FirstComponentSemilength on DyckPath
The position of the first return to height 0, counted in up steps: half the steps of the first primitive component.
FindStat's "position of the first return" counts up steps, not steps: UD is 1.
Called as CombinatorialStat(x, "FirstComponentSemilength"). The head's page has its examples.
Live
Distribution
Over DyckPaths at each size, tallied here in your browser; sizes past 10 are not offered.
FindStat St000026
The position of the first return of a Dyck path.
| Object | Value |
|---|---|
[1,0] | 1 |
[1,0,1,0] | 1 |
[1,1,0,0] | 2 |
[1,0,1,0,1,0] | 1 |
[1,0,1,1,0,0] | 1 |
[1,1,0,0,1,0] | 2 |
[1,1,0,1,0,0] | 3 |
[1,1,1,0,0,0] | 3 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000026 has the rest.