cp's OEIS Frontend

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d. A graph with positive integer vertices is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.

A set partition is totally crossing if every pair of distinct blocks is of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y.

A multiset is normal if its union is an initial interval of positive integers.

A multiset partition is crossing if it has a 2-element submultiset of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected, where two edges cross each other if they are of the form {{x,y},{z,t}} with x < z < y < t or z < x < t < y.

Covering means there are no isolated vertices.

A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected, where two edges cross each other if they are of the form {{x,y},{z,t}} with x < z < y < t or z < x < t < y.

An antichain is non-crossing if no pair of distinct parts is of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

For the definition of a purely crossing partition refer to Dykema link (see PC(n) Definition 1.2 and Table 2).

From Gus Wiseman, Feb 23 2019: (Start)

For n >= 1, a set partition of {1,...,n} is purely crossing if it is topologically connected (A099947), has no successive elements in the same block (A000110(n - 1)), and the first and last vertices belong to different blocks (A005493(n - 2)). For example, the a(4) = 1, a(6) = 5, and a(7) = 14 purely crossing set partitions are:

{{13}{24}} {{135}{246}} {{13}{246}{57}}

{{13}{25}{46}} {{13}{257}{46}}

{{14}{25}{36}} {{135}{26}{47}}

{{14}{26}{35}} {{135}{27}{46}}

{{15}{24}{36}} {{136}{24}{57}}

{{136}{25}{47}}

{{14}{257}{36}}

{{14}{26}{357}}

{{146}{25}{37}}

{{146}{27}{35}}

{{15}{246}{37}}

{{15}{247}{36}}

{{16}{24}{357}}

{{16}{247}{35}}

(End)

If all subsets are allowed instead of just pairs (chords), we get A324173. The rightmost column is A000108 (see Riordan). - Gus Wiseman, Feb 27 2019

A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y.

Two blocks are nesting if they are of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. A set partition has its nesting blocks connected if the graph whose vertices are the blocks and whose edges are nesting pairs of blocks is connected.