cp's OEIS Frontend

Two edges (a,b), (c,d) are nesting if a < c and b > d or a > c and b < d.

Also unsortable digraphs with vertices {1..n}, where a digraph is sortable if, when the edges are listed in lexicographic order, their targets are weakly increasing.

Also the number of semicrossing digraphs with vertices {1..n}, where two edges (a,b), (c,d) are semicrossing if a < c and b < d or a > c and b > d. For example, the a(2) = 4 semicrossing digraph edge-sets are:

{11,22}

{11,12,22}

{11,21,22}

{11,12,21,22}

These are digraphs with the property that, if the edges are listed in lexicographic order, the sequence of targets is weakly increasing. For example, the digraph with lexicographically ordered edge set {(1,2),(2,1),(3,1),(3,2)} is nesting because the targets are (2,1,1,2), a sequence that is not weakly increasing.

Also the number of non-semicrossing digraphs with vertices {1..n}, where two edges (a,b), (c,d) are semicrossing if a < c and b < d or a > c and b > d. For example, the a(2) = 4 non-semicrossing digraph edge-sets are:

{}

{11}

{12}

{21}

{22}

{11,12}

{11,21}

{12,21}

{12,22}

{21,22}

{11,12,21}

{12,21,22}

Apparently a duplicate of A152254. - R. J. Mathar, Jul 12 2019

Includes disconnected graphs. - Felix A. Pahl, Jul 15 2018

Loops and parallel edges are not allowed, but 2-cycles (in other words, edges A --> B and B --> A existing simultaneously) are allowed. - Mikhail Lavrov, Mar 04 2025

A directed edge (a,b) is increasing if a < b. Two edges (a,b), (c,d) are crossing if a < c < b < d or c < a < d < b.

Eulerian means that for every vertex the in-degree equals the out-degree.

A directed edge (a,b) is increasing if a < b. Two edges (a,b), (c,d) are crossing if a < c < b < d or c < a < d < b.

Conjecture: Also the number of non-nesting digraphs with vertices {1..n} whose increasing edges are not crossing, where two edges (a,b), (c,d) are nesting if a < c < d < b or c < a < b < d.

Since separability is not invariant under graph isomorphism, all 2^(n(2n-1))-1 nonzero Laplacian matrices are treated as different. A nonzero Laplacian matrix different from the complete graph is separable in C^2 X C^n if and only if its complement is. Since the complete graph is separable, this means that a(n) is odd for all n.

Since entanglement is not invariant under graph isomorphism, all 2^(n(2n-1))-1 nonzero Laplacian matrices are treated as different. A nonzero Laplacian matrix not equal to the complete graph is entangled in C^2 x C^n if and only if its complement is. Since the complete graph is not entangled, this means that a(n) is even for all n.