Your data matches 58 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000740
(load all 75 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 2
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 3
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 3
[2,3,1] => [1,3,2] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => 4
[1,3,4,2] => [1,2,4,3] => 3
[1,4,2,3] => [1,4,2,3] => 3
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 4
[2,1,4,3] => [2,1,4,3] => 3
[2,3,1,4] => [1,3,2,4] => 4
[2,3,4,1] => [1,2,4,3] => 3
[2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => 4
[3,1,4,2] => [2,1,4,3] => 3
[3,2,1,4] => [3,2,1,4] => 4
[3,2,4,1] => [2,1,4,3] => 3
[3,4,1,2] => [2,4,1,3] => 3
[3,4,2,1] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 3
[4,2,3,1] => [4,1,3,2] => 2
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => 5
[1,2,4,5,3] => [1,2,3,5,4] => 4
[1,2,5,3,4] => [1,2,5,3,4] => 4
[1,2,5,4,3] => [1,2,5,4,3] => 3
[1,3,2,4,5] => [1,3,2,4,5] => 5
[1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,2,4,3,5] => 5
[1,3,4,5,2] => [1,2,3,5,4] => 4
[1,3,5,2,4] => [1,3,5,2,4] => 4
[1,3,5,4,2] => [1,2,5,4,3] => 3
[1,4,2,3,5] => [1,4,2,3,5] => 5
[1,4,2,5,3] => [1,3,2,5,4] => 4
[1,4,3,2,5] => [1,4,3,2,5] => 5
[1,4,3,5,2] => [1,3,2,5,4] => 4
[1,4,5,2,3] => [1,3,5,2,4] => 4
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St000734
(load all 6 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
 => 1
[1,2] => [1,2] => [[1,2]]
 => 2
[2,1] => [2,1] => [[1],[2]]
 => 1
[1,2,3] => [1,2,3] => [[1,2,3]]
 => 3
[1,3,2] => [1,3,2] => [[1,2],[3]]
 => 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
 => 3
[2,3,1] => [1,3,2] => [[1,2],[3]]
 => 2
[3,1,2] => [3,1,2] => [[1,2],[3]]
 => 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 4
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 4
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
 => 3
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 4
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 3
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 4
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
 => 3
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
 => 4
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
 => 3
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 4
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
 => 3
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
 => 3
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
 => 3
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
 => 2
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
 => 3
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
 => 2
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
 => 2
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 5
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => 4
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 5
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 4
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 5
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 4
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
 => 5
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 4
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 5
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 4
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 4
Description
The last entry in the first row of a standard tableau.
Matching statistic: St001007
(load all 13 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 1
[1,2] => [2,1] => [1,1,0,0]
 => 2
[2,1] => [1,2] => [1,0,1,0]
 => 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 3
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 3
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 4
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 4
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 4
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 4
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000024
(load all 13 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
 => 1 = 2 - 1
[2,1] => [1,2] => [1,0,1,0]
 => 0 = 1 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 2 = 3 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 2 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 3 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 3 = 4 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 3 = 4 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 4 - 1
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000013
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
 => 2
[2,1] => [2,1] => [1,2] => [1,0,1,0]
 => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 3
[2,3,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[3,1,2] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000025
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
 => 2
[2,1] => [2,1] => [1,2] => [1,0,1,0]
 => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 3
[2,3,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[3,1,2] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000676
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 1
[1,2] => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2,1] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000738
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
 => [[1]]
 => 1
[1,2] => [1,2] => [[1,2]]
 => [[1],[2]]
 => 2
[2,1] => [2,1] => [[1],[2]]
 => [[1,2]]
 => 1
[1,2,3] => [1,2,3] => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[1,3,2] => [1,3,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
 => [[1,2],[3]]
 => 3
[2,3,1] => [1,3,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[3,1,2] => [3,1,2] => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 4
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 4
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 3
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 4
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 3
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 4
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 3
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 4
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 3
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 3
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 3
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 5
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 4
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => 5
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 4
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 5
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 4
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 5
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 4
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => 5
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 4
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 4
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St001068
(load all 5 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 1
[1,2] => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2
[2,1] => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001809
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
 => 2
[2,1] => [2,1] => [1,2] => [1,0,1,0]
 => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 3
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
 => 3
[2,3,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[3,1,2] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 4
Description
The index of the step at the first peak of maximal height in a Dyck path.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000439The position of the first down step of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St001497The position of the largest weak excedence of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000702The number of weak deficiencies of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St000332The positive inversions of an alternating sign matrix. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000316The number of non-left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000051The size of the left subtree of a binary tree. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000216The absolute length of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000155The number of exceedances (also excedences) of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St001769The reflection length of a signed permutation.