A351978 Number of integer partitions of n for which the number of even parts, the number of odd parts, the number of even conjugate parts, and the number of odd conjugate parts are all equal.

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 6, 1, 3, 1, 8, 5, 3, 5, 7, 14, 2, 13, 9, 28, 5, 22, 26, 44, 17, 30, 60, 59, 42, 41, 120, 84, 84, 66, 204, 143, 144, 131, 325, 268, 226, 261, 486, 498, 344, 488, 739, 874

Offset: 0

Author

Gus Wiseman, Mar 15 2022

Keywords

• nonn

Examples

			The a(n) partitions for selected n (A = 10):
n = 3    12     19       21       23       24         27
   --------------------------------------------------------------
    21   4332   633322   643332   644333   84332211   655443
         4431   643321   654321   654332   84441111   655542
                644311   665211   654431   85322211   665541
                653221            655322   86322111   666333
                654211            655421   86421111   666531
                664111            664331              A522221111
                                  665321              A622211111
                                  666311
		

Crossrefs

The strict case appears to be the indicator function for A014105.

These partitions are ranked by A350947.

There are four statistics:

- A257991 = # of odd parts, conjugate A344616.

- A257992 = # of even parts, conjugate A350847.

There are six pairings of statistics:

- A045931: # of even parts = # of odd parts:

- ordered A098123

- strict A239241

- ranked by A325698

- A045931: # even conj = # odd conj, ranked by A350848, strict A352129.

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

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

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

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

There are three double-pairings of statistics:

- A351976, ranked by A350949.

- A351977, ranked by A350946.

- A351981, ranked by A351980.

A000041 counts integer partitions, strict A000009.

A103919 and A116482 count partitions by sum and number of odd/even parts.

A195017 = # of even parts - # of odd parts.

Cf. A000070, A122111, A130780, A171966, A236559, A236914, A350849, A350941, 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],Count[#,?EvenQ]==Count[#,?OddQ]==Count[conj[#],?EvenQ]==Count[conj[#],?OddQ]&]],{n,0,30}]