A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 5, 5, 6, 6, 9, 8, 10, 12, 13, 15, 17, 20, 20, 26, 26, 32, 35, 39, 44, 50, 55, 61, 71, 76, 87, 96, 108, 117, 135, 145, 164, 181, 200, 222, 246, 272, 298, 334, 363, 404, 443

Offset

0,14

Links

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

Example

The a(n) strict partitions for selected n:
n = 3      13         15         18         20           22
   ------------------------------------------------------------------
    (2,1)  (6,5,2)    (10,5)     (12,6)     (12,7,1)     (12,8,2)
           (6,4,2,1)  (6,4,3,2)  (8,7,3)    (8,5,4,3)    (8,6,5,3)
                      (6,5,3,1)  (8,5,3,2)  (8,6,4,2)    (8,7,5,2)
                                 (8,6,3,1)  (8,7,4,1)    (12,7,2,1)
                                            (8,6,3,2,1)  (8,6,4,3,1)
                                                         (8,7,4,2,1)

Mathematica

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

Crossrefs

This is the strict case of A045931, ranked by A350848 (zeros of A350941).

The conjugate version is A239241, non-strict A045931 (ranked by A325698).

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, ranked by A349157, strict A352131.

- A277103, ranked by A350944.

- A277579, ranked by A350943, strict A352130.

- A350948, ranked by A350945.

There are three double-pairings of statistics:

- A351976, ranked by A350949.

- A351977, ranked by A350946, strict A352128.

- A351981, ranked by A351980.

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

Cf. A000070, A098123, A195017, A236559, A241638, A350839, A350849, A350942.

Sequence in context: A234361 A350013 A240450 * A340351 A115872 A133926

Adjacent sequences: A352126 A352127 A352128 * A352130 A352131 A352132

Keyword

nonn

Author

Gus Wiseman, Mar 15 2022

Status

approved