Dinv on DyckPath
The dinv statistic, read from the area sequence.
dinv = #{i<j : a_i=a_j} + #{i<j : a_i=a_j+1}; the two counts never overlap (a_i=a_j and a_i=a_j+1 can't both hold), so this is one pass over the pairs.
Called as CombinatorialStat(x, "Dinv"). 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 St000006
The dinv of a Dyck path.
| Object | Value |
|---|---|
[1,0] | 0 |
[1,0,1,0] | 1 |
[1,1,0,0] | 0 |
[1,0,1,0,1,0] | 3 |
[1,0,1,1,0,0] | 1 |
[1,1,0,0,1,0] | 2 |
[1,1,0,1,0,0] | 1 |
[1,1,1,0,0,0] | 0 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000006 has the rest.