A322400
Heinz numbers of integer partitions with vertex-connectivity 1.
Original entry on oeis.org
3, 5, 7, 9, 11, 17, 19, 21, 23, 25, 27, 31, 41, 49, 53, 57, 59, 63, 67, 81, 83, 97, 103, 109, 115, 121, 125, 127, 131, 133, 147, 157, 159, 171, 179, 189, 191, 211, 227, 241, 243, 277, 283, 289, 311, 331, 343, 353, 361, 367, 371, 377, 393, 399, 401, 419, 431
Offset: 1
The sequence of all integer partitions with vertex-connectivity 1 begins: (2), (3), (4), (2,2), (5), (7), (8), (4,2), (9), (3,3), (2,2,2), (11), (13), (4,4), (16), (8,2), (17), (4,2,2), (19), (2,2,2,2), (23), (25), (27), (29), (9,3), (5,5), (3,3,3), (31), (32), (8,4), (4,4,2), (37), (16,2), (8,2,2), (41), (4,2,2,2), (43).
Cf.
A003963,
A013922,
A056239,
A112798,
A302242,
A304716,
A305078,
A305079,
A322387,
A322388,
A322389,
A322390,
A322390,
A322394.
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primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
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]]];
Select[Range[100],vertConn[primeMS[#]]==1&]
A327375
Number of set-systems with n vertices and vertex-connectivity 2.
Original entry on oeis.org
0, 0, 0, 72, 4752
Offset: 0
BII-numbers for vertex-connectivity 2 are
A327374.
BII-numbers for cut-connectivity 2 are
A327082.
BII-numbers for spanning edge-connectivity 2 are
A327108.
BII-numbers for non-spanning edge-connectivity 2 are
A327097.
Labeled graphs with vertex-connectivity 2 are
A327198.
The vertex-connectivity of the set-system with BII-number n is
A327051(n).
The enumeration of labeled graphs by vertex-connectivity is
A327334.
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csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],vertConnSys[Range[n],#]==2&]],{n,0,3}]
A322401
Number of strict integer partitions of n with edge-connectivity 1.
Original entry on oeis.org
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, 33, 19, 38, 22, 54, 28, 71, 44, 83, 53, 110, 68, 134, 98, 154, 120, 209, 145, 253, 191, 302, 244, 385, 299, 459, 390, 553, 483, 693, 578
Offset: 0
The a(30) = 11 strict integer partitions with edge-connectivity 1:
(30),
(10,9,6,5), (12,10,5,3), (14,7,6,3), (15,6,5,4), (15,10,3,2),
(9,8,6,4,3), (10,9,6,3,2), (12,9,4,3,2), (15,6,4,3,2),
(10,6,5,4,3,2).
Cf.
A095983,
A218970,
A304714,
A304716,
A305078,
A305079,
A322335,
A322337,
A322387,
A322390,
A322391,
A322393,
A322394,
A322400.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[primeMS/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[primeMS/@#]]!=1&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&edgeConn[#]==1&]],{n,30}]
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