A263296
Triangle read by rows: T(n,k) is the number of graphs with n vertices with edge connectivity k.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 5, 3, 2, 1, 13, 10, 8, 2, 1, 44, 52, 41, 15, 3, 1, 191, 351, 352, 121, 25, 3, 1, 1229, 3714, 4820, 2159, 378, 41, 4, 1, 13588, 63638, 113256, 68715, 14306, 1095, 65, 4, 1, 288597, 1912203, 4602039, 3952378, 1141575, 104829, 3441, 100, 5, 1
Offset: 1
Triangle begins:
1;
1, 1;
2, 1, 1;
5, 3, 2, 1;
13, 10, 8, 2, 1;
44, 52, 41, 15, 3, 1;
191, 351, 352, 121, 25, 3, 1;
1229, 3714, 4820, 2159, 378, 41, 4, 1;
...
Columns k=0..10 are
A000719,
A052446,
A052447,
A052448,
A241703,
A241704,
A241705,
A324096,
A324097,
A324098,
A324099.
Cf.
A002494,
A095983,
A259862,
A327076,
A327108,
A327109,
A327111,
A327144,
A327145,
A327147,
A327236.
A327144
Spanning edge-connectivity of the set-system with BII-number n.
Original entry on oeis.org
0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2
Offset: 0
Positions of first appearances of each integer together with the corresponding set-systems:
0: {}
1: {{1}}
52: {{1,2},{1,3},{2,3}}
116: {{1,2},{1,3},{2,3},{1,2,3}}
3952: {{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4}}
8052: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4}}
The same for cut-connectivity is
A326786.
The same for non-spanning edge-connectivity is
A326787.
The same for vertex-connectivity is
A327051.
Positions of first appearance of each integer are
A327147.
Cf.
A000120,
A048793,
A070939,
A322338,
A323818,
A326031,
A327041,
A327069,
A327076,
A327130,
A327145.
-
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[spanEdgeConn[Union@@bpe/@bpe[n],bpe/@bpe[n]],{n,0,100}]
A327102
BII-numbers of set-systems with non-spanning edge-connectivity >= 2.
Original entry on oeis.org
5, 6, 17, 20, 21, 24, 34, 36, 38, 40, 48, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 72, 80, 81, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121
Offset: 1
The sequence of all set-systems with non-spanning edge-connectivity >= 2 together with their BII-numbers begins:
5: {{1},{1,2}}
6: {{2},{1,2}}
17: {{1},{1,3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
24: {{3},{1,3}}
34: {{2},{2,3}}
36: {{1,2},{2,3}}
38: {{2},{1,2},{2,3}}
40: {{3},{2,3}}
48: {{1,3},{2,3}}
52: {{1,2},{1,3},{2,3}}
53: {{1},{1,2},{1,3},{2,3}}
54: {{2},{1,2},{1,3},{2,3}}
55: {{1},{2},{1,2},{1,3},{2,3}}
56: {{3},{1,3},{2,3}}
60: {{1,2},{3},{1,3},{2,3}}
61: {{1},{1,2},{3},{1,3},{2,3}}
62: {{2},{1,2},{3},{1,3},{2,3}}
63: {{1},{2},{1,2},{3},{1,3},{2,3}}
Graphs with spanning edge-connectivity >= 2 are counted by
A095983.
Graphs with non-spanning edge-connectivity >= 2 are counted by
A322395.
Also positions of terms >=2 in
A326787.
BII-numbers for non-spanning edge-connectivity 2 are
A327097.
BII-numbers for non-spanning edge-connectivity 1 are
A327099.
BII-numbers for spanning edge-connectivity >= 2 are
A327109.
Cf.
A000120,
A048793,
A059166,
A070939,
A263296,
A326031,
A326749,
A327076,
A327101,
A327102,
A327108,
A327148.
-
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
Select[Range[0,100],edgeConn[bpe[#]]>=2&]
A327201
Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs covering n vertices with non-spanning edge-connectivity k.
Original entry on oeis.org
1, 0, 1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 3, 7, 5, 4, 1, 1
Offset: 0
Triangle begins:
1
{}
0 1
0 0 1 1
1 1 2 2 1
2 3 7 5 4 1 1
Spanning edge-connectivity is
A263296.
The non-covering version is
A327236 (partial sums).
Cf.
A000088,
A322338,
A322396,
A326787,
A327076,
A327077,
A327079,
A327126,
A327129,
A327148,
A327235.
A327200
Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.
Original entry on oeis.org
0, 0, 0, 4, 42, 718, 26262, 1878422, 256204460, 67525498676, 34969833809892, 35954978661632864, 73737437034063350534, 302166248212488958298674, 2475711390267267917290354410, 40563960064630744031043287569378, 1329219366981359393514586291328267704
Offset: 0
Row sums of
A327148 if the first two columns are removed.
BII-numbers of set-systems with non-spanning edge-connectivity >= 2 are
A327102.
Graphs with non-spanning edge-connectivity 1 are
A327231.
Cf.
A001187,
A006129,
A095983,
A182100,
A322395,
A326787,
A327076,
A327079,
A327097,
A327099,
A327236.
-
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]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],eConn[#]>=2&]],{n,0,5}]
A329552
Smallest MM-number of a connected set of n sets.
Original entry on oeis.org
1, 2, 39, 195, 5655, 62205, 2674815
Offset: 0
The sequence of terms together with their corresponding systems begins:
1: {}
2: {{}}
39: {{1},{1,2}}
195: {{1},{2},{1,2}}
5655: {{1},{2},{1,2},{1,3}}
62205: {{1},{2},{3},{1,2},{1,3}}
2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}
MM-numbers of connected set-systems are
A328514.
The weight of the system with MM-number n is
A302242(n).
Maximum connected divisor is
A327076.
BII-numbers of connected sets of sets are
A326749.
The smallest BII-number of a connected set of n sets is
A329625(n).
Allowing edges to have repeated vertices gives
A329553.
Requiring the edges to form an antichain gives
A329555.
The smallest MM-number of a set of n nonempty sets is
A329557(n).
Cf.
A048143,
A056239,
A112798,
A302494,
A304714,
A304716,
A305079,
A322389,
A328513,
A329554,
A329556,
A329558.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
da=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[da[[Position[PrimeOmega/@da,n][[1,1]]]],{n,First[Split[Union[PrimeOmega/@da],#2==#1+1&]]}]
A329555
Smallest MM-number of a clutter (connected antichain) of n distinct sets.
Original entry on oeis.org
1, 2, 377, 16211, 761917
Offset: 0
The sequence of terms together with their corresponding systems begins:
1: {}
2: {{}}
377: {{1,2},{1,3}}
16211: {{1,2},{1,3},{1,4}}
761917: {{1,2},{1,3},{1,4},{2,3}}
Spanning cutters of distinct sets are counted by
A048143.
MM-numbers of connected weak-antichains are
A329559.
MM-numbers of sets of sets are
A302494.
The smallest BII-number of a clutter with n edges is
A329627.
Not requiring the edges to form an antichain gives
A329552.
Cf.
A056239,
A112798,
A302242,
A319837,
A320275,
A322113,
A327076,
A328514,
A329552,
A329558,
A329560,
A329561.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&&stableQ[primeMS[#],Divisible]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]
A327517
Number of factorizations of n that are empty or have at least two factors, all of which are > 1 and pairwise coprime.
Original entry on oeis.org
1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 4, 0, 1, 1, 0, 1, 4, 0, 1, 1, 4, 0, 1, 0, 1, 1, 1, 1, 4, 0, 1, 0, 1, 0, 4, 1, 1, 1
Offset: 1
See link for additional cross-references.
-
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],#=={}||CoprimeQ@@#&]],{n,100}]
A327199
Number of labeled simple graphs with n vertices whose edge-set is not connected.
Original entry on oeis.org
1, 1, 1, 1, 4, 56, 1031, 27189, 1165424, 89723096, 13371146135, 3989665389689, 2388718032951812, 2852540291841718752, 6768426738881535155247, 31870401029679493862010949, 297787425565749788134314214272
Offset: 0
The a(4) = 4 edge-sets: {}, {12,34}, {13,24}, {14,23}.
Cf.
A001187,
A006129,
A322395,
A326787,
A327075,
A327076,
A327079,
A327129,
A327200,
A327201,
A327231,
A327236.
-
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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]!=1&]],{n,0,5}]
A328513
Connected squarefree numbers.
Original entry on oeis.org
1, 2, 3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 173, 179, 181, 183, 185, 191, 193, 195
Offset: 1
The sequence of all connected sets of multisets together with their MM-numbers (A302242) begins:
1: {}
2: {{}}
3: {{1}}
5: {{2}}
7: {{1,1}}
11: {{3}}
13: {{1,2}}
17: {{4}}
19: {{1,1,1}}
21: {{1},{1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
37: {{1,1,2}}
39: {{1},{1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
53: {{1,1,1,1}}
57: {{1},{1,1,1}}
These are Heinz numbers of the partitions counted by
A304714.
The maximum connected squarefree divisor of n is
A327398(n).
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[100],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&]
Showing 1-10 of 18 results.
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