then develop computational techniques for enumerating misere quotients of small order and apply them to count the number of non-isomorphic quotients of order at most 18. We also include a manual proof that there is exactly one quotient of order 8." [Quotation corrected by Thane Plambeck, Jul 08 2014]

References

• E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see pp. 89 and 102.
• J. H. Conway, On Numbers and Games, Second Edition. A. K. Peters, Ltd, 2001, p. 128.
• T. E. Plambeck, Advances in Losing, in M. Albert and M. J. Nowakowski, eds., Games of No Chance 3, Cambridge University Press, forthcoming.

Links

• Achim Flammenkamp, Sparse- and Common-Positions of Sprague-Grundy Values of Octal-Games
• Aaron N. Siegel, The structure and classification of misere quotients, figure 1, p. 3, 2 Mar 2007.

Crossrefs

Cf. A071074, A071434.