Andrew Howroyd, <a href="/A368836/b368836_1.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

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Yes. When k = 1 there is one loop. Remove the vertex with the loop and add loops to its neighbors. This process is reversible so there is a bijection. - Andrew Howroyd, Jan 13 2024

1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 2, 6, 6, 2, 1, 6, 17, 18, 8, 2, 1, 21, 52, 58, 30, 9, 2, 1, 65, 173, 191, 107, 37, 9, 2, 1, 221, 585, 666, 393, 148, 39, 9, 2, 1, 771, 2064, 2383, 1493, 589, 168, 40, 9, 2, 1, 2769, 7520, 8847, 5765, 2418, 718, 176, 40, 9, 2, 1

Andrew Howroyd, <a href="/A368836/b368836_1.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

(PARI)

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}

row(n) = {my(s=0, A=1+O(x*x^n)); forpart(p=n, s+=permcount(p) * polcoef(edges(p, i->A + x^i)*prod(i=1, #p, A + (x*y)^p[i]), n)); Vecrev(s/n!)} \\ Andrew Howroyd, Jan 13 2024

nonn,tabl,more,new

a(28) onwards from Andrew Howroyd, Jan 13 2024

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Column k = 1 appears to be A368598 shifted left.

A000085 counts , A100861, A111924 count set partitions into singletons or pairs.

A058891 counts set-systems (without singletons A016031), , unlabeled A000612.

A100861 counts set partitions into singletons or pairs by number of pairs.

A111924 counts set partitions into singletons or pairs by length.

A322661 counts labeled covering half-loop-graphs, connected A062740.

Cf. `A000272, A007717, A062740, A322661, A333331, A368596, `A368600, `A368601, A368730, A368835, `A368927.

allocated for Gus WisemanTriangle read by rows where T(n,k) is the number of unlabeled loop-graphs on up to n vertices with k loops and n-k non-loops.

1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 2, 6, 6, 2, 1, 6, 17, 18, 8, 2, 1, 21, 52, 58, 30, 9, 2, 1

0,8

Are the row sums the same as column k = 1 (shifted left)?

Triangle begins:

1

0 1

0 1 1

1 2 2 1

2 6 6 2 1

6 17 18 8 2 1

21 52 58 30 9 2 1

Representatives of the loop-graphs counted by row n = 4:

{12}{13}{14}{23} {1}{12}{13}{14} {1}{2}{12}{13} {1}{2}{3}{12} {1}{2}{3}{4}

{12}{13}{24}{34} {1}{12}{13}{23} {1}{2}{12}{34} {1}{2}{3}{14}

{1}{12}{13}{24} {1}{2}{13}{14}

{1}{12}{23}{24} {1}{2}{13}{23}

{1}{12}{23}{34} {1}{2}{13}{24}

{1}{23}{24}{34} {1}{2}{13}{34}

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], {1, 2}], {n}], Count[#, {_}]==k&]]], {n, 0, 4}, {k, 0, n}]

Column k = 0 is A001434.

Column k = 1 appears to be A368598 shifted left.

Row sums are A368598.

The labeled version is A368928.

A000085 counts set partitions into singletons or pairs.

A006125 counts graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

A014068 counts loop-graphs, unlabeled A000666.

A058891 counts set-systems (without singletons A016031), unlabeled A000612.

A100861 counts set partitions into singletons or pairs by number of pairs.

A111924 counts set partitions into singletons or pairs by length.

A322661 counts labeled covering half-loop-graphs, connected A062740.

Cf. `A000272, A007717, A368596, `A368600, `A368601, A368730, A368835, `A368927.

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Gus Wiseman, Jan 11 2024

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