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A114639
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Number of partitions of n such that the set of parts and the set of multiplicities of parts are disjoint.
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22
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1, 0, 2, 2, 2, 3, 5, 4, 7, 7, 13, 16, 19, 23, 33, 34, 44, 58, 63, 80, 101, 112, 139, 171, 196, 234, 288, 328, 394, 478, 545, 658, 777, 881, 1050, 1236, 1414, 1666, 1936, 2216, 2592, 3018, 3428, 3992, 4604, 5243, 6069, 6986, 7951, 9139, 10447, 11892, 13625
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OFFSET
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0,3
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COMMENTS
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LINKS
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EXAMPLE
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The a(2) = 2 through a(9) = 7 partitions:
(2) (3) (4) (5) (6) (7) (8) (9)
(11) (111) (1111) (32) (33) (43) (44) (54)
(11111) (42) (52) (53) (63)
(222) (1111111) (62) (72)
(111111) (2222) (432)
(3311) (222111)
(11111111) (111111111)
(End)
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MAPLE
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b:= proc(n, i, p, m) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1, p, select(x-> x<i, m))+
add(`if`(i=j or i in m or j in p, 0, b(n-i*j, i-1,
select(x-> x<=n-i*j, p union {i}),
select(x-> x<i, m union {j}))), j=1..n/i)))
end:
a:= n-> b(n$2, {}$2):
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MATHEMATICA
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b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, 1, If[i<1, 0, b[n, i-1, p, Select[m, #<i&]] + Sum[If[i == j || MemberQ[m, i] || MemberQ[p, j], 0, b[n-i*j, i-1, Select[ p ~Union~ {i}, # <= n-i*j&], Select[m ~Union~ {j}, #<i&]]], {j, 1, n/i}]]]; a[n_] := b[n, n, {}, {}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], Intersection[#, Length/@Split[#]]=={}&]], {n, 0, 30}] (* Gus Wiseman, Apr 02 2019 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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