A352128 Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.

1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 3, 0, 3, 0, 2, 2, 5, 2, 5, 4, 6, 7, 7, 8, 8, 9, 9, 13, 9, 14, 12, 20, 13, 25, 17, 33, 23, 40, 26, 50, 33, 59, 39, 68, 45, 84, 58, 92, 70, 115, 88, 132, 109, 156, 139, 182, 172, 212, 211

Offset

0,19

Links

Table of n, a(n) for n=0..70.

Example

The a(n) strict partitions for selected n:
n = 3      18         22          28           31              32
   -----------------------------------------------------------------------
    (2,1)  (8,5,3,2)  (8,6,5,3)   (12,7,5,4)   (10,7,5,4,3,2)  (12,8,7,5)
           (8,6,3,1)  (8,7,5,2)   (12,8,5,3)   (10,7,6,5,2,1)  (12,9,7,4)
                      (12,7,2,1)  (12,9,5,2)   (10,8,5,4,3,1)  (16,9,4,3)
                                  (16,9,2,1)   (10,9,6,3,2,1)  (12,10,7,3)
                                  (12,10,5,1)                  (12,11,7,2)
                                                               (16,11,4,1)

Mathematica

conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Count[#, _?OddQ]==Count[#, _?EvenQ]&&Count[conj[#], _?OddQ]==Count[conj[#], _?EvenQ]&]], {n, 0, 30}]

Crossrefs

The first condition is A239241, non-strict A045931 (ranked by A325698).

This is the strict version of A351977, ranked by A350946.

The second condition is A352129, non-strict A045931 (ranked by A350848).

A000041 counts integer partitions, strict A000009.

A130780 counts partitions with no more even than odd parts, strict A239243.

A171966 counts partitions with no more odd than even parts, strict A239240.

There are four statistics:

- A257991 = # of odd parts, conjugate A344616.

- A257992 = # of even parts, conjugate A350847.

There are four other pairings of statistics:

- A277579, strict A352131.

- A277103, ranked by A350944, strict A000700.

- A277579, ranked by A350943, strict A352130.

- A350948, ranked by A350945.

There are two other double-pairings of statistics:

- A351976, ranked by A350949.

- A351981, ranked by A351980.

The case of all four statistics equal is A351978, ranked by A350947.

Cf. A000070, A014105, A088218, A098123, A195017, A236559, A236914, A241638, A325700, A350839, A350941.

Sequence in context: A103863 A166395 A061199 * A334203 A309107 A366353

Adjacent sequences: A352125 A352126 A352127 * A352129 A352130 A352131

Keyword

nonn

Author

Gus Wiseman, Mar 15 2022

Status

approved