A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 10, 10, 12, 14, 15, 14, 17, 21, 20, 20, 25, 28, 28, 29, 34, 39, 39, 40, 47, 52, 53, 56, 64, 70, 71, 77, 86, 92, 97, 104, 114, 122

Offset: 0

Author

Gus Wiseman, Mar 15 2022

Keywords

• nonn

Examples

			The a(n) strict partitions for selected n:
n = 3      10         14         18         21             24
   ----------------------------------------------------------------------
    (2,1)  (6,4)      (8,6)      (10,8)     (11,10)        (8,7,5,4)
           (4,3,2,1)  (5,4,3,2)  (6,5,4,3)  (8,6,4,3)      (9,8,4,3)
                      (6,5,2,1)  (7,6,3,2)  (8,7,4,2)      (10,8,4,2)
                                 (8,7,2,1)  (10,8,2,1)     (10,9,3,2)
                                            (6,5,4,3,2,1)  (11,10,2,1)
                                                           (8,6,4,3,2,1)
		

Crossrefs

This is the strict case of A277579, ranked by A349157 (zeros of A350849).

The conjugate version is A352130, non-strict A277579 (ranked by A350943).

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.

- 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. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350942, A350950, A350951.

Programs

Mathematica

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