A327106 -id:A327106 - 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).

A327104 Maximum vertex-degree of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 4, 3

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) = 3.

Positions of 1's are A326701 (BII-numbers of set-partitions).
The minimum vertex-degree is A327103.
Positions of 2's are A327106.
Cf. A000120, A048793, A058891, A070939, A326031, A326701, A326786, A327041.

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

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

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