A327228 -id:A327228 - cp's OEIS frontend

A245797 The number of labeled graphs of n vertices that have endpoints, where an endpoint is a vertex with degree 1.

0, 1, 6, 49, 710, 19011, 954184, 90154415, 16108626420, 5481798833245, 3582369649269620, 4532127781040045649, 11177949079089720090800, 54050029251399545975868271, 514598463471970554205910304780, 9677402372862708729859372687791391

Offset: 1

Chai Wah Wu, Aug 01 2014

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50

Equal to row sums of A245796.
The covering case is A327227.
The connected case is A327362.
The generalization to set-systems is A327228.
BII-numbers of set-systems with minimum degree 1 are A327105.
Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327229, A327230.

m = 16;
egf = Exp[x^2/2]*Sum[2^Binomial[n, 2]*(x/Exp[x])^n/n!, {n, 0, m}];
A059167[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!;
a[n_] := 2^(n(n-1)/2) - A059167[n];
Array[a, m] (* Jean-François Alcover, Feb 23 2019 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}] (* Gus Wiseman, Sep 11 2019 *)

a(n) = 2^(n*(n+1)/2) - A059167(n).

Binomial transform of A327227 (assuming a(0) = 0).

a(9)-a(16) from Andrew Howroyd, Oct 26 2017

A327227 Number of labeled simple graphs covering n vertices with at least one endpoint/leaf.

Original entry on oeis.org

0, 0, 1, 3, 31, 515, 15381, 834491, 83016613, 15330074139, 5324658838645, 3522941267488973, 4489497643961740521, 11119309286377621015089, 53893949089393110881259181, 513788884660608277842596504415, 9669175277199248753133328740702449

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

Covering means there are no isolated vertices.
A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.
Also graphs with minimum vertex-degree 1.

			The a(4) = 31 edge-sets:
  {12,34}  {12,13,14}  {12,13,14,23}
  {13,24}  {12,13,24}  {12,13,14,24}
  {14,23}  {12,13,34}  {12,13,14,34}
           {12,14,23}  {12,13,23,24}
           {12,14,34}  {12,13,23,34}
           {12,23,24}  {12,14,23,24}
           {12,23,34}  {12,14,24,34}
           {12,24,34}  {12,23,24,34}
           {13,14,23}  {13,14,23,34}
           {13,14,24}  {13,14,24,34}
           {13,23,24}  {13,23,24,34}
           {13,23,34}  {14,23,24,34}
           {13,24,34}
           {14,23,24}
           {14,23,34}
           {14,24,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k=1 of A327366.
The non-covering version is A245797.
The unlabeled version is A324693.
The generalization to set-systems is A327229.
BII-numbers of set-systems with minimum degree 1 are A327105.
Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327228, A327230.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

Inverse binomial transform of A245797, if we assume A245797(0) = 0.

A327103 Minimum vertex-degree in the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2

Offset: 0

Gus Wiseman, Aug 26 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

In a set-system, the degree of a vertex is the number of edges containing it.

			The BII-number of {{2},{3},{1,2},{1,3},{2,3}} is 62, and its degrees are (2,3,3), so a(62) = 2.

The maximum vertex-degree is A327104.
Positions of 1's are A327105.
Positions of terms > 1 are A327107.
Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326783, A326786, A327041, A327228, A327229.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[If[n==0,0,Min@@Length/@Split[Sort[Join@@bpe/@bpe[n]]]],{n,0,100}]

A327369 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 0, 6, 0, 15, 12, 30, 4, 3, 314, 320, 260, 80, 50, 0, 13757, 10890, 5445, 1860, 735, 66, 15, 1142968, 640836, 228564, 64680, 16800, 2772, 532, 0, 178281041, 68362504, 17288852, 3666600, 702030, 115416, 17892, 1016, 105

Offset: 0

Gus Wiseman, Sep 04 2019

			Triangle begins:
      1
      1     0
      1     0     1
      2     0     6     0
     15    12    30     4     3
    314   320   260    80    50     0
  13757 10890  5445  1860   735    66    15

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Row sums are A006125.
Row sums without the first column are A245797.
Column k = 0 is A059167.
Column k = 1 is A277072.
Column k = 2 is A277073.
Column k = 3 is A277074.
Column k = n is A123023.
Column k = n - 1 is A327370.
The unlabeled version is A327371.
The covering version is A327377.
Cf. A004110, A055302, A055314, A059166, A059167, A100743, A327227, A327228, A327362, A327364.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Count[Length/@Split[Sort[Join@@#]],1]==k&]],{n,0,5},{k,0,n}]

Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.
  my(R=sum(n=1, n, x^n*sum(k=1, n, stirling(n-1, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(U=sum(n=2, n, x^n*sum(k=1, n, stirling(n-2, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
  my(A=exp(x + U + subst(B-x, x, x*(1-y) + R)));
  Vecrev(n!*polcoef(A, n), n + 1);
}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Oct 05 2019

Column-wise binomial transform of A327377.

E.g.f.: exp(x + U(x,y) + B(x*(1-y) + R(x,y))), where R(x,y) is the e.g.f. of A055302, U(x,y) is the e.g.f. of A055314 and B(x) + x is the e.g.f. of A059167. - Andrew Howroyd, Oct 05 2019

Terms a(28) and beyond from Andrew Howroyd, Sep 09 2019

A327229 Number of set-systems covering n vertices with at least one endpoint/leaf.

Original entry on oeis.org

0, 1, 4, 50, 3069, 2521782, 412169726428, 4132070622008664529903, 174224571863520492185852863478334475199686, 133392486801388257127953774730008469744261637221272599199572772174870315402893538

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

Covering means there are no isolated vertices.
A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.
Also covering set-systems with minimum vertex-degree 1.

			The a(2) = 4 set-systems:
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..12

The non-covering version is A327228.
The specialization to simple graphs is A327227.
The unlabeled version is A327230.
BII-numbers of these set-systems are A327105.
Cf. A003465, A245797, A327079, A327098, A327103, A327107, A327197.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,3}]

Inverse binomial transform of A327228.

Terms a(5) and beyond from Andrew Howroyd, Jan 21 2023

A327105 BII-numbers of set-systems with minimum degree 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 48, 49, 50, 56, 57, 58, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 88, 89, 96, 98, 104, 106, 128

Offset: 1

Gus Wiseman, Aug 26 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

In a set-system, the degree of a vertex is the number of edges containing it.

			The sequence of all set-systems with minimum degree 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  18: {{2},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}

Positions of 1's in A327103.
BII-numbers for minimum degree > 1 are A327107.
Graphs with minimum degree 1 are counted by A245797, with covering case A327227.
Set-systems with minimum degree 1 are counted by A327228, with covering case A327229.
Cf. A000120, A029931, A048793, A058891, A070939, A326031, A326701, A326786, A327041, A327104, A327230.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],If[#==0,0,Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]]==1&]

A327230 Number of non-isomorphic set-systems covering n vertices with at least one endpoint/leaf.

Original entry on oeis.org

0, 1, 3, 14, 198

Offset: 0

Gus Wiseman, Sep 01 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also covering set-systems with minimum vertex-degree 1.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 14 set-systems:
  {{1}}  {{1,2}}      {{1,2,3}}
         {{1},{2}}    {{1},{2,3}}
         {{2},{1,2}}  {{1},{2},{3}}
                      {{1,3},{2,3}}
                      {{3},{1,2,3}}
                      {{1},{3},{2,3}}
                      {{2,3},{1,2,3}}
                      {{2},{1,3},{2,3}}
                      {{2},{3},{1,2,3}}
                      {{3},{1,3},{2,3}}
                      {{1},{2},{3},{2,3}}
                      {{3},{2,3},{1,2,3}}
                      {{2},{3},{1,3},{2,3}}
                      {{2},{3},{2,3},{1,2,3}}

Unlabeled covering set-systems are A055621.
The labeled version is A327229.
The non-covering version is A327335 (partial sums).
Cf. A002494, A245797, A261919, A283877, A327103, A327105, A327197, A327227, A327228.

A327335 Number of non-isomorphic set-systems with n vertices and at least one endpoint/leaf.

Original entry on oeis.org

0, 1, 4, 18, 216

Offset: 0

Gus Wiseman, Sep 02 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also covering set-systems with minimum covered vertex-degree 1.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 18 set-systems:
  {{1}}  {{1}}        {{1}}
         {{1,2}}      {{1,2}}
         {{1},{2}}    {{1},{2}}
         {{1},{1,2}}  {{1,2,3}}
                      {{1},{1,2}}
                      {{1},{2,3}}
                      {{1},{1,2,3}}
                      {{1,2},{1,3}}
                      {{1},{2},{3}}
                      {{1,2},{1,2,3}}
                      {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1},{2},{3},{1,2}}
                      {{1},{2},{1,2},{1,3}}
                      {{1},{2},{1,2},{1,2,3}}

Unlabeled set-systems are A000612.
The labeled version is A327228.
The covering version is A327230 (first differences).
Cf. A002494, A245797, A261919, A283877, A327103, A327105, A327197, A327227, A327229, A327336.

cp's OEIS Frontend

A245797 The number of labeled graphs of n vertices that have endpoints, where an endpoint is a vertex with degree 1.

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A327227 Number of labeled simple graphs covering n vertices with at least one endpoint/leaf.

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A327103 Minimum vertex-degree in the set-system with BII-number n.

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A327369 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

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A327229 Number of set-systems covering n vertices with at least one endpoint/leaf.

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A327105 BII-numbers of set-systems with minimum degree 1.

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A327230 Number of non-isomorphic set-systems covering n vertices with at least one endpoint/leaf.

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A327335 Number of non-isomorphic set-systems with n vertices and at least one endpoint/leaf.

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