A373457 Number of losing integer partitions of n in the impartial combinatorial game LCTR (left column, top row).

0, 0, 1, 3, 3, 4, 7, 11, 12, 17, 24, 34, 40, 54, 73, 100, 125, 164, 208, 270, 337, 428, 534, 673, 828, 1033, 1276, 1584, 1938, 2385, 2909, 3559, 4318, 5252, 6346, 7678, 9230, 11108, 13309, 15953, 19034, 22719, 27019, 32132, 38084, 45129, 53326, 62988, 74200, 87371

Offset: 1

Author

Eric Gottlieb, Jun 06 2024

Keywords

• nonn

Examples

			For n = 8, the a(8) = 11 losing partitions are the six nondegenerate hooks (7,1), (6, 1, 1), (5, 1, 1, 1), (4, 1, 1, 1, 1), (3, 1, 1, 1, 1, 1), (2, 1, 1, 1, 1, 1, 1) and (5, 3), (4, 4), (3, 3, 2), (2, 2, 2, 2), (2, 2, 2, 1, 1).
		

Links

• Eric Gottlieb, Matjaž Krnc, and Peter Muršič, Sprague-Grundy values and complexity for LCTR, Discrete Applied Mathematics, Discrete Applied Mathematics, Volume 346, 2024, Pages 154-169.
• Eric Gottlieb, Jelena Ilić, and Matjaž Krnc, Some results on LCTR, an impartial game on partitions, Involve, Vol. 16 2023, No. 3, pages 529-546.

Crossrefs

Cf. A000041.

Programs

Mathematica

<< "Combinatorica`"
Mex[Ls_] :=
 If[Ls == {}, 0, Min[Complement[Table[n, {n, 0, Length[Ls]}], Ls]]]
LCTRMoves[Pttn_] :=
 Union[{Rest[Pttn],
   TransposePartition[Rest[TransposePartition[Pttn]]]}]
LCTRSG[Pttn_] :=
 If[Pttn == {}, 0, LCTRSG[Pttn] = Mex[LCTRSG /@ LCTRMoves[Pttn]]]
NumLosingPttns[n_] :=
 Table[{k,
    Length[Select[IntegerPartitions[k], LCTRSG[#] == 0 &]]}, {k, 1,
    n}] // TableForm