A327107 -id:A327107 - cp's OEIS frontend

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

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}]

A100743 Number of labeled n-vertex graphs without vertices of degree <=1.

Original entry on oeis.org

1, 0, 0, 1, 10, 253, 12068, 1052793, 169505868, 51046350021, 29184353055900, 32122563615242615, 68867440268165982320, 290155715157676330952559, 2417761590648159731258579164, 40013923822242935823157820555477, 1318910080336893719646370269435043184

Offset: 0

Goran Kilibarda, Zoran Maksimovic, Vladeta Jovovic, Jan 03 2005

nonn

			From _Gus Wiseman_, Aug 18 2019: (Start)
The a(4) = 10 edge-sets:
  {12,13,24,34}
  {12,14,23,34}
  {13,14,23,24}
  {12,13,14,23,24}
  {12,13,14,23,34}
  {12,13,14,24,34}
  {12,13,23,24,34}
  {12,14,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
(End)

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

Graphs without isolated nodes are A006129.
The connected case is A059166.
Graphs without endpoints are A059167.
Labeled graphs with endpoints are A245797.
The unlabeled version is A261919.
Cf. A095983, A136284, A322395, A327079, A327107, A327227, A327229, A327230.
Cf. A095983, A322395.

m = 13;
egf = Exp[-x + x^2/2]*Sum[2^(n (n-1)/2)*(x/Exp[x])^n/n!, {n, 0, m+1}];
s = egf + O[x]^(m+1);
a[n_] := n!*SeriesCoefficient[s, n];
Table[a[n], {n, 0, m}] (* Jean-François Alcover, Feb 23 2019 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}] (* Gus Wiseman, Aug 18 2019 *)

seq(n)={Vec(serlaplace(exp(-x + x^2/2 + O(x*x^n))*sum(k=0, n, 2^(k*(k-1)/2)*(x/exp(x + O(x^n)))^k/k!)))} \\ Andrew Howroyd, Sep 04 2019

E.g.f.: exp(-x+x^2/2)*(Sum_{n>=0} 2^(n*(n-1)/2)*(x/exp(x))^n/n!). - Vladeta Jovovic, Jan 26 2006
Exponential transform of A059166. - Gus Wiseman, Aug 18 2019
Inverse binomial transform of A059167. - Gus Wiseman, Sep 02 2019

Terms a(14) and beyond from Andrew Howroyd, Sep 04 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&]

A327228 Number of set-systems with n vertices and at least one endpoint/leaf.

Original entry on oeis.org

0, 1, 6, 65, 3297, 2537672, 412184904221, 4132070624893905681577, 174224571863520492218909428465944685216436, 133392486801388257127953774730008469745829658368044283629394202488602260177922751

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

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 set-systems with minimum covered vertex-degree 1.

			The a(2) = 6 set-systems:
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}

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

The covering version is A327229.
The specialization to simple graphs is A245797.
BII-numbers of these set-systems are A327105.
Cf. A058891, A059167, A327098, A327103, A327104, A327107, A327197, A327227, A327230, A330059.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,4}]

Binomial transform of A327229.

a(n) = A058891(n+1) - A330059(n). - Andrew Howroyd, Jan 21 2023

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

cp's OEIS Frontend

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

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A100743 Number of labeled n-vertex graphs without 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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A327228 Number of set-systems with n vertices and at least one endpoint/leaf.

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