Runs on Permutation

Maximal increasing runs — one more than the number of descents.

On Permutation

Called as CombinatorialStat(x, "Runs"). The head's page has its examples.

Live

FindStat St000325

The width of the tree associated to a permutation.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000325 has the rest.

FindStat St000470

The number of runs in a permutation.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000470 has the rest.