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Also the number of set-systems with n vertices, -element sets of finite nonempty subsets of {1..n edges, }, including a unique singleton, and only such that there is exactly one way to choose a different vertex element from each edge. For example, the a(0) = 0 through a(3) = 15 set-systems are:

a(n)/n = (-1)^(n-1) * n * A134531(n). - Gus Wiseman, Jan 02 2024

For a fixed sink we have A134531 (up to sign), column k=1 of A368602.

A058891 counts set-systems, unlabeled A000612.

A059201 counts covering T_0 set-systems.

A323818 counts covering connected set-systems, unlabeled A323819.

A361718 counts acyclic digraphs by number of sinks, fixed A368602.

Cf. A000169, ~A003465, A058891, A059201, A082402, `A088957, ~A133686, A323818, A334282, `A367862, A367904, `A367908, A368600, A368601.

From Gus Wiseman, Jan 02 2024: (Start)

Also the number of set-systems with n vertices, n edges, a unique singleton, and only one way to choose a different vertex from each edge. For example, the a(0) = 0 through a(3) = 15 set-systems are:

. {{1}} {{1},{1,2}} {{1},{1,2},{1,3}}

{{2},{1,2}} {{1},{1,2},{2,3}}

{{1},{1,3},{2,3}}

{{2},{1,2},{1,3}}

{{2},{1,2},{2,3}}

{{2},{1,3},{2,3}}

{{3},{1,2},{1,3}}

{{3},{1,2},{2,3}}

{{3},{1,3},{2,3}}

{{1},{1,2},{1,2,3}}

{{1},{1,3},{1,2,3}}

{{2},{1,2},{1,2,3}}

{{2},{2,3},{1,2,3}}

{{3},{1,3},{1,2,3}}

{{3},{2,3},{1,2,3}}

These set-systems are all connected.

The case of labeled graphs is A000169.

(End)

a(n)/n = (-1)^(n-1) * A134531(n). - Gus Wiseman, Jan 02 2024

Table[Length[Select[Subsets[Subsets[Range[n]], {n}], Count[#, {_}]==1&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]], {n, 0, 4}] (* Gus Wiseman, Jan 02 2024 *)

For any number of sinks we have A003024, unlabeled A003087.

For n-1 sinks we have A058877.

For a fixed sink we have A134531 (up to sign).

A058891 counts set-systems, unlabeled A000612.

A059201 counts covering T_0 set-systems.

A323818 counts covering connected set-systems, unlabeled A323819.

A361718 counts acyclic digraphs by number of sinks, fixed A368602.

Cf. A000169, ~A003465, A082402, `A088957, ~A133686, A334282, `A367862, A367904, `A367908, A368600, A368601.

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Column k=1 of A361718.

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