searching the database
Your data matches 35 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000745
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[2]]
=> 2
[1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[1,2,5,7,8],[3,4,6,9,10]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[1,2,5,6,9],[3,4,7,8,10]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[1,2,4,5,6],[3,7,8,9,10]]
=> 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000326
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ 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%
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]
=> [1,2] => [1,2] => 0 => 2
[1,1,0,0]
=> [2,1] => [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 01 => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 01 => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 10 => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 11 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 011 => 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 011 => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 011 => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 011 => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 011 => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 101 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 100 => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 101 => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 101 => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 110 => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 110 => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 110 => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 111 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 0111 => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 0111 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 0111 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 0111 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 1011 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 1010 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 1010 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 1001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 1011 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 1011 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 1001 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 1011 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 1011 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 1011 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,5,4] => 1101 => 1
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: St000297
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ 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%
Mp00066: Permutations —inverse⟶ 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] => [1] => => ? = 2 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 10 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 01 => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 01 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 101 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 100 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 100 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 010 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 010 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 010 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 010 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => 010 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 001 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => 001 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 001 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => 1010 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 1010 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 1010 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 1001 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 1001 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => 0100 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => 0100 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => 0100 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => 0100 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => 0100 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => 0101 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => 0101 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 0101 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 0100 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 0100 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0100 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,4,5,1,3] => 0010 => 0 = 1 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000390
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000390: 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]
=> [2,1] => [1,2] => 1 => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [2,1] => 0 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => 11 => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,3,2] => 10 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,2,1] => 00 => 0 = 1 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 00 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 00 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => 111 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,2,4,3] => 110 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,4,3,2] => 100 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,4,2,3] => 100 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => 100 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [4,2,3,1] => 000 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,2,4,1] => 000 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => 000 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => 000 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,4,3,1] => 000 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,2,1] => 000 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => 000 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 1111 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => 1110 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => 1100 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => 1100 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => 1100 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => 1000 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => 1000 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => 1000 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,5,2,3,4] => 1000 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => 1000 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => 1000 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,4,5,3,2] => 1000 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => 1000 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => 0000 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => 0000 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 0000 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => 0000 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => 0000 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 0000 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,5,2] => 0000 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,1,2,4,3] => 0000 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 0000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,2,5,3] => 0000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => 0000 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => 0000 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,1,5,2,3] => 0000 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,4,5,2] => 0000 => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,5,3,4,1] => 0000 => 0 = 1 - 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001217
(load all 182 compositions to match this statistic)
(load all 182 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 1 - 1
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 2
[1,0,1,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,7,8,1,6] => [7,6,5,4,2,1,8,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,8,7] => [7,6,5,4,3,8,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,8,1,7] => [7,6,5,4,3,1,8,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,8] => [7,6,5,4,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => [7,6,5,4,3,2,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,8] => [4,8,7,6,5,3,2,1] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [3,8,7,6,5,4,1,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,8] => [3,8,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,1,2,3,4,8,5,6] => [2,8,7,6,5,1,4,3] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [2,8,7,6,5,4,1,3] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [2,8,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,10,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [2,9,8,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,9] => [8,7,6,5,4,3,2,9,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [9,8,7,6,5,4,3,2,1,10] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,11,10,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [2,10,9,8,7,6,5,4,3,1] => [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => [2,9,8,7,6,5,4,1,3] => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,9] => [3,9,8,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8,9] => [8,7,6,5,4,3,9,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,9,1,8] => [8,7,6,5,4,3,1,9,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1,10] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000990
(load all 69 compositions to match this statistic)
(load all 69 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => [] => ? = 2
[1,0,1,0]
=> [2,1] => [1] => [1] => ? = 2
[1,1,0,0]
=> [1,2] => [1] => [1] => ? = 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1] => [2,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1] => [2,1] => 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => [1,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,2] => [1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2] => [1,2] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => [2,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => [1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => [4,1,2,3] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => [3,1,2,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => [3,1,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,4,1,2] => [3,4,1,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => [3,4,1,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => [3,4,1,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,8,5] => [6,7,1,2,3,4,5] => [3,4,5,6,7,1,2] => ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => ? = 2
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,9,1] => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,8] => [7,1,2,3,4,5,6,8] => [2,3,4,5,6,7,1,8] => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,1,9,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => ? = 2
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [3,1,4,5,6,7,8,2] => ? = 2
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,10,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => ? = 2
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => ? = 2
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: St001204
(load all 176 compositions to match this statistic)
(load all 176 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001204: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [] => []
=> ? = 2 - 1
[1,0,1,0]
=> [1,2] => [1] => [1,0]
=> ? = 2 - 1
[1,1,0,0]
=> [2,1] => [1] => [1,0]
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => [1,4,7,6,5,3,2] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,8,7,4,5,6,3,2] => [1,7,4,5,6,3,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => [1,7,4,6,5,3,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,8,6,5,4,7,3,2] => [1,6,5,4,7,3,2] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,8,7,5,4,6,3,2] => [1,7,5,4,6,3,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,8,7,5,6,4,3,2] => [1,7,5,6,4,3,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [7,3,6,5,4,2,1,8] => [7,3,6,5,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [7,6,3,4,5,2,1,8] => [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,6,3,5,4,2,1,8] => [7,6,3,5,4,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,5,4,3,6,2,1,8] => [7,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,6,4,3,5,2,1,8] => [7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => [7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,7,6,4,5,3,2,1,9] => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,9,8,7,5,6,4,3,2] => [1,8,7,5,6,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,6,4,3,2,1,10] => [9,8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,7,5,4,6,3,2,1,9] => [8,7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,4,6,5,3,2,1,9] => [8,7,4,6,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,9,8,6,5,7,4,3,2] => [1,8,6,5,7,4,3,2] => ?
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,9,8,5,7,6,4,3,2] => [1,8,5,7,6,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,6,7,5,4,3,2] => [1,9,8,6,7,5,4,3,2] => ?
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
Description
Call 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.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra.
The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.
Matching statistic: St000541
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [[1]]
=> [1] => ? = 2 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [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 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [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 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [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 - 1
[1,0,1,1,0,1,0,0]
=> [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 - 1
[1,0,1,1,1,0,0,0]
=> [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 - 1
[1,1,0,0,1,0,1,0]
=> [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] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [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] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [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] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [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] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [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] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [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] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,1,2,3,5] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [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 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [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 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [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 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [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 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [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 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [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 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [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] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [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] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [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] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [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] => 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [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] => 0 = 1 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,3,4,7,5,6] => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [2,1,4,3,5,7,6] => ? = 2 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,-1,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,4,3,7,5,6] => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [2,1,3,5,4,7,6] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,3,4,7,5,6] => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [2,1,3,4,5,7,6] => ? = 2 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,3,4,7,5,6] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [2,1,3,6,4,5,7] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,0,1,0],[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]]
=> [2,1,3,7,4,5,6] => ? = 2 - 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [2,1,3,5,4,7,6] => ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,-1,1,0],[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]]
=> [2,1,3,7,4,5,6] => ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [2,1,3,4,7,5,6] => ? = 2 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,8,6,7] => ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,-1,1,0],[0,0,1,0,0,0,-1,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,8,4,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,-1,1,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,5,4,8,6,7] => ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,6,5,8,7] => ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,0,1,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,8,5,6,7] => ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,8,6,7] => ? = 2 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,-1,1,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,8,5,6,7] => ? = 2 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,8,5,6,7] => ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0,0],[1,0,0,0,-1,1,0,0],[0,1,0,0,0,-1,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,8,3,4,5,6,7] => ? = 1 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,-1,1,0,0,0],[0,1,0,0,-1,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,8,5,6,7] => ? = 1 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,8,6,7] => ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,0,1,0,0,0],[0,1,0,0,-1,1,0,0],[0,0,1,0,0,-1,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,8,4,5,6,7] => ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,8,5,6,7] => ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [[0,0,0,1,0,0,0,0],[1,0,0,-1,1,0,0,0],[0,1,0,0,-1,1,0,0],[0,0,1,0,0,-1,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,8,4,5,6,7] => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [[0,0,0,1,0,0,0,0,0],[1,0,0,-1,1,0,0,0,0],[0,1,0,0,-1,1,0,0,0],[0,0,1,0,0,-1,1,0,0],[0,0,0,1,0,0,-1,0,1],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,9,5,6,7,8] => ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
=> [[0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,-1,1,0,0,0],[0,0,1,0,0,-1,1,0,0],[0,0,0,1,0,0,-1,1,0],[0,0,0,0,1,0,0,-1,1],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,9,6,7,8] => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0,0,0,0],[1,0,0,0,-1,1,0,0,0,0],[0,1,0,0,0,-1,1,0,0,0],[0,0,1,0,0,0,-1,1,0,0],[0,0,0,1,0,0,0,-1,0,1],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,10,5,6,7,8,9] => ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0,0,0],[1,0,0,0,-1,1,0,0,0],[0,1,0,0,0,-1,1,0,0],[0,0,1,0,0,0,-1,0,1],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,9,4,5,6,7,8] => ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,-1,1,0,0],[0,0,1,0,0,0,-1,1,0],[0,0,0,1,0,0,0,-1,1],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,9,5,6,7,8] => ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,-1,1,0,0,0],[0,0,1,0,0,0,-1,1,0,0],[0,0,0,1,0,0,0,-1,1,0],[0,0,0,0,1,0,0,0,-1,1],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,10,6,7,8,9] => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,-1,1,0,0,0,0,0],[0,1,0,0,0,-1,1,0,0,0,0],[0,0,1,0,0,0,-1,1,0,0,0],[0,0,0,1,0,0,0,-1,1,0,0],[0,0,0,0,1,0,0,0,-1,0,1],[0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,11,6,7,8,9,10] => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [[0,0,0,1,0,0,0,0,0,0],[1,0,0,-1,1,0,0,0,0,0],[0,1,0,0,-1,1,0,0,0,0],[0,0,1,0,0,-1,1,0,0,0],[0,0,0,1,0,0,-1,1,0,0],[0,0,0,0,1,0,0,-1,0,1],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,10,6,7,8,9] => ? = 1 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [[0,0,0,1,0,0,0,0,0],[1,0,0,-1,1,0,0,0,0],[0,1,0,0,-1,1,0,0,0],[0,0,1,0,0,-1,1,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,-1,0,1],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,9,6,7,8] => ? = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [[0,0,1,0,0,0,0,0,0],[1,0,-1,0,1,0,0,0,0],[0,1,0,0,-1,1,0,0,0],[0,0,1,0,0,-1,1,0,0],[0,0,0,1,0,0,-1,0,1],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,9,5,6,7,8] => ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0,0]
=> [[0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,-1,1,0,0,0],[0,0,1,0,0,-1,1,0,0],[0,0,0,1,0,0,-1,1,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,6,5,9,7,8] => ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,-1,0,1,0,0,0],[0,0,1,0,0,-1,1,0,0],[0,0,0,1,0,0,-1,1,0],[0,0,0,0,1,0,0,-1,1],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,9,6,7,8] => ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [[0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,-1,1,0,0,0,0],[0,0,1,0,0,-1,1,0,0,0],[0,0,0,1,0,0,-1,1,0,0],[0,0,0,0,1,0,0,-1,1,0],[0,0,0,0,0,1,0,0,-1,1],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,10,7,8,9] => ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,-1,1,0,0,0,0],[0,0,1,0,0,0,-1,1,0,0,0],[0,0,0,1,0,0,0,-1,1,0,0],[0,0,0,0,1,0,0,0,-1,1,0],[0,0,0,0,0,1,0,0,0,-1,1],[0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,11,7,8,9,10] => ? = 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: St001498
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 2 - 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? = 1 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => [6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => [6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [1,4,5,6,2,7,8,3] => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [1,5,6,7,8,2,3,4] => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [1,4,5,6,7,2,8,3] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [1,3,5,6,7,8,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [1,4,5,6,7,8,3,2] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [1,4,5,6,7,8,2,3] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [1,3,4,5,6,7,8,2] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,8,7] => [3,4,5,1,6,8,7,2] => [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,8,5] => [4,5,6,8,1,2,7,3] => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [3,4,5,6,1,8,7,2] => [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,2,7,3,4,5,8,6] => [2,4,5,6,8,3,7,1] => [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,7,2,3,4,5,8,6] => [3,4,5,6,8,2,7,1] => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [3,4,5,6,8,1,7,2] => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => [2,3,4,5,6,8,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => [2,3,4,5,6,7,9,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => [2,3,4,5,6,7,8,10,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => [3,4,5,6,7,9,1,8,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,9,1,3,4,5,6,7,8] => [1,4,5,6,7,8,9,2,3] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => [1,3,4,5,6,7,8,9,10,2] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [2,3,4,5,6,7,8,9,11,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,10,8] => [3,4,5,6,7,8,10,1,9,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,8,2,3,4,5,6,9,7] => [3,4,5,6,7,9,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,9,8] => [3,4,5,6,7,1,9,8,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,1,9,3,4,5,6,7,8] => [1,4,5,6,7,8,9,3,2] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7,9] => [1,4,5,6,7,8,2,9,3] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,10,1,3,4,5,6,7,8,9] => [1,4,5,6,7,8,9,10,2,3] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [1,3,4,5,6,7,8,9,10,11,2] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St000264The girth of a graph, which is not a tree. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000877The depth of the binary word interpreted as a path. St000456The monochromatic index of a connected graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. 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 $Af$. St000382The first part of an integer composition. St001545The second Elser number of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000392The length of the longest run of ones in a binary word. St000383The last part of an integer composition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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 $eA$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
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!