login
Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.
15

%I #8 Mar 18 2022 00:20:54

%S 1,1,0,1,1,1,1,1,2,2,2,4,5,5,5,6,9,11,11,16,21,22,24,31,41,46,48,64,

%T 82,91,98,120,155,175,188,237,297,329,357,437,544,607,658,803,987,

%U 1098,1196,1432,1749,1955,2126,2541,3071,3417,3729,4406,5291,5890,6426

%N Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.

%e The a(n) partitions for selected n:

%e n = 3 8 11 12 15 16

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

%e (21) (332) (4322) (4332) (4443) (4444)

%e (4211) (4331) (4422) (54321) (53332)

%e (4421) (4431) (632211) (55222)

%e (611111) (53211) (633111) (55411)

%e (621111) (642111) (633211)

%e (81111111) (642211)

%e (643111)

%e (7321111)

%e (82111111)

%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[conj[#],_?OddQ]&&Count[#,_?EvenQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]

%Y The first condition alone is A277103, ranked by A350944, strict A000700.

%Y The second condition alone is A350948, ranked by A350945.

%Y These partitions are ranked by A350949.

%Y A000041 counts integer partitions.

%Y A122111 represents partition conjugation using Heinz numbers.

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

%Y There are four statistics:

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

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

%Y There are four other possible pairings of statistics:

%Y - A045931: # even = # odd, ranked by A325698, strict A239241.

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

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

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

%Y There are two other possible double-pairings of statistics:

%Y - A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.

%Y - A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.

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

%Y Cf. A088218, A098123, A130780, A171966, A236559, A236914, A241638, A350849, A350941, A350942, A350950, A350951.

%K nonn

%O 0,9

%A _Gus Wiseman_, Mar 14 2022