NorthStepOrderingCount on DyckPath
The total orderings of the up steps such that the up steps after the k-th down step are never among the first k in the order.
The coefficient of z_1^a_1 ... z_n^a_n in the product of (z_1 + ... + z_k), by Stanley's product formula: each factor is the height before a down step over the factorial of the up steps since the last one.
Called as CombinatorialStat(x, "NorthStepOrderingCount"). The head's page has its examples.
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Distribution
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FindStat St001786
The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order.
| Object | Value |
|---|---|
[1,0] | 1 |
[1,0,1,0] | 1 |
[1,1,0,0] | 1 |
[1,0,1,0,1,0] | 1 |
[1,0,1,1,0,0] | 1 |
[1,1,0,0,1,0] | 1 |
[1,1,0,1,0,0] | 2 |
[1,1,1,0,0,0] | 1 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001786 has the rest.