DoubleStepVisiblePoints on DyckPath
Points below the path, at the parity of the path's height there, from which the north-east diagonal meets a double fall and the north-west diagonal meets a double rise.
A point (v, j) has j = h - 2m under the path's vertex v of height h. Its north-east diagonal meets the path at a double fall (v, h') with v - j = v' - h'; its north-west diagonal at a double rise with v + j = v' + h'.
Called as CombinatorialStat(x, "DoubleStepVisiblePoints"). The head's page has its examples.
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Distribution
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FindStat St000683
The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.
| Object | Value |
|---|---|
[1,0,1,0] | 0 |
[1,1,0,0] | 1 |
[1,0,1,0,1,0] | 0 |
[1,0,1,1,0,0] | 1 |
[1,1,0,0,1,0] | 1 |
[1,1,0,1,0,0] | 0 |
[1,1,1,0,0,0] | 3 |
[1,0,1,0,1,0,1,0] | 0 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000683 has the rest.