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User: Eric Gottlieb

Eric Gottlieb has authored 2 sequences.

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

Eric Gottlieb, Jun 06 2024

nonn

			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).

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.

Cf. A000041.

<< "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

A364686 a(n) is the number of parity self-conjugate partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 4, 2, 2, 2, 7, 5, 3, 4, 11, 11, 5, 10, 17, 18, 8, 17, 29, 30, 16, 28, 46, 45, 28, 42, 77, 69, 48, 65, 119, 103, 77, 97, 182, 157, 118, 149, 267, 236, 176, 222, 389, 353, 258, 335, 551, 515, 373, 494, 785, 746, 534, 718, 1099, 1061, 764, 1021, 1538, 1494

Offset: 1

Eric Gottlieb, Aug 02 2023

nonn

A partition p is parity self-conjugate if the j-th parts of p and p' have the same parity for every j. If p and p' have different numbers of parts, include terminal 0's as needed.

Such partition p of n has exactly A110654(n) parts for n != 2 and so the largest part is at most A110654(n). - David A. Corneth, Dec 09 2023

			The seven parity self-conjugate partitions of 12 are (6,6), (5, 5, 2), (4, 4, 2, 2), (3, 3, 2, 2, 2), (5, 3, 2, 1, 1), (2, 2, 2, 2, 2, 2), and (6, 2, 1, 1, 1, 1).
From _David A. Corneth_, Dec 09 2023: (Start)
Read as digits these are, with the conjugates, (66, 222222), (552, 33222), (4422, 4422), (33222, 552), (53211, 53211), (22222, 66), (621111, 621111).
66 is extended to 660000 to then check parity of terms in the conjugate 222222.
Note that for example (552, 33222) and (33222, 552) are both counted even though they hold the same partitions, just in a different order. (End)

Cf. A000700, A110654.

<< "Combinatorica`"
Zs[n_] := Table[0, n]
PadDiff[{L1_, L2_}] := Block[{n1 = Length[L1], n2 = Length[L2]},
  Which[n1 < n2, Join[L1, Zs[n2 - n1]] - L2, n1 > n2,
   L1 - Join[L2, Zs[n1 - n2]], n1 == n2, L1 - L2 ]]
PSC1[n_] :=
 Block[{Pttns = IntegerPartitions[n]},
  Union[Flatten[
    Select[Transpose[{Pttns, TransposePartition /@ Pttns}],
     AllTrue[PadDiff[#], EvenQ] &], 1]]]
Table[Length[PSC1[n]], {n, 1, 50}]

a(n) >= A000700(n). - David A. Corneth, Dec 09 2023

More terms from David A. Corneth, Dec 09 2023

cp's OEIS Frontend

User: Eric Gottlieb

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

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