A327104 -id:A327104 - cp's OEIS frontend

A094574 Number of (<=2)-covers of an n-set.

1, 1, 5, 40, 457, 6995, 136771, 3299218, 95668354, 3268445951, 129468914524, 5868774803537, 301122189141524, 17327463910351045, 1109375488487304027, 78484513540137938209, 6098627708074641312182, 517736625823888411991202, 47791900951140948275632148

Offset: 0

Goran Kilibarda, Vladeta Jovovic, May 12 2004

nonn

Also the number of strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. For example, the a(2) = 5 strict multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (11)(22), (1)(2)(12). - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Sep 02 2019: (Start)
These are set-systems covering {1..n} with vertex-degrees <= 2. For example, the a(3) = 40 covers are:
  {123}  {1}{23}    {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}    {1}{2}{13}    {1}{2}{3}{13}
         {3}{12}    {1}{2}{23}    {1}{2}{3}{23}
         {1}{123}   {1}{3}{12}    {1}{2}{13}{23}
         {12}{13}   {1}{3}{23}    {1}{2}{3}{123}
         {12}{23}   {2}{3}{12}    {1}{3}{12}{23}
         {13}{23}   {2}{3}{13}    {2}{3}{12}{13}
         {2}{123}   {1}{12}{23}
         {3}{123}   {1}{13}{23}
         {12}{123}  {1}{2}{123}
         {13}{123}  {1}{3}{123}
         {23}{123}  {2}{12}{13}
                    {2}{13}{23}
                    {2}{3}{123}
                    {3}{12}{13}
                    {3}{12}{23}
                    {12}{13}{23}
                    {1}{23}{123}
                    {2}{13}{123}
                    {3}{12}{123}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row n=2 of A219585. - Alois P. Heinz, Nov 23 2012
Dominated by A003465.
Graphs with vertex-degrees <= 2 are A136281.
Cf. A002718, A007716, A020554, A020555, A050535, A094574, A136284, A316974, A327104, A327106, A327229.
Main diagonal of A346517.

facs[n_]:=facs[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[Array[Prime,n,1,Times]^2],UnsameQ@@#&]],{n,0,6}] (* Gus Wiseman, Jul 18 2018 *)
m = 20;
a094577[n_] := Sum[Binomial[n, k]*BellB[2 n - k], {k, 0, n}];
egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
CoefficientList[egf + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, May 13 2019 *)

Row sums of A094573.

E.g.f: exp(-1-1/2*(exp(x)-1))*Sum(exp(x*binomial(n+1, 2))/n!, n=0..infinity) or exp((1-exp(x))/2)*Sum(A094577 (n)*(x/2)^n/n!, n=0..infinity).

A327108 BII-numbers of set-systems with spanning edge-connectivity 2.

Original entry on oeis.org

52, 53, 54, 55, 60, 61, 62, 63, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113, 114, 115, 120, 121, 122, 123, 772, 773, 774, 775, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 848, 849, 850

Offset: 1

Gus Wiseman, Aug 23 2019

nonn

Differs from A327109 in lacking 116, 117, 118, 119, 124, 125, 126, 127, ...
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.
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 disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-connectivity 2 together with their BII-numbers begins:
   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}}
   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}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   86: {{2},{1,2},{1,3},{1,2,3}}
   87: {{1},{2},{1,2},{1,3},{1,2,3}}
   92: {{1,2},{3},{1,3},{1,2,3}}
   93: {{1},{1,2},{3},{1,3},{1,2,3}}
   94: {{2},{1,2},{3},{1,3},{1,2,3}}
   95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  101: {{1},{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}
  103: {{1},{2},{1,2},{2,3},{1,2,3}}

Positions of 2's in A327144.
Graphs with spanning edge-connectivity >= 2 are counted by A095983.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A326749, A326787, A327041, A327069, A327075, A327102, A327103, A327104.

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&];
Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==2&]

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}]

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

A327107 BII-numbers of set-systems with minimum vertex-degree > 1.

Original entry on oeis.org

7, 25, 30, 31, 42, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 99, 100, 101, 102, 103, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

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 maximum degree > 1 together with their BII-numbers begins:
   7: {{1},{2},{1,2}}
  25: {{1},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  42: {{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  51: {{1},{2},{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}}
  59: {{1},{2},{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}}
  75: {{1},{2},{3},{1,2,3}}
  76: {{1,2},{3},{1,2,3}}

Positions of terms > 1 in A327103.
BII-numbers for minimum degree 1 are A327105.
Graphs with minimum degree > 1 are counted by A059167.
Cf. A000120, A029931, A048793, A058891, A070939, A245797, A326031, A326701, A326786, A327041, A327104, A327227-A327230.

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

A327106 BII-numbers of set-systems with maximum degree 2.

Original entry on oeis.org

5, 6, 7, 13, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27, 28, 30, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 65, 66, 67, 68, 72, 73, 74, 75, 76, 80, 82, 96, 97, 133, 134, 135, 141, 142, 143, 145, 147, 148, 150, 152, 153, 154, 155, 156, 158, 162

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 maximum degree 2 together with their BII-numbers begins:
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  26: {{2},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}

Positions of 2's in A327104.
Graphs with maximum degree 2 are counted by A136284.
Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326786, A327041, A327103.

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

cp's OEIS Frontend

A094574 Number of (<=2)-covers of an n-set.

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A327108 BII-numbers of set-systems with spanning edge-connectivity 2.

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

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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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A327107 BII-numbers of set-systems with minimum vertex-degree > 1.

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A327106 BII-numbers of set-systems with maximum degree 2.

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