A351978 - OEIS

%I #6 Mar 18 2022 00:21:09

%S 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,

%T 5,22,26,44,17,30,60,59,42,41,120,84,84,66,204,143,144,131,325,268,

%U 226,261,486,498,344,488,739,874

%N 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.

%e The a(n) partitions for selected n (A = 10):

%e n = 3    12     19       21       23       24         27

%e    --------------------------------------------------------------

%e     21   4332   633322   643332   644333   84332211   655443

%e          4431   643321   654321   654332   84441111   655542

%e                 644311   665211   654431   85322211   665541

%e                 653221            655322   86322111   666333

%e                 654211            655421   86421111   666531

%e                 664111            664331              A522221111

%e                                   665321              A622211111

%e                                   666311

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[Length[Select[IntegerPartitions[n],Count[#,_?EvenQ]==Count[#,_?OddQ]==Count[conj[#],_?EvenQ]==Count[conj[#],_?OddQ]&]],{n,0,30}]

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

%Y These partitions are ranked by A350947.

%Y There are four statistics:

%Y - A257991 = # of odd parts, conjugate A344616.

%Y - A257992 = # of even parts, conjugate A350847.

%Y There are six pairings of statistics:

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

%Y   - ordered A098123

%Y   - strict A239241

%Y   - ranked by A325698

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

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

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

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

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

%Y There are three double-pairings of statistics:

%Y - A351976, ranked by A350949.

%Y - A351977, ranked by A350946.

%Y - A351981, ranked by A351980.

%Y A000041 counts integer partitions, strict A000009.

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

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

%Y Cf. A000070, A122111, A130780, A171966, A236559, A236914, A350849, A350941, A350942, A350950, A350951.

%K nonn

%O 0,13

%A _Gus Wiseman_, Mar 15 2022