A373457 - OEIS

%I #21 Sep 03 2024 01:16:08

%S 0,0,1,3,3,4,7,11,12,17,24,34,40,54,73,100,125,164,208,270,337,428,

%T 534,673,828,1033,1276,1584,1938,2385,2909,3559,4318,5252,6346,7678,

%U 9230,11108,13309,15953,19034,22719,27019,32132,38084,45129,53326,62988,74200,87371

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

%H Eric Gottlieb, Matjaž Krnc, and Peter Muršič, <a href="https://doi.org/10.1016/j.dam.2023.11.036">Sprague-Grundy values and complexity for LCTR, Discrete Applied Mathematics</a>, Discrete Applied Mathematics, Volume 346, 2024, Pages 154-169.

%H Eric Gottlieb, Jelena Ilić, and Matjaž Krnc, <a href="https://doi.org/10.2140/involve.2023.16.529">Some results on LCTR, an impartial game on partitions</a>, Involve, Vol. 16 2023, No. 3, pages 529-546.

%e 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).

%t << "Combinatorica`"

%t Mex[Ls_] :=

%t  If[Ls == {}, 0, Min[Complement[Table[n, {n, 0, Length[Ls]}], Ls]]]

%t LCTRMoves[Pttn_] :=

%t  Union[{Rest[Pttn],

%t    TransposePartition[Rest[TransposePartition[Pttn]]]}]

%t LCTRSG[Pttn_] :=

%t  If[Pttn == {}, 0, LCTRSG[Pttn] = Mex[LCTRSG /@ LCTRMoves[Pttn]]]

%t NumLosingPttns[n_] :=

%t  Table[{k,

%t     Length[Select[IntegerPartitions[k], LCTRSG[#] == 0 &]]}, {k, 1,

%t     n}] // TableForm

%Y Cf. A000041.

%K nonn

%O 1,4

%A _Eric Gottlieb_, Jun 06 2024