A088314 Cardinality of set of sets of parts of all partitions of n.

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 51, 61, 79, 96, 124, 148, 186, 222, 275, 326, 400, 473, 575, 673, 811, 946, 1132, 1317, 1558, 1813, 2138, 2463, 2893, 3323, 3882, 4461, 5177, 5917, 6847, 7818, 8994, 10251, 11766, 13334, 15281, 17309, 19732, 22307

Offset

0,3

Comments

Number of different values of A007947(m) when A056239(m) is equal to n.

From Gus Wiseman, Sep 11 2023: (Start)

Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:

{1} {1} {1} {1} {1} {1} {1}

{2} {3} {2} {5} {2} {7}

{1,2} {4} {1,2} {3} {1,2}

{1,2} {1,3} {6} {1,3}

{1,3} {1,4} {1,2} {1,4}

{2,3} {1,3} {1,5}

{1,4} {1,6}

{1,5} {2,3}

{2,4} {2,5}

{1,2,3} {3,4}

{1,2,3}

{1,2,4}

Cf. A365073, A365311, A365312, A365322, A365380.

(End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Formula

a(n) = 2^(n-1) - A070880(n). - Alois P. Heinz, Feb 08 2019

a(n) = A365042(n) + 1. - Gus Wiseman, Sep 13 2023

Example

The 7 partitions of 5 and their sets of parts are
[ #]  partition      set of parts
[ 1]  [ 1 1 1 1 1 ]  {1}
[ 2]  [ 2 1 1 1 ]    {1, 2}
[ 3]  [ 2 2 1 ]      {1, 2}  (same as before)
[ 4]  [ 3 1 1 ]      {1, 3}
[ 5]  [ 3 2 ]        {2, 3}
[ 6]  [ 4 1 ]        {1, 4}
[ 7]  [ 5 ]          {5}
so we have a(5) = |{{1}, {1, 2}, {1, 3}, {2, 3}, {1, 4}, {5}}| = 6.

Maple

list2set := L -> {op(L)};
a:= N -> list2set(map( list2set, combinat[partition](N) ));
seq(nops(a(n)), n=0..30);
#  Yogy Namara (yogy.namara(AT)gmail.com), Jan 13 2010
b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x->{x[], i}, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..40);
# Alois P. Heinz, Aug 09 2012

Mathematica

Table[Length[Union[Map[Union, IntegerPartitions[n]]]], {n, 1, 30}] (* Geoffrey Critzer, Feb 19 2013 *)
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i < 1, {},
     Union@Flatten@{b[n, i - 1], Table[If[Head[#] == List,
     Append[#, i]]& /@ b[n - i*j, i - 1], {j, 1, n/i}]}]];
a[n_] := Length[b[n, n]];
a /@ Range[0, 40] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
combp[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 1, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions, n], UnsameQ@@#&&combp[n, #]!={}&]], {n, 0, 15}] (* Gus Wiseman, Sep 11 2023 *)

Prog

(Haskell)
a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013
(Python)
from sympy.utilities.iterables import partitions
def A088314(n): return len({tuple(sorted(set(p))) for p in partitions(n)}) # Chai Wah Wu, Sep 10 2023

Crossrefs

Cf. A182410.

The complement in subsets of {1..n-1} is A070880(n) = A365045(n) - 1.

The case of pairs is A365315, see also A365314, A365320, A365321.

A116861 and A364916 count linear combinations of strict partitions.

A179822 and A326080 count sum-closed subsets.

A326083 and A124506 appear to count combination-free subsets.

A364914 and A365046 count combination-full subsets.

Cf. A000009, A007865, A088528, A088809, A093971, A326020, A364272, A364534, A365043, A364350.

Sequence in context: A023025 A130898 A199016 * A304405 A347446 A097071

Adjacent sequences: A088311 A088312 A088313 * A088315 A088316 A088317

Keyword

easy,nonn

Author

Naohiro Nomoto, Nov 05 2003

Extensions

More terms and clearer definition from Vladeta Jovovic, Apr 21 2005

Status

approved