Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000745
(load all 23 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [[1],[2]]
 => 2
[1,0,1,0]
 => [3,1,2] => [[1,2],[3]]
 => 1
[1,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [[1,2,3],[4]]
 => 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [[1,2],[3,4]]
 => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [[1,2],[3],[4]]
 => 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [[1,2,3,4],[5]]
 => 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [[1,2,3],[4,5]]
 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [[1,2,4],[3,5]]
 => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [[1,2,3],[4],[5]]
 => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [[1,2,5],[3,4]]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [[1,3,4],[2,5]]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [[1,3,5],[2,4]]
 => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [[1,2,4],[3],[5]]
 => 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [[1,2,3],[4],[5]]
 => 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [[1,2],[3,5],[4]]
 => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [[1,3,4],[2,5]]
 => 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [[1,3],[2,4],[5]]
 => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [[1,2],[3,4],[5]]
 => 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,3,4,5],[2]]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [[1,2,3,4,5],[6]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [[1,2,3,4],[5,6]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [[1,2,3,5],[4,6]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [[1,2,3,4],[5],[6]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [[1,2,3,6],[4,5]]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [[1,2,4,5],[3,6]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [[1,2,4],[3,5,6]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [[1,2,3,5],[4],[6]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [[1,2,3,4],[5],[6]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [[1,2,3],[4,6],[5]]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [[1,2,5],[3,4,6]]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [[1,2,4],[3,5],[6]]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [[1,2,3],[4,5],[6]]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [[1,2,5,6],[3,4]]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [[1,3,4,5],[2,6]]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [[1,3,4],[2,5,6]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [[1,3,5],[2,4,6]]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [[1,3,4],[2,5],[6]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [[1,3,5,6],[2,4]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [[1,2,4,5],[3],[6]]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [[1,2,4],[3,6],[5]]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [[1,2,3,5],[4],[6]]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [[1,2,3,4],[5,6]]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [[1,2,3],[4,6],[5]]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [[1,2,5],[3,6],[4]]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [[1,2,4],[3,5],[6]]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [[1,2,3],[4,5],[6]]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [[1,2,6],[3,5],[4]]
 => 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000678
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => [1,0,1,0]
 => 2
[1,0,1,0]
 => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,0]
 => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,6,5,1,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [1,6,5,2,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,6,1,5,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [4,6,2,5,1,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [2,6,3,5,1,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [4,6,1,2,5,3] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [4,6,3,2,1,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [5,1,6,4,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [5,3,6,1,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [5,1,6,2,4,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [5,3,6,2,1,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,4,6,1,5,2] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,4,6,2,1,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000297
(load all 9 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => => ? = 2 - 1
[1,0,1,0]
 => [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => 1 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => 01 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => 01 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => 10 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 001 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 010 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 010 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 001 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 010 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 100 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 101 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 010 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 100 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0010 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0010 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0001 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0010 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0100 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 0101 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 0010 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0100 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0101 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0010 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0001 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0010 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 0100 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 0101 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 0010 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 0100 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1000 => 1 = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 6 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => => ? = 2
[1,0,1,0]
 => [2,1] => [2,1] => 1 => 1
[1,1,0,0]
 => [1,2] => [1,2] => 0 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,3,1] => 10 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => 01 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => 10 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,3,2] => 01 => 2
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,4,3,1] => 101 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,4,1,3] => 100 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 101 => 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,4,1,3] => 100 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,4,3] => 101 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => 011 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,4,3,2] => 011 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [3,1,4,2] => 110 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => 100 => 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,4,2] => 110 => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,4,3,2] => 011 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,3,2] => 011 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,3,2] => 110 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,4,3,2] => 011 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,5,4,3,1] => 1011 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,5,4,1,3] => 1001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,5,1,4,3] => 1010 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,5,4,1,3] => 1001 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,5,1,4,3] => 1010 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,5,4,3] => 1011 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,5,1,4,3] => 1010 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,5,4,1,3] => 1001 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,5,1,4,3] => 1010 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,5,4,3] => 1011 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,4,3] => 1011 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,5,1,4,3] => 1010 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,5,4,3,2] => 0111 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [3,1,5,4,2] => 1101 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [3,1,5,4,2] => 1101 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,5,1,4,2] => 1100 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [3,5,4,1,2] => 1001 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,5,1,4,2] => 1100 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,5,4,2] => 1101 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,1,5,4,2] => 1101 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [3,5,1,4,2] => 1100 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,5,4,2] => 1101 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,5,4,3,2] => 0111 => 2
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000504
(load all 2 compositions to match this statistic)
Mapping
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [1] => {{1}}
 => ? = 2
[1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => {{1},{2}}
 => 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => {{1,2}}
 => 2
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => {{1},{2},{3}}
 => 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => {{1},{2,3}}
 => 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => {{1,2},{3}}
 => 2
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => {{1},{2,3}}
 => 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => {{1,3},{2}}
 => 2
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 2
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 2
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => {{1,3},{2},{4}}
 => 2
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 2
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => {{1},{2,4},{3}}
 => 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => {{1,2},{3,5},{4}}
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
 => 2
Description
The cardinality of the first block of a set partition. The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St000541
(load all 5 compositions to match this statistic)
Mapping
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [1] => [1] => ? = 2 - 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => [2,3,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => [1,3,4,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => [1,3,4,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => [2,1,4,3,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => [2,1,4,5,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => [1,3,4,2,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => [1,2,4,5,3] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => [1,3,4,5,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => [2,3,1,4,5] => 1 = 2 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000877
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => => ? = 2 - 1
[1,0,1,0]
 => [1,2] => [2,1] => 1 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [1,2] => 0 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => 11 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 10 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,3,1] => 01 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1,3] => 10 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => 01 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => 111 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,3,1,2] => 110 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => 101 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [4,2,1,3] => 110 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [4,1,3,2] => 101 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,4,2,1] => 011 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => 010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,4,1] => 101 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,2,1,4] => 110 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => 101 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,4,3,1] => 011 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 010 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,4,3] => 101 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,4,3,2] => 011 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => 1111 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,4,3,1,2] => 1110 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,4,2,3,1] => 1101 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [5,4,2,1,3] => 1110 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [5,4,1,3,2] => 1101 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,3,4,2,1] => 1011 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,4,1,2] => 1010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,3,2,4,1] => 1101 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [5,3,2,1,4] => 1110 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [5,3,1,4,2] => 1101 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [5,2,4,3,1] => 1011 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [5,2,4,1,3] => 1010 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [5,2,1,4,3] => 1101 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [5,1,4,3,2] => 1011 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [4,5,3,2,1] => 0111 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,5,2,3,1] => 0101 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,5,2,1,3] => 0110 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [4,5,1,3,2] => 0101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [4,3,5,2,1] => 1011 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,3,5,1,2] => 1010 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,5,1] => 1101 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 1110 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => 1101 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,5,3,1] => 1011 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => 1010 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,2,1,5,3] => 1101 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => 1011 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,5,4,2,1] => 0111 => 1 = 2 - 1
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000932
(load all 2 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => ? = 2 - 1
[1,0,1,0]
 => [1,2] => [2,1] => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [1,2] => [1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
Description
The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000990
(load all 60 compositions to match this statistic)
Mapping
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [1] => ? = 2
[1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 2
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 2
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,5,2,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [[0,0,0,1,0,0,0],[1,0,0,-1,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [[0,0,0,1,0,0,0],[1,0,0,-1,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [1,5,2,3,4,7,6] => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,-1,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [[0,0,0,0,1,0,0],[1,0,0,0,-1,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
 => [1,5,2,3,4,7,6] => ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [[0,0,0,0,0,1,0],[1,0,0,0,0,-1,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
 => [1,7,2,3,4,5,6] => ? = 1
Description
The first ascent of a permutation. For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$. For the first descent, see [[St000654]].
Matching statistic: St000390
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => 1 => 1 = 2 - 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => 00 => 0 = 1 - 1
[1,1,0,0]
 => [2,3,1] => [1,2,3] => 11 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => 000 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,1,3,2] => 000 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [1,4,2,3] => 100 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,1,3] => 000 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 111 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => 0000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,5,1,4,2] => 0000 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [4,1,5,3,2] => 0000 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [4,5,3,1,2] => 0000 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => 0000 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,4,5,2,3] => 1000 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [1,5,2,4,3] => 1000 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,2,5,1,3] => 0000 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,5,2,1,3] => 0000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,1,4,3] => 0000 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,2,5,3,4] => 1100 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,5,3,2,4] => 1000 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,1,4] => 0000 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 1111 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [3,4,5,6,1,2] => 00000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [3,4,6,1,5,2] => 00000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,5,1,6,4,2] => 00000 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [3,5,6,4,1,2] => 00000 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,6,1,4,5,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,1,5,6,3,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [4,1,6,3,5,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,5,3,6,1,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,5,6,3,1,2] => 00000 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [4,6,3,1,5,2] => 00000 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [5,1,3,6,4,2] => 00000 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,1,6,4,3,2] => 00000 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [5,6,3,4,1,2] => 00000 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,1,3,4,5,2] => 00000 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,4,5,6,2,3] => 10000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [1,4,6,2,5,3] => 10000 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [1,5,2,6,4,3] => 10000 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [1,5,6,4,2,3] => 10000 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [1,6,2,4,5,3] => 10000 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [4,2,5,6,1,3] => 00000 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [4,2,6,1,5,3] => 00000 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,5,2,6,1,3] => 00000 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [4,5,6,1,2,3] => 00000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,6,2,1,5,3] => 00000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [5,2,1,6,4,3] => 00000 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [5,2,6,4,1,3] => 00000 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,6,2,4,1,3] => 00000 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,1,4,5,3] => 00000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [8,1,2,3,6,4,5,7] => [3,4,6,7,5,8,1,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,1,2,6,3,4,5,7] => [3,5,6,7,4,8,1,2] => ? => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [8,1,2,6,3,7,4,5] => [3,5,7,8,4,6,1,2] => ? => ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [8,1,5,2,3,7,4,6] => [4,5,7,3,8,6,1,2] => ? => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [8,1,5,2,6,7,3,4] => [4,7,8,3,5,6,1,2] => ? => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [8,1,4,5,6,2,3,7] => [6,7,3,4,5,8,1,2] => ? => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [8,3,1,2,7,4,5,6] => [4,2,6,7,8,5,1,3] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [8,4,1,2,3,7,5,6] => [4,5,2,7,8,6,1,3] => ? => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [8,3,1,5,2,4,6,7] => [5,2,6,4,7,8,1,3] => ? => ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [8,4,1,5,2,3,6,7] => [5,6,2,4,7,8,1,3] => ? => ? = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [8,4,1,5,6,2,3,7] => [6,7,2,4,5,8,1,3] => ? => ? = 1 - 1
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
 => [8,3,4,1,2,7,5,6] => [5,2,3,7,8,6,1,4] => ? => ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [8,3,5,1,2,4,6,7] => [5,2,6,3,7,8,1,4] => ? => ? = 1 - 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [8,5,4,1,2,3,6,7] => [5,6,3,2,7,8,1,4] => ? => ? = 1 - 1
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [8,3,4,1,6,2,5,7] => [6,2,3,7,5,8,1,4] => ? => ? = 1 - 1
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [8,3,5,1,6,2,4,7] => [6,2,7,3,5,8,1,4] => ? => ? = 1 - 1
Description
The number of runs of ones in a binary word.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000237The number of small exceedances. St000989The number of final rises of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000654The first descent of a permutation. St000061The number of nodes on the left branch of a binary tree. St000352The Elizalde-Pak rank of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000234The number of global ascents of a permutation. St000542The number of left-to-right-minima of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001545The second Elser number of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000056The decomposition (or block) number of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001948The number of augmented double ascents of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.