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Your data matches 95 different statistics following compositions of up to 3 maps.
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Matching statistic: St000052
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(load all 3 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 2
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000157
(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
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => [[1,3],[2]]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => [[1,4,5],[2],[3]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => [[1,4,5],[2],[3]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [[1,3,5],[2,4]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,1,8,2] => [8,2,4,6,1,3,5,7] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,8,1,9,2] => [7,1,3,5,9,2,4,6,8] => ?
=> ? = 2
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,1,3,5,7,9] => [6,3,8,7,5,1,2,4,9] => ?
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,6,2,7,3,8,4,9,5] => [1,7,4,9,5,3,2,6,8] => ?
=> ? = 4
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 10 => 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 01 => 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => 11 => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 100 => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 100 => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => 101 => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => 010 => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 110 => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 110 => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => 110 => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 011 => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => 111 => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => 1001 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => 1010 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 1010 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => 1010 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 1011 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => 1011 => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 0100 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => 0101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => 1100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => 1100 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => 1100 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => 1100 => 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 1100 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => 1101 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => 1101 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 1101 => 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 1100 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[]
=> [] => [] => => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,8,1,9,2] => [7,1,3,5,9,2,4,6,8] => ? => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,1,3,5,7,9] => [6,3,8,7,5,1,2,4,9] => ? => ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,6,2,7,3,8,4,9,5] => [1,7,4,9,5,3,2,6,8] => ? => ? = 4
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000394
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 4
[]
=> [] => .
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,7,9,1,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? = 2
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? = 8
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,1,3,5,7,9] => [[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,6,2,7,3,8,4,9,5] => [.,[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]]
=> ?
=> ? = 4
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000012
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[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] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 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] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[]
=> [] => .
=> ?
=> ? = 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6,9] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,1,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,7,9,1,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,9,2] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,8,1,9,2] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? = 8
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,1,3,5,7,9] => [[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,6,2,7,3,8,4,9,5] => [.,[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]]
=> ?
=> ? = 4
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000340
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 91%●distinct values known / distinct values provided: 80%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 91%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[]
=> []
=> []
=> []
=> ? = 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 8
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ?
=> ? = 4
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000984
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000984: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000984: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 91%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[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] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 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] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[]
=> [] => .
=> ?
=> ? = 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6,9] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,1,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,7,9,1,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,9,2] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,8,1,9,2] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? = 8
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,1,3,5,7,9] => [[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,6,2,7,3,8,4,9,5] => [.,[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]]
=> ?
=> ? = 4
Description
The number of boxes below precisely one peak.
Imagine that each peak of the Dyck path, drawn with north and east steps, casts a shadow onto the triangular region between it and the diagonal. This statistic is the number of cells which are in the shade of precisely one peak.
Matching statistic: St000476
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,6,2,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,7,3,8] => [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 6
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 6
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7,8] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[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] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[]
=> [] => .
=> ?
=> ? = 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6,9] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,1,9] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,7,9,1,8] => [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ?
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,9,2] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,8,1,9,2] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ? = 8
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,1,3,5,7,9] => [[.,.],[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,6,2,7,3,8,4,9,5] => [.,[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]]
=> ?
=> ? = 4
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000925
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1,0]
=> [1,0]
=> {{1}}
=> ? = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,6,7},{5},{8}}
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6,7}}
=> ? = 2 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6,7}}
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 6 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 1 + 1
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 4 + 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 5 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 6 + 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 6 + 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 5 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 6 + 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 6 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 7 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> ? = 1 + 1
[]
=> []
=> []
=> {}
=> ? = 0 + 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 6 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 7 + 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 7 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 8 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9},{8}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,8,9},{7}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,10},{9}}
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,9},{7,8}}
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,7,9},{6},{8}}
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 9 + 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 8 + 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> {{1,7},{2,6},{3,5},{4},{8,9}}
=> ? = 4 + 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ?
=> ? = 4 + 1
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000024
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 87%●distinct values known / distinct values provided: 70%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 87%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,6,1,8,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,7,1,8,3] => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 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] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,6,2,7,8] => [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,8,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,1,7,8,2] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,1,8,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,7,3,8] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ? = 4
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 5
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 6
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[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] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 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] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6,9] => ?
=> ? = 6
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7,9] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,1,9] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,7,9,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,6,7,8,1,9,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 9
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [9,10,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 8
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,1,3,5,7,9] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,6,2,7,3,8,4,9,5] => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
The following 85 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000167The number of leaves of an ordered tree. St000053The number of valleys of the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000211The rank of the set partition. St001512The minimum rank of a graph. St000105The number of blocks in the set partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000912The number of maximal antichains in a poset. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000159The number of distinct parts of the integer partition. St000245The number of ascents of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000246The number of non-inversions of a permutation. St000702The number of weak deficiencies of a permutation. St000292The number of ascents of a binary word. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000354The number of recoils of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000809The reduced reflection length of the permutation. St000332The positive inversions of an alternating sign matrix. St000021The number of descents of a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000316The number of non-left-to-right-maxima of a permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St000740The last entry of a permutation. St000991The number of right-to-left minima of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St000216The absolute length of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001480The number of simple summands of the module J^2/J^3. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001346The number of parking functions that give the same permutation. St001427The number of descents of a signed permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000647The number of big descents of a permutation. St000710The number of big deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000711The number of big exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000358The number of occurrences of the pattern 31-2. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001905The number of preferred parking spots in a parking function less than the index of the car. St001712The number of natural descents of a standard Young tableau. St001946The number of descents in a parking function. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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