LongTunnels on DyckPath

The longest sequences of consecutive tunnels: the semilength less the valleys.

On DyckPath

Tunnels meet end to start exactly at valleys, and each valley joins one pair, so merging them leaves one long tunnel per tunnel not joined.

Called as CombinatorialStat(x, "LongTunnels"). 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 St000443

The number of long tunnels 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]	2
[1,1,0,0,1,0]	2
[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/St000443 has the rest.

FindStat St001007

The number of simple modules with projective dimension one in the linear Nakayama algebra corresponding to 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]	2
[1,1,0,0,1,0]	2
[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/St001007 has the rest.

FindStat St001187

The number of simple modules of grade at least one in the linear Nakayama algebra corresponding to 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]	2
[1,1,0,0,1,0]	2
[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/St001187 has the rest.

FindStat St001224

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.

Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the first Ext-group between X and the regular module.

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]	2
[1,1,0,0,1,0]	2
[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/St001224 has the rest.