Nestings on SetPartition
Pairs of arcs a < b < c < d with a~d and b~c.
The complementary case to Crossings: one arc's span strictly contains the other's. Equidistributed with Crossings over set partitions of [n] (Kasraoui–Zeng), and the noncrossing and nonnesting partitions are each counted by the Catalan numbers.
Called as CombinatorialStat(x, "Nestings"). The head's page has its examples.
Live
Distribution
Over SetPartitions at each size, tallied here in your browser; sizes past 10 are not offered.
Also on
The same name on other carriers, each its own implementation: Permutation.
FindStat St000233
The number of nestings of a set partition.
| Object | Value |
|---|---|
{{1}} | 0 |
{{1,2}} | 0 |
{{1,2,3}} | 0 |
{{1},{2}} | 0 |
{{1,2,3,4}} | 0 |
{{1,2},{3}} | 0 |
{{1,3},{2}} | 0 |
{{1},{2,3}} | 0 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000233 has the rest.