A352131
Number of strict integer partitions of n with same number of even parts as odd conjugate parts.
Original entry on oeis.org
1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 10, 10, 12, 14, 15, 14, 17, 21, 20, 20, 25, 28, 28, 29, 34, 39, 39, 40, 47, 52, 53, 56, 64, 70, 71, 77, 86, 92, 97, 104, 114, 122
Offset: 0
The a(n) strict partitions for selected n:
n = 3 10 14 18 21 24
----------------------------------------------------------------------
(2,1) (6,4) (8,6) (10,8) (11,10) (8,7,5,4)
(4,3,2,1) (5,4,3,2) (6,5,4,3) (8,6,4,3) (9,8,4,3)
(6,5,2,1) (7,6,3,2) (8,7,4,2) (10,8,4,2)
(8,7,2,1) (10,8,2,1) (10,9,3,2)
(6,5,4,3,2,1) (11,10,2,1)
(8,6,4,3,2,1)
A130780 counts partitions with no more even than odd parts, strict
A239243.
A171966 counts partitions with no more odd than even parts, strict
A239240.
There are four statistics:
There are four other pairings of statistics:
There are three double-pairings of statistics:
Cf.
A027187,
A027193,
A103919,
A122111,
A236559,
A325039,
A344607,
A344651,
A345196,
A350942,
A350950,
A350951.
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?EvenQ]==Count[conj[#],?OddQ]&]],{n,0,30}]
A352128
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.
Original entry on oeis.org
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, 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, 92, 70, 115, 88, 132, 109, 156, 139, 182, 172, 212, 211
Offset: 0
The a(n) strict partitions for selected n:
n = 3 18 22 28 31 32
-----------------------------------------------------------------------
(2,1) (8,5,3,2) (8,6,5,3) (12,7,5,4) (10,7,5,4,3,2) (12,8,7,5)
(8,6,3,1) (8,7,5,2) (12,8,5,3) (10,7,6,5,2,1) (12,9,7,4)
(12,7,2,1) (12,9,5,2) (10,8,5,4,3,1) (16,9,4,3)
(16,9,2,1) (10,9,6,3,2,1) (12,10,7,3)
(12,10,5,1) (12,11,7,2)
(16,11,4,1)
A130780 counts partitions with no more even than odd parts, strict
A239243.
A171966 counts partitions with no more odd than even parts, strict
A239240.
There are four statistics:
There are four other pairings of statistics:
There are two other double-pairings of statistics:
Cf.
A000070,
A014105,
A088218,
A098123,
A195017,
A236559,
A236914,
A241638,
A325700,
A350839,
A350941.
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?OddQ]==Count[#,?EvenQ]&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]