Area on DyckPath

The whole squares between the path and the diagonal below it: the sum of the area sequence.

On DyckPath

Haglund's area (FindStat St000012). Half of: the total of the heights after each step, less the semilength.

Called as CombinatorialStat(x, "Area"). 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.

Sizes 1 to

FindStat St000012

The area of a Dyck path.

Object	Value
[1,0]	0
[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]	2
[1,1,1,0,0,0]	3

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000012 has the rest.

FindStat St001295

The vector space dimension of $\operatorname{Hom}_A(J^2,J^2)$ for the linear Nakayama algebra $A$ corresponding to a Dyck path.

Object	Value
[1,0]	0
[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]	2
[1,1,1,0,0,0]	3

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001295 has the rest.