OFFSET
1,6
COMMENTS
Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(LCM(S)), where omega = A001221 and LCM is least common multiple. A z-forest is a finite set of pairwise indivisible positive integers greater than 1 such that all connected components are z-trees, meaning they have clutter density -1.
This is a generalization to multiset systems of the usual definition of hyperforest (viz. hypergraph F such that two distinct hyperedges of F intersect in at most a common vertex and such that every cycle of F is contained in a hyperedge).
If n is squarefree with k prime factors, then a(n) = A134954(k).
LINKS
Gus Wiseman, Table of n, a(n) for n = 1..250
R. Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO].
EXAMPLE
The a(60) = 16 z-forests together with the corresponding multiset systems (see A112798, A302242) are the following.
(60): {{1,1,2,3}}
(3,20): {{2},{1,1,3}}
(4,15): {{1,1},{2,3}}
(4,30): {{1,1},{1,2,3}}
(5,12): {{3},{1,1,2}}
(6,20): {{1,2},{1,1,3}}
(10,12): {{1,3},{1,1,2}}
(12,15): {{1,1,2},{2,3}}
(12,20): {{1,1,2},{1,1,3}}
(15,20): {{2,3},{1,1,3}}
(3,4,5): {{2},{1,1},{3}}
(3,4,10): {{2},{1,1},{1,3}}
(4,5,6): {{1,1},{3},{1,2}}
(4,6,10): {{1,1},{1,2},{1,3}}
(4,6,15): {{1,1},{1,2},{2,3}}
(4,10,15): {{1,1},{1,3},{2,3}}
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];
Table[Length[Select[Rest[Subsets[Rest[Divisors[n]]]], Function[s, LCM@@s==n&&And@@Table[zensity[Select[s, Divisible[m, #]&]]==-1, {m, zsm[s]}]&&Select[Tuples[s, 2], UnsameQ@@#&&Divisible@@#&]=={}]]], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 19 2018
STATUS
approved