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A351978
Number of integer partitions of n for which the number of even parts, the number of odd parts, the number of even conjugate parts, and the number of odd conjugate parts are all equal.
15
1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 6, 1, 3, 1, 8, 5, 3, 5, 7, 14, 2, 13, 9, 28, 5, 22, 26, 44, 17, 30, 60, 59, 42, 41, 120, 84, 84, 66, 204, 143, 144, 131, 325, 268, 226, 261, 486, 498, 344, 488, 739, 874
OFFSET
0,13
EXAMPLE
The a(n) partitions for selected n (A = 10):
n = 3 12 19 21 23 24 27
--------------------------------------------------------------
21 4332 633322 643332 644333 84332211 655443
4431 643321 654321 654332 84441111 655542
644311 665211 654431 85322211 665541
653221 655322 86322111 666333
654211 655421 86421111 666531
664111 664331 A522221111
665321 A622211111
666311
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[#, _?OddQ]==Count[conj[#], _?EvenQ]==Count[conj[#], _?OddQ]&]], {n, 0, 30}]
CROSSREFS
The strict case appears to be the indicator function for A014105.
These partitions are ranked by A350947.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are six pairings of statistics:
- A045931: # of even parts = # of odd parts:
- ordered A098123
- strict A239241
- ranked by A325698
- A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
- A277579: # even = # odd conj, ranked by A349157, strict A352131.
- A277103: # odd = # odd conj, ranked by A350944, strict A000700.
- A277579: # even conj = # odd, ranked by A350943, strict A352130.
- A350948: # even = # even conj, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946.
- A351981, ranked by A351980.
A000041 counts integer partitions, strict A000009.
A103919 and A116482 count partitions by sum and number of odd/even parts.
A195017 = # of even parts - # of odd parts.
Sequence in context: A271223 A260944 A101670 * A219491 A305714 A324730
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 15 2022
STATUS
approved