A327201 -id:A327201 - cp's OEIS frontend

A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

1, 1, 1, 1, 1, 3, 3, 1, 4, 18, 27, 14, 1, 56, 250, 402, 240, 65, 10, 1, 1031, 5475, 11277, 9620, 4282, 921, 146, 15, 1

Offset: 0

Gus Wiseman, Aug 27 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any isolated vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   1
   1   1
   1   3   3   1
   4  18  27  14   1
  56 250 402 240  65  10   1

Row sums are A006125.
Column k = 0 is A327199.
Column k = 1 is A327231.
The corresponding triangle for vertex-connectivity is A327125.
The corresponding triangle for spanning edge-connectivity is A327069.
The covering version is A327149.
The unlabeled version is A327236, with covering version A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327079, A327097, A327099, A327102, A327126, A327144, A327196, A327200, A327201.

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]]]]]]]]];
edgeConnSys[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}]],edgeConnSys[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

T(n,k) = Sum_{m = 0..n} binomial(n,m) A327149(m,k). In words, column k is the binomial transform of column k of A327149.

a(20)-a(28) from Robert Price, May 25 2021

A327075 Number of non-connected unlabeled simple graphs covering n vertices.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 10, 35, 185, 1242, 13929, 292131, 12344252, 1032326141, 166163019475, 50671385831320, 29105332577409883, 31455744378606296280, 64032559078724993894492, 245999991257359808853560276, 1787823917424909126688749033668, 24639597815428343970034635549911427

Offset: 0

Gus Wiseman, Aug 26 2019

nonn

We consider the empty graph to be neither connected (one component) nor disconnected (more than one component).

			Non-isomorphic representatives of the a(0) = 1 through a(6) = 10 graphs (empty columns not shown):
  {}  {12,34}  {12,35,45}     {12,34,56}
               {12,34,35,45}  {12,35,46,56}
                              {12,36,46,56}
                              {13,23,46,56}
                              {12,34,35,46,56}
                              {12,36,45,46,56}
                              {13,23,45,46,56}
                              {12,13,23,45,46,56}
                              {12,35,36,45,46,56}
                              {12,34,35,36,45,46,56}

Gus Wiseman, The a(4) = 1 through a(6) = 10 non-connected covering graphs.

Column k = 0 of A327201.
The labeled version is A327070.
Disconnected graphs are A000719.
Cf. A000088, A001187, A001349, A002494, A006129, A054592.

from functools import lru_cache
from itertools import combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A327075(n):
    if n <= 1: return 1-n
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return b(n)-b(n-1)-sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n # Chai Wah Wu, Jul 03 2024

a(n) = A002494(n) - A001349(n), if we assume A001349(0) = A001349(1) = 0.

a(20)-a(21) from Chai Wah Wu, Jul 03 2024

A327236 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs with n vertices whose edge-set has non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 1, 4, 5, 10, 8, 5, 1, 1

Offset: 0

Gus Wiseman, Sep 03 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a disconnected or empty graph, ignoring isolated vertices.

			Triangle begins:
  1
  1
  1  1
  1  1  1  1
  2  2  3  3  1
  4  5 10  8  5  1  1

Gus Wiseman, Unlabeled graphs with 5 vertices, organized by non-spanning edge-connectivity (isolated vertices not shown).

Row sums are A000088.
Column k = 0 is A327235.
The labeled version is A327148.
The covering version is A327201.
Spanning edge-connectivity is A263296.
Vertex-connectivity is A259862.
Cf. A322338, A322396, A326787, A327069, A327077, A327097, A327099, A327102, A327200, A327231.

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]]]]]]]]];
edgeConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Union[normclut/@Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==k&]]],{n,0,5},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 3, 12, 15, 10, 1, 40, 180, 297, 180, 60, 10, 1

Offset: 0

Gus Wiseman, Aug 27 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   {}
   0   1
   0   0   3   1
   3  12  15  10   1
  40 180 297 180  60  10   1

Row sums are A006129.
Column k = 0 is A327070.
Column k = 1 is A327079.
The corresponding triangle for vertex-connectivity is A327126.
The corresponding triangle for spanning edge-connectivity is A327069.
The non-covering version is A327148.
The unlabeled version is A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327097, A327099, A327102, A327125, A327129, A327144.

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]]]]]]]]];
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}]],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

A327148(n,k) = Sum_{m = 0..n} binomial(n,m) T(m,k). In words, column k is the inverse binomial transform of column k of A327148.

A327353 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 7, 3, 1, 53, 27, 45, 36, 6, 747, 511, 1497, 2085, 1540, 693, 316, 135, 45, 10, 1

Offset: 0

Gus Wiseman, Sep 10 2019

An antichain is a set of sets, none of which is a subset of any other.

The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

			Triangle begins:
    1
    1    1
    2    3
    8    7    3    1
   53   27   45   36    6
  747  511 1497 2085 1540  693  316  135   45   10    1
Row n = 3 counts the following antichains:
  {}             {{1}}      {{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
  {{1},{2}}      {{2}}      {{1,2},{2,3}}
  {{1},{3}}      {{3}}      {{1,3},{2,3}}
  {{2},{3}}      {{1,2}}
  {{1},{2,3}}    {{1,3}}
  {{2},{1,3}}    {{2,3}}
  {{3},{1,2}}    {{1,2,3}}
  {{1},{2},{3}}

Row sums are A014466.
Column k = 0 is A327354.
The covering case is A327357.
The version for spanning edge-connectivity is A327352.
The specialization to simple graphs is A327148, with covering case A327149, unlabeled version A327236, and unlabeled covering case A327201.
Cf. A052446, A307249, A326704, A326787, A327071, A327351, A327355.

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]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],eConn[#]==k&]],{n,0,4},{k,0,2^n}]//.{foe___,0}:>{foe}

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

Gus Wiseman, Sep 01 2019

nonn

Also graphs with non-spanning edge-connectivity 0.

			The a(4) = 4 edge-sets: {}, {12,34}, {13,24}, {14,23}.

Column k = 0 of A327148.
The covering case is A327070.
The unlabeled version is A327235.
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}]

Binomial transform of A327070.

A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 0, 1, 1, 1, 4, 1, 3, 1, 30, 13, 33, 32, 6, 546, 421, 1302, 1915, 1510, 693, 316, 135, 45, 10, 1

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

			Triangle begins:
    1
    0    1
    1    1
    4    1    3    1
   30   13   33   32    6
  546  421 1302 1915 1510  693  316  135   45   10    1
Row n = 3 counts the following antichains:
  {{1},{2,3}}    {{1,2,3}}  {{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
  {{2},{1,3}}               {{1,2},{2,3}}
  {{3},{1,2}}               {{1,3},{2,3}}
  {{1},{2},{3}}

Row sums are A307249.
Column k = 0 is A120338.
The non-covering version is A327353.
The version for spanning edge-connectivity is A327352.
The specialization to simple graphs is A327149, with unlabeled version A327201.
Cf. A014466, A293606, A326704, A326787, A327071, A327148, A327236, A327351.

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]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,5},{k,0,2^n}]//.{foe___,0}:>{foe}

A327437 Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

Original entry on oeis.org

1, 1, 3, 6, 15, 52, 410, 32697

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 antichains:
  {}  {}         {}             {}
      {{1}}      {{1}}          {{1}}
      {{1},{2}}  {{1,2}}        {{1,2}}
                 {{1},{2}}      {{1},{2}}
                 {{1},{2,3}}    {{1,2,3}}
                 {{1},{2},{3}}  {{1},{2,3}}
                                {{1,2},{1,3}}
                                {{1},{2},{3}}
                                {{1},{2,3,4}}
                                {{1,2},{3,4}}
                                {{1},{2},{3,4}}
                                {{1},{2},{3},{4}}
                                {{2},{1,3},{1,4}}
                                {{1,2},{1,3},{2,3}}
                                {{4},{1,2},{1,3},{2,3}}

Column k = 0 of A327438.
The labeled version is A327355.
The covering case is A327426.
Cf. A014466 A052446, A120338, A261005, A293606, A327201, A327236, A327352, A327353, A327354, A327438.

a(n > 0) = A306505(n) - A261006(n).

cp's OEIS Frontend

A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A327075 Number of non-connected unlabeled simple graphs covering n vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

Extensions

A327236 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs with n vertices whose edge-set has non-spanning edge-connectivity k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A327353 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327199 Number of labeled simple graphs with n vertices whose edge-set is not connected.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327437 Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula