A351981
Number of integer partitions of n with as many even parts as odd conjugate parts, and as many odd parts as even conjugate parts.
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 2, 2, 4, 2, 1, 6, 8, 7, 9, 13, 14, 15, 19, 21, 23, 32, 40, 41, 45, 66, 81, 80, 96, 124, 139, 160, 194, 221, 246, 303, 360, 390, 446, 546, 634, 703, 810, 971, 1115, 1250, 1448, 1685, 1910
Offset: 0
The a(n) partitions for selected n:
n = 3 9 15 18 19 20 21
-----------------------------------------------------------
21 4221 622221 633222 633322 644321 643332
4311 632211 643221 643321 653321 654321
642111 643311 644221 654221 665211
651111 644211 644311 654311 82222221
653211 653221 82222211 83222211
663111 653311 84221111 84222111
654211 86111111 85221111
664111 86211111
87111111
For example, the partition (6,6,3,1,1,1) has conjugate (6,3,3,2,2,2), and has 2 even, 4 odd, 4 even conjugate, and 2 odd conjugate parts, so is counted under a(18).
These partitions are ranked by
A351980.
There are four statistics:
There are four other pairings of statistics:
-
A045931: # of even parts = # of odd parts:
-
A277103: # of odd parts = # of odd conjugate parts, ranked by
A350944.
-
A350948: # of even parts = # of even conjugate parts, ranked by
A350945.
There are two other double-pairings of statistics:
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Count[#,?EvenQ]==Count[conj[#],?OddQ]&&Count[#,?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]
A352129
Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 5, 5, 6, 6, 9, 8, 10, 12, 13, 15, 17, 20, 20, 26, 26, 32, 35, 39, 44, 50, 55, 61, 71, 76, 87, 96, 108, 117, 135, 145, 164, 181, 200, 222, 246, 272, 298, 334, 363, 404, 443
Offset: 0
The a(n) strict partitions for selected n:
n = 3 13 15 18 20 22
------------------------------------------------------------------
(2,1) (6,5,2) (10,5) (12,6) (12,7,1) (12,8,2)
(6,4,2,1) (6,4,3,2) (8,7,3) (8,5,4,3) (8,6,5,3)
(6,5,3,1) (8,5,3,2) (8,6,4,2) (8,7,5,2)
(8,6,3,1) (8,7,4,1) (12,7,2,1)
(8,6,3,2,1) (8,6,4,3,1)
(8,7,4,2,1)
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:
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]
A352130
Number of strict integer partitions of n with as many odd parts as even conjugate parts.
Original entry on oeis.org
1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 6, 7, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 23, 25, 28, 31, 34, 37, 41, 45, 50, 55, 60, 65, 72, 79, 86, 93, 102, 111, 121, 132, 143, 155, 169, 183, 197, 213, 231, 251, 271, 292, 315, 340, 367, 396
Offset: 0
The a(n) strict partitions for selected n:
n = 2 7 9 13 14 15 16
--------------------------------------------------------------------
(2) (6,1) (8,1) (12,1) (14) (14,1) (16)
(4,2,1) (4,3,2) (6,4,3) (6,5,3) (6,5,4) (8,5,3)
(6,2,1) (8,3,2) (10,3,1) (8,4,3) (12,3,1)
(10,2,1) (6,4,3,1) (10,3,2) (6,5,4,1)
(8,3,2,1) (12,2,1) (8,4,3,1)
(6,5,3,1) (10,3,2,1)
(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,
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[#,?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]
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}]