Crossings on SetPartition
Pairs of arcs a < b < c < d with a~c and b~d.
Read off the standard arc representation: within each block, consecutive elements are linked, and a crossing is two arcs whose spans interleave rather than nest or sit apart.
Called as CombinatorialStat(x, "Crossings"). 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 St000232
The number of crossings 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/St000232 has the rest.