A322390 - OEIS

%I #5 Dec 06 2018 06:52:17

%S 0,1,1,2,1,3,1,4,2,5,1,8,1,7,3,11,1,14,2,18,7,21,6,35,14,43,28,65,42,

%T 96,70,141,120,205,187,315,286,445,445,657

%N Number of integer partitions of n with vertex-connectivity 1.

%C The vertex-connectivity of an integer partition is the minimum number of primes that must be divided out (and any parts then equal to 1 removed) so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/K-vertex-connected_graph">k-vertex-connected graph</a>

%e The a(14) = 7 integer partitions are (842), (8222), (77), (4442), (44222), (422222), (2222222).

%e The a(18) = 14 integer partitions:

%e   (9,9), (16,2),

%e   (8,8,2), (10,6,2),

%e   (8,4,4,2), (9,3,3,3),

%e   (4,4,4,4,2), (8,4,2,2,2),

%e   (3,3,3,3,3,3), (4,4,4,2,2,2), (8,2,2,2,2,2),

%e   (4,4,2,2,2,2,2),

%e   (4,2,2,2,2,2,2,2),

%e   (2,2,2,2,2,2,2,2,2).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];

%t vertConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@primeMS/@y],Function[del,Length[csm[DeleteCases[DeleteCases[primeMS/@y,Alternatives@@del,{2}],{}]]]!=1]]];

%t Table[Length[Select[IntegerPartitions[n],vertConn[#]==1&]],{n,20}]

%Y Cf. A013922, A054921, A095983, A304714, A304716, A305078, A305079, A322335, A322338, A322387, A322389, A322391.

%K nonn,more

%O 1,4

%A _Gus Wiseman_, Dec 05 2018