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

1,4

Links

Table of n, a(n) for n=1..50.

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.

Example

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).

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

Crossrefs

Cf. A000041.

Sequence in context: A087329 A298914 A205116 * A011959 A246023 A049927

Adjacent sequences: A373454 A373455 A373456 * A373458 A373459 A373460

Keyword

nonn

Author

Eric Gottlieb, Jun 06 2024

Status

approved