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Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000745
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [[1]]
 => 1
[1,0,1,0]
 => [1,2] => [1,2] => [[1,2]]
 => 1
[1,1,0,0]
 => [2,1] => [2,1] => [[1],[2]]
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [[1,2],[3]]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [[1,3],[2]]
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => [[1,2],[3]]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [[1,3,4],[2]]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [[1,3,4],[2]]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => [[1,2,4],[3]]
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => [[1,2],[3,4]]
 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => [[1,3],[2],[4]]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => [[1,3],[2],[4]]
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => [[1,2,3],[4,5]]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => [[1,2,4],[3],[5]]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [[1,3,5],[2,4]]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [[1,3,4],[2,5]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [[1,3,4,5],[2]]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => [[1,2,5],[3,4]]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,1,2,4,3] => [[1,3,4],[2],[5]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
 => 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000297
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 99%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1] => => ? = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => 0 => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => 1 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 01 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 10 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => 01 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 001 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => 010 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => 001 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => 100 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => 100 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => 010 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => 010 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => 010 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => 101 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => 101 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => 0100 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => 0100 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => 0010 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => 0010 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => 0010 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,2,4,3] => 0101 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => 0101 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => 1001 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => 0100 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => 0100 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => 0100 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,1,2,4,3] => 1001 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => 1010 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => 0100 => 0 = 1 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,3,6,7,5,8,4,2] => [1,5,7,4,8,2,3,6] => ? => ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,6,5,4,7,8,3,2] => [1,4,7,3,5,8,2,6] => ? => ? = 1 - 1
[1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [2,1,5,7,6,8,4,3] => [2,1,7,4,8,3,5,6] => ? => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,3,1,5,6,7,4,8] => [3,1,2,7,4,5,6,8] => ? => ? = 2 - 1
[1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,4,7,8,6,5,2,1] => [6,5,8,1,3,7,2,4] => ? => ? = 2 - 1
[1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
 => [3,6,5,4,7,8,2,1] => [4,8,1,3,5,7,2,6] => ? => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => [5,4,3,2,6,7,1,8] => [3,4,2,7,1,5,6,8] => ? => ? = 1 - 1
[]
 => [] => [] => ? => ? = 0 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,4,5,7,6,8,9,1] => [6,9,1,2,3,4,5,7,8] => ? => ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,1,4,5,6,8,9,7] => [3,1,2,4,5,6,9,7,8] => ? => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,4,5,7,6,8,9,10,1] => [6,10,1,2,3,4,5,7,8,9] => ? => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,1,5,7,8,9,6] => [4,1,2,3,5,9,6,7,8] => ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000990
(load all 3 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1] => [1] => ? = 1
[1,0,1,0]
 => [1,2] => [1,2] => [1,2] => 1
[1,1,0,0]
 => [2,1] => [2,1] => [2,1] => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [3,1,2] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => [2,1,4,3] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => [1,3,2,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,3,5,4] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => [3,5,1,2,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => [3,5,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,1,2,4,3] => [5,1,2,4,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => [3,2,5,1,4] => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => [1,5,2,3,4,7,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,3,4,5,7,6,2] => [1,6,7,2,3,4,5] => [1,5,7,2,3,4,6] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,3,4,6,5,7,2] => [1,5,7,2,3,4,6] => [1,5,7,2,3,4,6] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,3,4,6,7,5,2] => [1,7,2,3,4,6,5] => [1,7,2,3,4,6,5] => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => [1,5,4,7,2,3,6] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,3,5,4,6,7,2] => [1,4,7,2,3,5,6] => [1,4,7,2,3,5,6] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,3,5,6,4,2,7] => [1,6,2,3,5,4,7] => [1,6,2,3,5,4,7] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,3,5,6,4,7,2] => [1,7,2,3,5,4,6] => [1,7,2,3,5,4,6] => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => [1,5,4,7,2,3,6] => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,3,5,7,6,4,2] => [1,7,2,3,5,6,4] => [1,7,2,3,4,6,5] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => [1,5,4,7,2,3,6] => ? = 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,3,6,5,7,4,2] => [1,7,2,3,6,4,5] => [1,7,2,3,6,4,5] => ? = 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,3,6,7,5,4,2] => [1,5,7,2,3,6,4] => [1,4,7,2,3,6,5] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,4,5,3,2,6,7] => [1,5,2,4,3,6,7] => [1,5,2,4,3,6,7] => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,4,5,3,2,7,6] => [1,5,2,4,3,7,6] => [1,5,2,4,3,7,6] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,4,5,3,6,2,7] => [1,6,2,4,3,5,7] => [1,6,2,4,3,5,7] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,4,5,3,6,7,2] => [1,7,2,4,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,4,5,3,7,6,2] => [1,6,7,2,4,3,5] => [1,5,7,2,4,3,6] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,4,5,6,3,7,2] => [1,5,3,7,2,4,6] => [1,5,3,7,2,4,6] => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,4,5,6,7,3,2] => [1,7,2,4,6,3,5] => [1,7,2,4,6,3,5] => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,4,5,7,6,3,2] => [1,6,3,5,7,2,4] => [1,5,2,4,7,3,6] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,4,6,5,3,2,7] => [1,6,2,4,5,3,7] => [1,6,2,3,5,4,7] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,4,6,5,3,7,2] => [1,7,2,4,5,3,6] => [1,7,2,3,5,4,6] => ? = 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,4,6,5,7,3,2] => [1,6,3,7,2,4,5] => [1,5,3,7,2,4,6] => ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,4,7,6,5,3,2] => [1,5,7,2,4,6,3] => [1,4,7,2,3,6,5] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,4,6,3,2,7] => [1,6,2,5,3,4,7] => [1,6,2,5,3,4,7] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,5,4,6,3,7,2] => [1,7,2,5,3,4,6] => [1,7,2,5,3,4,6] => ? = 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,5,4,6,7,3,2] => [1,6,3,4,7,2,5] => [1,5,2,4,7,3,6] => ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,5,4,7,6,3,2] => [1,7,2,5,6,3,4] => [1,7,2,4,6,3,5] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,5,7,6,4,3,2] => [1,7,2,5,4,6,3] => [1,7,2,4,3,6,5] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,6,5,4,7,3,2] => [1,4,7,2,6,3,5] => [1,4,7,2,6,3,5] => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6,5,7,4,3,2] => [1,7,2,6,3,5,4] => [1,7,2,6,3,5,4] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [2,1,3,4,6,7,5] => [2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,6,4,7] => [2,1,3,6,4,5,7] => [2,1,3,6,4,5,7] => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,1,3,5,6,7,4] => [2,1,3,7,4,5,6] => [2,1,3,7,4,5,6] => ? = 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [2,1,3,5,7,6,4] => [2,1,3,6,7,4,5] => [2,1,3,5,7,4,6] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,5,4,7] => [2,1,3,5,6,4,7] => [2,1,3,4,6,5,7] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
 => [2,1,3,6,5,7,4] => [2,1,3,5,7,4,6] => [2,1,3,5,7,4,6] => ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [2,1,3,6,7,5,4] => [2,1,3,7,4,6,5] => [2,1,3,7,4,6,5] => ? = 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,6,5,4] => [2,1,3,6,5,7,4] => [2,1,3,5,4,7,6] => ? = 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: St001640
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => [1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => [3,2,5,4,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => [4,2,5,3,1] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,2,4,3] => [3,4,2,5,1] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => [2,5,3,4,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [4,3,5,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => [3,5,4,1,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [4,5,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => [3,2,1,5,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => [5,2,1,4,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => [4,2,1,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,1,2,4,3] => [3,4,2,1,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => [2,1,5,3,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [7,5,4,3,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [5,4,3,7,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [5,6,4,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [4,3,7,5,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,4,6,3,7] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,5,6,4,3,7] => [1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,4,7,3] => [1,2,7,3,5,4,6] => [6,4,5,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,4,3] => [1,2,6,4,7,3,5] => [5,3,7,4,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,5,7,6,4,3] => [1,2,7,3,5,6,4] => [4,6,5,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,5,4,3,7] => [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,5,4,7,3] => [1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,6,5,7,4,3] => [1,2,7,3,6,4,5] => [5,4,6,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,6,7,5,4,3] => [1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,6,5,4,3] => [1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [6,5,7,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => [1,3,2,4,6,7,5] => [5,7,6,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => [7,5,4,6,2,3,1] => ? = 1 - 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000654
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1] => [1] => ? = 1
[1,0,1,0]
 => [1,2] => [1,2] => [2,1] => 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,2] => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [3,1,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,3,1] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => [2,1,3] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [4,5,1,3,2] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => [4,5,2,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => [2,1,5,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => [3,2,5,4,1] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => [3,1,5,4,2] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,1,2,4,3] => [1,5,4,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => [2,3,1,5,4] => 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => [4,3,5,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [7,6,2,1,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [7,6,3,2,5,4,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [7,6,3,1,5,4,2] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [7,6,2,3,1,5,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,4,6,3,7] => [1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,5,6,4,3,7] => [1,2,6,3,5,4,7] => [7,6,2,5,3,4,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,4,7,3] => [1,2,7,3,5,4,6] => [7,6,1,5,3,4,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,4,3] => [1,2,6,4,7,3,5] => [7,6,2,4,1,5,3] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,5,7,6,4,3] => [1,2,7,3,5,6,4] => [7,6,1,5,3,2,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,5,4,3,7] => [1,2,5,4,6,3,7] => [7,6,3,4,2,5,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,5,4,7,3] => [1,2,5,4,7,3,6] => [7,6,3,4,1,5,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,6,5,7,4,3] => [1,2,7,3,6,4,5] => [7,6,1,5,2,4,3] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,6,7,5,4,3] => [1,2,5,7,3,6,4] => [7,6,3,1,5,2,4] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,6,5,4,3] => [1,2,5,6,4,7,3] => [7,6,3,2,4,1,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [7,5,6,4,3,1,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => [1,3,2,4,6,7,5] => [7,5,6,4,2,1,3] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => ? = 1
Description
The first descent of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St000989
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1] => [1] => ? = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [2,3,1] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => [1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => [2,3,1,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => [3,2,5,4,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => [4,2,5,3,1] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,2,4,3] => [3,4,2,5,1] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => [2,5,3,4,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [4,3,5,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => [3,5,4,1,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [4,5,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [4,3,2,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => [3,2,1,5,4] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => [5,2,1,4,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => [4,2,1,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,1,2,4,3] => [3,4,2,1,5] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => [2,1,5,3,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => [5,4,1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [7,5,4,3,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [5,4,3,7,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [5,6,4,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [4,3,7,5,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,5,4,6,3,7] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,5,4,7,6,3] => [1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,2,5,6,4,3,7] => [1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,5,6,4,7,3] => [1,2,7,3,5,4,6] => [6,4,5,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,4,3] => [1,2,6,4,7,3,5] => [5,3,7,4,6,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,2,5,7,6,4,3] => [1,2,7,3,5,6,4] => [4,6,5,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,5,4,3,7] => [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,6,5,4,7,3] => [1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,6,5,7,4,3] => [1,2,7,3,6,4,5] => [5,4,6,3,7,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,2,6,7,5,4,3] => [1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,6,5,4,3] => [1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [6,5,7,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,6,5] => [1,3,2,4,6,7,5] => [5,7,6,4,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ? = 1 - 1
Description
The number of final rises of a permutation. For a permutation $\pi$ of length $n$, this is the maximal $k$ such that $$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$ Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St001948
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1] => [1] => ? = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => [3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => [2,1,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [4,5,1,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => [4,5,2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,5,1,2,3] => [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,4,1,2,5] => [3,2,5,4,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,1,2,4] => [3,1,5,4,2] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [5,1,2,4,3] => [1,5,4,2,3] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [4,3,5,1,2] => [2,3,1,5,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [2,3,1,4,5] => [4,3,5,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,5,4,3,1,2] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [6,5,4,2,3,1] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,6,4,5] => [6,5,4,1,3,2] => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [6,5,3,4,1,2] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => [1,2,5,3,4,6] => [6,5,2,4,3,1] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,6,3,4,5] => [6,5,1,4,3,2] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,2,4,6,5,3] => [1,2,5,6,3,4] => [6,5,2,1,4,3] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => [1,2,4,5,3,6] => [6,5,3,2,4,1] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [1,2,4,6,3,5] => [6,5,3,1,4,2] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,4,3] => [1,2,6,3,5,4] => [6,5,1,4,2,3] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [1,2,5,4,6,3] => [6,5,2,3,1,4] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [6,4,5,3,2,1] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [6,4,5,3,1,2] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,3,2,5,6,4] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => [1,3,2,5,6,4] => [6,4,5,2,1,3] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,4,2,5,6] => [1,4,2,3,5,6] => [6,3,5,4,2,1] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5] => [1,4,2,3,6,5] => [6,3,5,4,1,2] => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,3,4,5,2,6] => [1,5,2,3,4,6] => [6,2,5,4,3,1] => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,5,2] => [1,5,6,2,3,4] => [6,2,1,5,4,3] => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,3,5,4,2,6] => [1,4,5,2,3,6] => [6,3,2,5,4,1] => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,3,5,4,6,2] => [1,4,6,2,3,5] => [6,3,1,5,4,2] => ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,3,5,6,4,2] => [1,6,2,3,5,4] => [6,1,5,4,2,3] => ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,6,5,4,2] => [1,5,4,6,2,3] => [6,2,3,1,5,4] => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => [1,3,4,2,5,6] => [6,4,3,5,2,1] => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => [1,3,4,2,6,5] => [6,4,3,5,1,2] => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,3,5,2,6] => [1,3,5,2,4,6] => [6,4,2,5,3,1] => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => [1,3,6,2,4,5] => [6,4,1,5,3,2] => ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,4,3,6,5,2] => [1,3,5,6,2,4] => [6,4,2,1,5,3] => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,5,3,2,6] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,3,6,2] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,4,5,6,3,2] => [1,5,3,6,2,4] => [6,2,4,1,5,3] => ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,5,3,2] => [1,6,2,4,5,3] => [6,1,5,3,2,4] => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => [1,4,3,5,2,6] => [6,3,4,2,5,1] => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,4,3,6,2] => [1,4,3,6,2,5] => [6,3,4,1,5,2] => ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,4,6,3,2] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,4,3,2] => [1,4,6,2,5,3] => [6,3,1,5,2,4] => ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [6,3,2,4,1,5] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [5,6,4,3,2,1] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [5,6,4,3,1,2] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [5,6,4,2,3,1] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,3,5,6,4] => [2,1,3,6,4,5] => [5,6,4,1,3,2] => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,5,4] => [2,1,3,5,6,4] => [5,6,4,2,1,3] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [5,6,3,4,2,1] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => ? = 2 - 1
Description
The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Matching statistic: St001621
(load all 2 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => ([],1)
 => ([(0,1)],2)
 => 1
[1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1
[1,1,0,0]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,1,1,0,0,0]
 => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
 => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
 => ? = 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ? = 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
 => ? = 2
[1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? = 2
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? = 2
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ?
 => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => ([],5)
 => ?
 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
 => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ([(0,5),(1,8),(1,9),(2,7),(2,9),(3,7),(3,8),(4,6),(5,4),(6,1),(6,2),(6,3),(7,10),(8,10),(9,10)],11)
 => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ? = 1
Description
The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.