|
| |
|
|
A088314
|
|
Cardinality of set of sets of parts of all partitions of n.
|
|
33
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Number of different values of A007947(m) when A056239(m) is equal to n.
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}
(End)
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
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);
|
|
|
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]];
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)]
(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
|
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
