cp's OEIS Frontend

A304382 Number of z-trees summing to n. Number of connected strict integer partitions of n with pairwise indivisible parts and clutter density -1.

%I A304382 #25 Feb 03 2025 09:38:23
%S A304382 1,1,1,1,1,1,1,1,1,2,1,1,1,3,2,2,1,3,2,4,3,5,2,5,4,6,3,7,6,8,4,9,8,13,
%T A304382 9,15,8,14,12,16,12,20,20,24,15,27,20,33,27,35
%N A304382 Number of z-trees summing to n. Number of connected strict integer partitions of n with pairwise indivisible parts and clutter density -1.
%C A304382 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.
%C A304382 The clutter density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(LCM(S)).
%H A304382 Roland 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], 2011.
%e A304382 The a(30) = 8 z-trees together with the corresponding multiset systems are the following.
%e A304382        (30): {{1,2,3}}
%e A304382      (26,4): {{1,6},{1,1}}
%e A304382      (22,8): {{1,5},{1,1,1}}
%e A304382      (21,9): {{2,4},{2,2}}
%e A304382     (16,14): {{1,1,1,1},{1,4}}
%e A304382    (15,9,6): {{2,3},{2,2},{1,2}}
%e A304382   (14,10,6): {{1,4},{1,3},{1,2}}
%e A304382   (12,10,8): {{1,1,2},{1,3},{1,1,1}}
%t A304382 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 A304382 zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
%t A304382 zreeQ[s_]:=And[Length[s]>=2,zensity[s]==-1];
%t A304382 strConnAnti[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&&Length[zsm[#]]==1&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}&];
%t A304382 Table[Length[Select[strConnAnti[n],Length[#]==1||zreeQ[#]&]],{n,20}]
%Y A304382 Cf. A000009, A006126, A048143, A054921, A112798, A285572, A286518, A286520, A293993, A303362, A303837, A304714, A304716, A305194, A305195.
%K A304382 nonn,more
%O A304382 1,10
%A A304382 _Gus Wiseman_, May 21 2018