A351977 Number of integer partitions of n with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.

1, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 0, 2, 4, 2, 1, 6, 6, 7, 9, 11, 10, 13, 17, 17, 21, 28, 36, 35, 41, 58, 71, 72, 90, 106, 121, 142, 178, 191, 216, 269, 320, 344, 400, 486, 564, 633, 734, 867, 991, 1130, 1312, 1509, 1702, 1978, 2288, 2582, 2917, 3404

Offset: 0

Author

Gus Wiseman, Mar 14 2022

Keywords

• nonn

Examples

			The a(n) partitions for selected n (A..C = 10..12):
n = 3     9         15            18          20
   ----------------------------------------------------------
    (21)  (63)      (A5)          (8433)      (8543)
          (222111)  (632211)      (8532)      (8741)
                    (642111)      (8631)      (C611)
                    (2222211111)  (43322211)  (43332221)
                                  (44322111)  (44432111)
                                  (44421111)  (84221111)
                                              (422222111111)
		

Crossrefs

The first condition alone is A045931, ranked by A325698, strict A239241.

The second condition alone is A045931, ranked by A350848, strict A352129.

These partitions are ranked by A350946.

The strict case is A352128.

There are four statistics:

- A257991 = # of odd parts, conjugate A344616.

- A257992 = # of even parts, conjugate A350847.

There are four additional pairings of statistics:

- A277579: # even = # odd conj, ranked by A349157, strict A352131.

- A277579: # even conj = # odd, ranked by A350943, strict A352130.

- A277103: # odd = # odd conj, ranked by A350944, strict A000700.

- A350948: # even = # even conj, ranked by A350945.

There are two additional double-pairings of statistics:

- A351981, ranked by A351980.

- A351976, ranked by A350949.

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

Cf. A000041, A000070, A088218, A098123, A130780, A171966, A195017, A236559, A236914, A241638, A350849.

Programs

Mathematica

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