Andrew Howroyd, <a href="/A370167/b370167_1.txt">Table of n, a(n) for n = 0..1350</a> (rows 0..20)

reviewed

approved

proposed

reviewed

editing

proposed

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 0, 1, 4, 5, 5, 4, 2, 1, 1, 0, 0, 0, 1, 3, 9, 15, 20, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 1, 6, 20, 41, 73, 110, 133, 139, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 1, 3, 15, 50, 124, 271, 515, 832, 1181, 1460, 1581, 1516, 1291, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1

Andrew Howroyd, <a href="/A370167/b370167_1.txt">Table of n, a(n) for n = 0..1350</a> (rows 0..20)

(PARI) \\ G(n) defined in A008406.

row(n)={Vecrev(G(n)-if(n>0, G(n-1)), binomial(n, 2)+1)}

{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

nonn,tabf,more,new

a(42) onwards from Andrew Howroyd, Feb 19 2024

approved

editing

proposed

approved

editing

proposed

- or -

Triangle begins:

1

0

0 1

0 0 1 1

0 0 1 2 2 1 1

0 0 0 1 4 5 5 4 2 1 1

Cf. A000666, A006125, A006649, `A054547, A062740, A066383, A121252, A133686, A322700, A368597, A369196, A369197.

Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

allocated for Gus WisemanIrregular triangle read by rows where T(n,k) is the number of unlabeled graphs covering n vertices with k = 0..binomial(n,2) edges.

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 0, 1, 4, 5, 5, 4, 2, 1, 1, 0, 0, 0, 1, 3, 9, 15, 20, 22, 20, 14, 9, 5, 2, 1, 1

0,12

Triangle begins:

1

0

0 1

0 0 1 1

0 0 1 2 2 1 1

0 0 0 1 4 5 5 4 2 1 1

0 0 0 1 3 9 15 20 22 20 14 9 5 2 1 1

- or -

Triangle begins:

1

0

0 1

0 0 1 1

0 0 1 2 2 1 1

0 0 0 1 4 5 5 4 2 1 1

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]], p[[i]]}, {i, Length[p]}])], {p, Permutations[Range[Length[Union@@m]]]}]]];

Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n], {2}], {k}], Union@@#==Range[n]&]]], {n, 0, 5}, {k, 0, Binomial[n, 2]}]

Column sums are A000664.

Row sums are A002494.

This is the covering case of A008406, labeled A084546.

The labeled version is A054548, row sums A006129, column sums A121251.

The connected case is A054924, row sums A001349, column sums A002905.

The labeled connected case is A062734, with loops A369195.

The connected case with loops is A283755, row sums A054921.

The labeled version w/ loops is A369199, row sums A322661, col sums A173219.

Cf. A000666, A006125, A006649, `A054547, A062740, A066383, A121252, A133686, A322700, A368597, A369196, A369197.

allocated

nonn,tabf,more

Gus Wiseman, Feb 15 2024

approved

editing