DepthIndex on SetPartition
The sum of n - i over the arcs i, less the depth of each element, plus the depth of each arc.
An element's depth is the arcs strictly over it; an arc's, the arcs strictly over it.
Called as CombinatorialStat(x, "DepthIndex"). 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.
FindStat St001094
The depth index of a set partition.
| Object | Value |
|---|---|
{{1}} | 0 |
{{1,2}} | 1 |
{{1,2,3}} | 3 |
{{1},{2}} | 0 |
{{1,2,3,4}} | 6 |
{{1,2},{3}} | 2 |
{{1,3},{2}} | 1 |
{{1},{2,3}} | 2 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001094 has the rest.