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A351976
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Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.
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15
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1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 5, 6, 9, 11, 11, 16, 21, 22, 24, 31, 41, 46, 48, 64, 82, 91, 98, 120, 155, 175, 188, 237, 297, 329, 357, 437, 544, 607, 658, 803, 987, 1098, 1196, 1432, 1749, 1955, 2126, 2541, 3071, 3417, 3729, 4406, 5291, 5890, 6426
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OFFSET
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0,9
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LINKS
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EXAMPLE
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The a(n) partitions for selected n:
n = 3 8 11 12 15 16
----------------------------------------------------------
(21) (332) (4322) (4332) (4443) (4444)
(4211) (4331) (4422) (54321) (53332)
(4421) (4431) (632211) (55222)
(611111) (53211) (633111) (55411)
(621111) (642111) (633211)
(81111111) (642211)
(643111)
(7321111)
(82111111)
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MATHEMATICA
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conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], Count[#, _?OddQ]==Count[conj[#], _?OddQ]&&Count[#, _?EvenQ]==Count[conj[#], _?EvenQ]&]], {n, 0, 30}]
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CROSSREFS
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These partitions are ranked by A350949.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
There are four statistics:
There are four other possible pairings of statistics:
There are two other possible double-pairings of statistics:
- A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.
- A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.
Cf. A088218, A098123, A130780, A171966, A236559, A236914, A241638, A350849, A350941, A350942, A350950, A350951.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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