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

0,3

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

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

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

Adjacent sequences: A182407 A182408 A182409 * A182411 A182412 A182413

Keyword

nonn

Author

Olivier Gérard, May 09 2012

Status

approved