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

1,8

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

Links

Table of n, a(n) for n=1..65.

Formula

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

Example

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)

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}]

Crossrefs

Cf. A000700, A110654.

Sequence in context: A255909 A344903 A198101 * A341859 A058634 A037919

Adjacent sequences: A364683 A364684 A364685 * A364687 A364688 A364689

Keyword

nonn

Author

Eric Gottlieb, Aug 02 2023

Extensions

More terms from David A. Corneth, Dec 09 2023

Status

approved