cp's OEIS Frontend

A322401 Number of strict integer partitions of n with edge-connectivity 1.

%I A322401 #8 Jun 15 2023 07:59:16
%S A322401 0,1,1,1,1,1,1,1,1,1,1,2,1,2,1,2,1,5,1,6,2,7,2,13,3,14,6,18,8,28,11,
%T A322401 33,19,38,22,54,28,71,44,83,53,110,68,134,98,154,120,209,145,253,191,
%U A322401 302,244,385,299,459,390,553,483,693,578
%N A322401 Number of strict integer partitions of n with edge-connectivity 1.
%C A322401 The edge-connectivity of an integer partition is the minimum number of parts that must be removed so that the prime factorizations of the remaining parts form a disconnected (or empty) hypergraph.
%H A322401 Wikipedia, <a href="https://en.wikipedia.org/wiki/K-edge-connected_graph">k-edge-connected graph</a>
%e A322401 The a(30) = 11 strict integer partitions with edge-connectivity 1:
%e A322401   (30),
%e A322401   (10,9,6,5), (12,10,5,3), (14,7,6,3), (15,6,5,4), (15,10,3,2),
%e A322401   (9,8,6,4,3), (10,9,6,3,2), (12,9,4,3,2), (15,6,4,3,2),
%e A322401   (10,6,5,4,3,2).
%t A322401 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A322401 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 A322401 edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]];
%t A322401 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&edgeConn[#]==1&]],{n,30}]
%Y A322401 Cf. A095983, A218970, A304714, A304716, A305078, A305079, A322335, A322337, A322387, A322390, A322391, A322393, A322394, A322400.
%K A322401 nonn
%O A322401 0,12
%A A322401 _Gus Wiseman_, Dec 06 2018