OFFSET
0,7
COMMENTS
Also number of cliques in following graph: each strict partition of n represents a vertex, the relation "having no common integer" defines the edges connecting these. - Wouter Meeussen, May 27 2002
Conjecture: a(n) asymptotically grows as a*exp(bx^c), where a≈0.57, b≈0.062, c≈1.54. - Elijah Beregovsky, Nov 12 2022
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..46
Naohiro Nomoto, a(0)-a(13) [Broken link]
EXAMPLE
a(8) = 3: {8=1+7=2+6=3+5, 8=1+2+5, 8=1+3+4=2+6}.
a(11) = 8: {11=1+10=2+9=3+8=4+7=5+6, 11=1+2+8=4+7=5+6, 11=1+3+7=2+9=5+6, 11=1+4+6=3+8=2+9, 11=2+3+6=4+7=1+10, 11=2+4+5=1+10=3+8, 11=1+2+3+5=4+7, 11=2+4+5=1+3+7}.
MATHEMATICA
maximal[hit_List, candi_List] := Not[Or@@(UnsameQ@@Flatten[{candi, #}]&/@hit)]; (* write 'ListQPartitions[n]' to list all distinct partitions of n *) Table[it=ListQPartitions[n]; Length@DeleteCases[Backtrack[{#, {}}&/@it, UnsameQ@@Flatten[{#}]&, maximal[it, DeleteCases[ #, {}]]&, All], {}, -1], {n, 3, 14}]
Length[FindClique[RelationGraph[DisjointQ, Select[IntegerPartitions[#], DuplicateFreeQ]], Infinity, All]] & /@ Range[25] (* Elijah Beregovsky, Nov 12 2022 *)
PROG
(Python)
from itertools import combinations
from sympy.utilities.iterables import partitions
from networkx import empty_graph, find_cliques
def A068598(n):
if n == 0: return 1
v = tuple(tuple(p.keys()) for p in partitions(n) if max(p.values(), default=0)==1)
G = empty_graph(v)
G.add_edges_from((a, b) for a, b in combinations(v, 2) if set(a).isdisjoint(set(b)))
return sum(1 for c in find_cliques(G)) # Chai Wah Wu, Jan 11 2024
CROSSREFS
KEYWORD
hard,nonn,nice
AUTHOR
Naohiro Nomoto, Mar 28 2002
EXTENSIONS
More terms from Wouter Meeussen, May 27 2002
a(25) from Robert G. Wilson v, May 29 2002
a(26)-a(39) from Elijah Beregovsky, Nov 12 2022
a(40)-a(45) from Chai Wah Wu, Jan 11 2024
STATUS
approved