|
|
A365044
|
|
Number of subsets of {1..n} whose greatest element cannot be written as a (strictly) positive linear combination of the others.
|
|
9
|
|
|
1, 2, 3, 5, 9, 20, 43, 96, 207, 442, 925, 1913, 3911, 7947, 16061, 32350, 64995, 130384, 261271, 523194, 1047208, 2095459, 4192212, 8386044, 16774078, 33550622, 67104244, 134212163, 268428760, 536862900, 1073732255, 2147472267, 4294953778, 8589918612, 17179850312
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Sets of this type may be called "positive combination-free".
Also subsets of {1..n} such that no element can be written as a (strictly) positive linear combination of the others.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The subset S = {3,5,6,8} has 6 = 2*3 + 0*5 + 0*8 and 8 = 1*3 + 1*5 + 0*6 but neither of these is strictly positive, so S is counted under a(8).
The a(0) = 1 through a(5) = 20 subsets:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{2} {2} {2} {2}
{3} {3} {3}
{2,3} {4} {4}
{2,3} {5}
{3,4} {2,3}
{2,3,4} {2,5}
{1,2,3,4} {3,4}
{3,5}
{4,5}
{2,3,4}
{2,4,5}
{3,4,5}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
{1,2,3,4,5}
|
|
MATHEMATICA
|
combp[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 1, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]], And@@Table[combp[Last[#], Union[Most[#]]]=={}, {k, Length[#]}]&]], {n, 0, 10}]
|
|
PROG
|
(Python)
from itertools import combinations
from sympy.utilities.iterables import partitions
mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m, k=m-1)} for m in range(1, n+1))
return n+1+sum(1 for k in range(2, n+1) for w in combinations(range(1, n+1), k) if w[:-1] not in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023
|
|
CROSSREFS
|
A085489 and A364755 count subsets w/o the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A006951, A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364534, A365069.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|