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%I #33 Sep 10 2023 21:55:21
%S 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,
%T 138,164,175,204,222,252,274,317,332,385,403,466,500,563,592,674,720,
%U 799,854,957,994,1131,1196,1328,1395,1551,1627,1817,1912,2098,2197
%N Number of length sets of integer partitions of n.
%C 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.
%C a(n) is the number of distinct such sets in all integer partitions of n.
%H Alois P. Heinz, <a href="/A182410/b182410.txt">Table of n, a(n) for n = 0..170</a>
%e 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}.
%p b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i=1, {{n}},
%p {b(n, i-1)[], seq(map(x-> {x[], j}, b(n-i*j, i-1))[], j=1..n/i)}))
%p end:
%p a:= n-> nops(b(n, n)):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Aug 09 2012
%t Table[Length@ Union@ Map[Union@(Length /@ Split[#]) &, IntegerPartitions[n]], {n, 1, 20}]
%o (Python)
%o from sympy.utilities.iterables import partitions
%o def A182410(n): return len({tuple(sorted(set(p.values()))) for p in partitions(n)}) # _Chai Wah Wu_, Sep 10 2023
%Y Cf. A000041 (number of partitions).
%Y Cf. A088314 (number of different ordered lists of the c(i)).
%Y Cf. A088887 (number of different sorted lists of the c(i)).
%K nonn
%O 0,3
%A _Olivier Gérard_, May 09 2012
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