A352128 - OEIS

%I #8 Mar 18 2022 00:21:32

%S 1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,1,1,0,2,0,2,2,3,0,3,0,2,2,5,2,5,4,6,7,

%T 7,8,8,9,9,13,9,14,12,20,13,25,17,33,23,40,26,50,33,59,39,68,45,84,58,

%U 92,70,115,88,132,109,156,139,182,172,212,211

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

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

%e n = 3      18         22          28           31              32

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

%e     (2,1)  (8,5,3,2)  (8,6,5,3)   (12,7,5,4)   (10,7,5,4,3,2)  (12,8,7,5)

%e            (8,6,3,1)  (8,7,5,2)   (12,8,5,3)   (10,7,6,5,2,1)  (12,9,7,4)

%e                       (12,7,2,1)  (12,9,5,2)   (10,8,5,4,3,1)  (16,9,4,3)

%e                                   (16,9,2,1)   (10,9,6,3,2,1)  (12,10,7,3)

%e                                   (12,10,5,1)                  (12,11,7,2)

%e                                                                (16,11,4,1)

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

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

%Y The first condition is A239241, non-strict A045931 (ranked by A325698).

%Y This is the strict version of A351977, ranked by A350946.

%Y The second condition is A352129, non-strict A045931 (ranked by A350848).

%Y A000041 counts integer partitions, strict A000009.

%Y A130780 counts partitions with no more even than odd parts, strict A239243.

%Y A171966 counts partitions with no more odd than even parts, strict A239240.

%Y There are four statistics:

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

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

%Y There are four other pairings of statistics:

%Y - A277579, strict A352131.

%Y - A277103, ranked by A350944, strict A000700.

%Y - A277579, ranked by A350943, strict A352130.

%Y - A350948, ranked by A350945.

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

%Y - A351976, ranked by A350949.

%Y - A351981, ranked by A351980.

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

%Y Cf. A000070, A014105, A088218, A098123, A195017, A236559, A236914, A241638, A325700, A350839, A350941.

%K nonn

%O 0,19

%A _Gus Wiseman_, Mar 15 2022