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A182410 Number of length sets of integer partitions of n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
LINKS
EXAMPLE
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}.
MAPLE
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)):
seq(a(n), n=0..50); # Alois P. Heinz, Aug 09 2012
MATHEMATICA
Table[Length@ Union@ Map[Union@(Length /@ Split[#]) &, IntegerPartitions[n]], {n, 1, 20}]
PROG
(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
CROSSREFS
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)).
Sequence in context: A230167 A060028 A341951 * A341719 A099770 A099383
KEYWORD
nonn
AUTHOR
Olivier Gérard, May 09 2012
STATUS
approved

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Last modified March 29 09:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)