A305194 - OEIS

%I #5 May 27 2018 14:43:29

%S 1,1,1,1,2,1,3,2,3,4,5,4,6,7,7,9,11,12,13,15,17,20,23,25,27,32,35,40,

%T 45,49,54,58,67,78,82,95,99,111,123,135,150,164,177,194,214,236,260,

%U 282,309,330

%N Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.

%C 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. Then a z-forest is a strict integer partition with pairwise indivisible parts greater than 1 such that all connected components have clutter density -1.

%H R. Bacher, <a href="https://arxiv.org/abs/1102.2708">On the enumeration of labelled hypertrees and of labelled bipartite trees</a>, arXiv:1102.2708 [math.CO].

%e The a(17) = 11 z-forests together with the corresponding multiset systems:

%e        (17): {{7}}

%e      (15,2): {{2,3},{1}}

%e      (14,3): {{1,4},{2}}

%e      (13,4): {{6},{1,1}}

%e      (12,5): {{1,1,2},{3}}

%e      (11,6): {{5},{1,2}}

%e      (10,7): {{1,3},{4}}

%e       (9,8): {{2,2},{1,1,1}}

%e    (10,4,3): {{1,3},{1,1},{2}}

%e     (7,6,4): {{4},{1,2},{1,1}}

%e   (7,5,3,2): {{4},{3},{2},{1}}

%t 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]]]]]]]]];

%t zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];

%t zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];

%t Table[Length[Select[IntegerPartitions[n],Function[s,UnsameQ@@s&&And@@(Length[#]==1||zreeQ[#]&)/@Table[Select[s,Divisible[m,#]&],{m,zsm[s]}]&&Select[Tuples[s,2],UnsameQ@@#&&Divisible@@#&]=={}]]],{n,50}]

%Y Cf. A030019, A048143, A134954, A275307, A285572, A293510, A293993, A293994, A303362, A303837, A303838, A304118, A304382, A305078, A305195.

%K nonn

%O 1,5

%A _Gus Wiseman_, May 27 2018