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

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

Author

Eric Gottlieb, Aug 02 2023

Keywords

• nonn

Comments

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

Examples

			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)
		

Crossrefs

Cf. A000700, A110654.

Programs

Mathematica

<< "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}]

Formula

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

Extensions

More terms from David A. Corneth, Dec 09 2023