A351977 - OEIS

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.

%I #5 Mar 18 2022 00:21:02

%S 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,

%T 35,41,58,71,72,90,106,121,142,178,191,216,269,320,344,400,486,564,

%U 633,734,867,991,1130,1312,1509,1702,1978,2288,2582,2917,3404

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

%e The a(n) partitions for selected n (A..C = 10..12):

%e n = 3 9 15 18 20

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

%e (21) (63) (A5) (8433) (8543)

%e (222111) (632211) (8532) (8741)

%e (642111) (8631) (C611)

%e (2222211111) (43322211) (43332221)

%e (44322111) (44432111)

%e (44421111) (84221111)

%e (422222111111)

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

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

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

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

%Y These partitions are ranked by A350946.

%Y The strict case is A352128.

%Y There are four statistics:

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

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

%Y There are four additional pairings of statistics:

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

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

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

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

%Y There are two additional double-pairings of statistics:

%Y - A351981, ranked by A351980.

%Y - A351976, ranked by A350949.

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

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

%K nonn

%O 0,10

%A _Gus Wiseman_, Mar 14 2022