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A182410
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Number of length sets of integer partitions of n.
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2
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1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 15, 17, 24, 25, 31, 34, 45, 48, 59, 64, 77, 83, 99, 109, 131, 138, 164, 175, 204, 222, 252, 274, 317, 332, 385, 403, 466, 500, 563, 592, 674, 720, 799, 854, 957, 994, 1131, 1196, 1328, 1395, 1551, 1627, 1817, 1912, 2098, 2197
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OFFSET
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0,3
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COMMENTS
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For an integer partition n = c(1)*1 + c(2)*2 + ... + c(n)*n, construct the set of all positive c(i) occurring at least one time.
a(n) is the number of distinct such sets in all integer partitions of n.
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LINKS
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EXAMPLE
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For n=8 the 11 possible sets are {1}, {2}, {4}, {8}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3} and {2, 4}.
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i=1, {{n}},
{b(n, i-1)[], seq(map(x-> {x[], j}, b(n-i*j, i-1))[], j=1..n/i)}))
end:
a:= n-> nops(b(n, n)):
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MATHEMATICA
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Table[Length@ Union@ Map[Union@(Length /@ Split[#]) &, IntegerPartitions[n]], {n, 1, 20}]
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PROG
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(Python)
from sympy.utilities.iterables import partitions
def A182410(n): return len({tuple(sorted(set(p.values()))) for p in partitions(n)}) # Chai Wah Wu, Sep 10 2023
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CROSSREFS
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Cf. A000041 (number of partitions).
Cf. A088314 (number of different ordered lists of the c(i)).
Cf. A088887 (number of different sorted lists of the c(i)).
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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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