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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.
15
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
0,10
EXAMPLE
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).
MATHEMATICA
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}]
CROSSREFS
The first condition alone is A277579, ranked by A349157.
The second condition alone is A277579, ranked by A350943.
These partitions are ranked by A351980.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931: # of even parts = # of odd parts:
- conjugate also A045931
- ordered A098123
- strict A239241
- ranked by A325698
- conjugate ranked by A350848
- 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:
- A351976, ranked by A350949.
- A351977, ranked by A350946.
The case of all four statistics equal is A351978, ranked by A350947.
Sequence in context: A137992 A047654 A358477 * A058487 A062243 A128095
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 15 2022
STATUS
approved