A182410 Number of length sets of integer partitions of n.

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

Offset: 0

Author

Olivier Gérard, May 09 2012

Keywords

• nonn

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.

Examples

			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}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..170

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)).

Programs

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}]

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