A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 6, 7, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 23, 25, 28, 31, 34, 37, 41, 45, 50, 55, 60, 65, 72, 79, 86, 93, 102, 111, 121, 132, 143, 155, 169, 183, 197, 213, 231, 251, 271, 292, 315, 340, 367, 396

Offset

0,8

Links

Table of n, a(n) for n=0..60.

Example

The a(n) strict partitions for selected n:
n = 2    7        9        13        14         15         16
   --------------------------------------------------------------------
    (2)  (6,1)    (8,1)    (12,1)    (14)       (14,1)     (16)
         (4,2,1)  (4,3,2)  (6,4,3)   (6,5,3)    (6,5,4)    (8,5,3)
                  (6,2,1)  (8,3,2)   (10,3,1)   (8,4,3)    (12,3,1)
                           (10,2,1)  (6,4,3,1)  (10,3,2)   (6,5,4,1)
                                     (8,3,2,1)  (12,2,1)   (8,4,3,1)
                                                (6,5,3,1)  (10,3,2,1)
                                                           (6,4,3,2,1)

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[conj[#], _?EvenQ]&]], {n, 0, 30}]

Crossrefs

This is the strict case of A277579, ranked by A350943 (zeros of A350942).

The conjugate version is A352131, non-strict A277579 (ranked by A349157).

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:

- A045931, ranked by A325698, strict A239241.

- A045931, ranked by A350848, strict A352129.

- A277103, ranked by A350944, strict new.

- A350948, ranked by A350945, strict new.

There are three double-pairings of statistics:

- A351976, ranked by A350949, strict A010054?

- A351977, ranked by A350946, strict A352128.

- A351981, ranked by A351980. strict A014105?

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

Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350950, A350951.

Sequence in context: A284523 A034584 A359357 * A035430 A167227 A048280

Adjacent sequences: A352127 A352128 A352129 * A352131 A352132 A352133

Keyword

nonn

Author

Gus Wiseman, Mar 15 2022

Status

approved