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A340601
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Number of integer partitions of n of even rank.
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27
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1, 1, 0, 3, 1, 5, 3, 11, 8, 18, 16, 34, 33, 57, 59, 98, 105, 159, 179, 262, 297, 414, 478, 653, 761, 1008, 1184, 1544, 1818, 2327, 2750, 3480, 4113, 5137, 6078, 7527, 8899, 10917, 12897, 15715, 18538, 22431, 26430, 31805, 37403, 44766, 52556, 62620, 73379
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OFFSET
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0,4
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COMMENTS
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The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
(1) . (3) (22) (5) (42) (7) (44) (9)
(21) (41) (321) (43) (62) (63)
(111) (311) (2211) (61) (332) (81)
(2111) (322) (521) (333)
(11111) (331) (2222) (522)
(511) (4211) (531)
(2221) (32111) (711)
(4111) (221111) (4221)
(31111) (4311)
(211111) (6111)
(1111111) (32211)
(33111)
(51111)
(222111)
(411111)
(3111111)
(21111111)
(111111111)
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MAPLE
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b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),
`if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
`if`(r<0, irem(i, 2), r))))
end:
a:= n-> b(n$2, -1):
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MATHEMATICA
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Table[If[n==0, 1, Length[Select[IntegerPartitions[n], EvenQ[Max[#]-Length[#]]&]]], {n, 0, 30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1];
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CROSSREFS
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Note: Heinz numbers are given in parentheses below.
The Heinz numbers of these partitions are A340602.
- Rank -
A072233 counts partitions by sum and length.
A257541 gives the rank of the partition with Heinz number n.
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A034008 counts compositions of even length.
A052841 counts ordered set partitions of even length.
A339846 counts factorizations of even length.
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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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