Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)
Values
[1,0] => [2,1] => [1,1,0,0] => 1
[1,0,1,0] => [3,1,2] => [1,1,1,0,0,0] => 1
[1,1,0,0] => [2,3,1] => [1,1,0,1,0,0] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [1,1,1,1,0,0,0,0] => 1
[1,0,1,1,0,0] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [1,1,0,1,1,0,0,0] => 2
[1,1,0,1,0,0] => [4,3,1,2] => [1,1,1,1,0,0,0,0] => 1
[1,1,1,0,0,0] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => 3
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => 2
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => 3
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => 4
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0] => 2
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0] => 3
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0] => 3
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0] => 4
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0] => 2
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0] => 3
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0] => 4
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0] => 3
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0] => 4
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 5
[] => [1] => [1,0] => 1
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of peaks of a Dyck path.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!