OFFSET
1,10
COMMENTS
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.
The clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(LCM(S)).
LINKS
R. Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO].
EXAMPLE
The a(30) = 8 z-trees together with the corresponding multiset systems are the following.
(30): {{1,2,3}}
(26,4): {{1,6},{1,1}}
(22,8): {{1,5},{1,1,1}}
(21,9): {{2,4},{2,2}}
(16,14): {{1,1,1,1},{1,4}}
(15,9,6): {{2,3},{2,2},{1,2}}
(14,10,6): {{1,4},{1,3},{1,2}}
(12,10,8): {{1,1,2},{1,3},{1,1,1}}
MATHEMATICA
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
zreeQ[s_]:=And[Length[s]>=2, zensity[s]==-1];
strConnAnti[n_]:=Select[IntegerPartitions[n], UnsameQ@@#&&Length[zsm[#]]==1&&Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]=={}&];
Table[Length[Select[strConnAnti[n], Length[#]==1||zreeQ[#]&]], {n, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 21 2018
STATUS
approved