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Your data matches 164 different statistics following compositions of up to 3 maps.
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Matching statistic: St000011
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000382
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => 1
[1,1] => [2] => 2
[2] => [1,1] => 1
[1,1,1] => [3] => 3
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[3] => [1,1,1] => 1
[1,1,1,1] => [4] => 4
[1,1,2] => [1,3] => 1
[1,2,1] => [2,2] => 2
[1,3] => [1,1,2] => 1
[2,1,1] => [3,1] => 3
[2,2] => [1,2,1] => 1
[3,1] => [2,1,1] => 2
[4] => [1,1,1,1] => 1
[1,1,1,1,1] => [5] => 5
[1,1,1,2] => [1,4] => 1
[1,1,2,1] => [2,3] => 2
[1,1,3] => [1,1,3] => 1
[1,2,1,1] => [3,2] => 3
[1,2,2] => [1,2,2] => 1
[1,3,1] => [2,1,2] => 2
[1,4] => [1,1,1,2] => 1
[2,1,1,1] => [4,1] => 4
[2,1,2] => [1,3,1] => 1
[2,2,1] => [2,2,1] => 2
[2,3] => [1,1,2,1] => 1
[3,1,1] => [3,1,1] => 3
[3,2] => [1,2,1,1] => 1
[4,1] => [2,1,1,1] => 2
[5] => [1,1,1,1,1] => 1
[1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [1,5] => 1
[1,1,1,2,1] => [2,4] => 2
[1,1,1,3] => [1,1,4] => 1
[1,1,2,1,1] => [3,3] => 3
[1,1,2,2] => [1,2,3] => 1
[1,1,3,1] => [2,1,3] => 2
[1,1,4] => [1,1,1,3] => 1
[1,2,1,1,1] => [4,2] => 4
[1,2,1,2] => [1,3,2] => 1
[1,2,2,1] => [2,2,2] => 2
[1,2,3] => [1,1,2,2] => 1
[1,3,1,1] => [3,1,2] => 3
[1,3,2] => [1,2,1,2] => 1
[1,4,1] => [2,1,1,2] => 2
[1,5] => [1,1,1,1,2] => 1
[2,1,1,1,1] => [5,1] => 5
[2,1,1,2] => [1,4,1] => 1
[2,1,2,1] => [2,3,1] => 2
[1,1,1,1,1,1,3,1] => [2,1,7] => ? = 2
[1,1,6,2] => [1,2,1,1,1,1,3] => ? = 1
[2,1,1,1,1,4] => [1,1,1,6,1] => ? = 1
[2,1,7] => [1,1,1,1,1,1,3,1] => ? = 1
[2,6,2] => [1,2,1,1,1,1,2,1] => ? = 1
[10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 2
[1,1,1,1,1,1,1,1,1,1,1,1] => [12] => ? = 12
Description
The first part of an integer composition.
Matching statistic: St001777
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 97%●distinct values known / distinct values provided: 91%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1 = 2 - 1
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [2] => 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 2 = 3 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 3 = 4 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 0 = 1 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 3 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 0 = 1 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 0 = 1 - 1
[1,1,1,1,1] => [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] => 4 = 5 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 0 = 1 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 2 = 3 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 0 = 1 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 1 = 2 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 3 = 4 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 2 = 3 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 0 = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 1 = 2 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 5 = 6 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => 0 = 1 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => 1 = 2 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => 0 = 1 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => 2 = 3 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => 0 = 1 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => 1 = 2 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6] => 0 = 1 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => 3 = 4 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => 0 = 1 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,1] => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [6] => 0 = 1 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,1] => 2 = 3 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [6] => 0 = 1 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1] => 1 = 2 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6] => 0 = 1 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => 4 = 5 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => 0 = 1 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,1] => 1 = 2 - 1
[1,1,5,2] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
=> ? => ? = 1 - 1
[1,1,6,2] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0,0]
=> ? => ? = 1 - 1
[2,6,2] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0,0]
=> ? => ? = 1 - 1
[1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [11] => ? = 1 - 1
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ? = 12 - 1
[1,2,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [11,1] => ? = 2 - 1
[1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11] => ? = 1 - 1
Description
The number of weak descents in an integer composition.
A weak descent of an integer composition $\alpha=(a_1, \dots, a_n)$ is an index $1\leq i < n$ such that $a_i \geq a_{i+1}$.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => 1 => 0 => 2 = 1 + 1
[1,1] => [1,1] => 11 => 00 => 3 = 2 + 1
[2] => [2] => 10 => 01 => 2 = 1 + 1
[1,1,1] => [1,1,1] => 111 => 000 => 4 = 3 + 1
[1,2] => [2,1] => 101 => 010 => 2 = 1 + 1
[2,1] => [1,2] => 110 => 001 => 3 = 2 + 1
[3] => [3] => 100 => 011 => 2 = 1 + 1
[1,1,1,1] => [1,1,1,1] => 1111 => 0000 => 5 = 4 + 1
[1,1,2] => [2,1,1] => 1011 => 0100 => 2 = 1 + 1
[1,2,1] => [1,2,1] => 1101 => 0010 => 3 = 2 + 1
[1,3] => [3,1] => 1001 => 0110 => 2 = 1 + 1
[2,1,1] => [1,1,2] => 1110 => 0001 => 4 = 3 + 1
[2,2] => [2,2] => 1010 => 0101 => 2 = 1 + 1
[3,1] => [1,3] => 1100 => 0011 => 3 = 2 + 1
[4] => [4] => 1000 => 0111 => 2 = 1 + 1
[1,1,1,1,1] => [1,1,1,1,1] => 11111 => 00000 => 6 = 5 + 1
[1,1,1,2] => [2,1,1,1] => 10111 => 01000 => 2 = 1 + 1
[1,1,2,1] => [1,2,1,1] => 11011 => 00100 => 3 = 2 + 1
[1,1,3] => [3,1,1] => 10011 => 01100 => 2 = 1 + 1
[1,2,1,1] => [1,1,2,1] => 11101 => 00010 => 4 = 3 + 1
[1,2,2] => [2,2,1] => 10101 => 01010 => 2 = 1 + 1
[1,3,1] => [1,3,1] => 11001 => 00110 => 3 = 2 + 1
[1,4] => [4,1] => 10001 => 01110 => 2 = 1 + 1
[2,1,1,1] => [1,1,1,2] => 11110 => 00001 => 5 = 4 + 1
[2,1,2] => [2,1,2] => 10110 => 01001 => 2 = 1 + 1
[2,2,1] => [1,2,2] => 11010 => 00101 => 3 = 2 + 1
[2,3] => [3,2] => 10010 => 01101 => 2 = 1 + 1
[3,1,1] => [1,1,3] => 11100 => 00011 => 4 = 3 + 1
[3,2] => [2,3] => 10100 => 01011 => 2 = 1 + 1
[4,1] => [1,4] => 11000 => 00111 => 3 = 2 + 1
[5] => [5] => 10000 => 01111 => 2 = 1 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 111111 => 000000 => 7 = 6 + 1
[1,1,1,1,2] => [2,1,1,1,1] => 101111 => 010000 => 2 = 1 + 1
[1,1,1,2,1] => [1,2,1,1,1] => 110111 => 001000 => 3 = 2 + 1
[1,1,1,3] => [3,1,1,1] => 100111 => 011000 => 2 = 1 + 1
[1,1,2,1,1] => [1,1,2,1,1] => 111011 => 000100 => 4 = 3 + 1
[1,1,2,2] => [2,2,1,1] => 101011 => 010100 => 2 = 1 + 1
[1,1,3,1] => [1,3,1,1] => 110011 => 001100 => 3 = 2 + 1
[1,1,4] => [4,1,1] => 100011 => 011100 => 2 = 1 + 1
[1,2,1,1,1] => [1,1,1,2,1] => 111101 => 000010 => 5 = 4 + 1
[1,2,1,2] => [2,1,2,1] => 101101 => 010010 => 2 = 1 + 1
[1,2,2,1] => [1,2,2,1] => 110101 => 001010 => 3 = 2 + 1
[1,2,3] => [3,2,1] => 100101 => 011010 => 2 = 1 + 1
[1,3,1,1] => [1,1,3,1] => 111001 => 000110 => 4 = 3 + 1
[1,3,2] => [2,3,1] => 101001 => 010110 => 2 = 1 + 1
[1,4,1] => [1,4,1] => 110001 => 001110 => 3 = 2 + 1
[1,5] => [5,1] => 100001 => 011110 => 2 = 1 + 1
[2,1,1,1,1] => [1,1,1,1,2] => 111110 => 000001 => 6 = 5 + 1
[2,1,1,2] => [2,1,1,2] => 101110 => 010001 => 2 = 1 + 1
[2,1,2,1] => [1,2,1,2] => 110110 => 001001 => 3 = 2 + 1
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => 1001111111 => 0110000000 => ? = 1 + 1
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => 1010111111 => 0101000000 => ? = 1 + 1
[1,1,1,1,1,1,3,1] => [1,3,1,1,1,1,1,1] => 1100111111 => 0011000000 => ? = 2 + 1
[1,1,1,1,1,1,4] => [4,1,1,1,1,1,1] => 1000111111 => 0111000000 => ? = 1 + 1
[1,1,1,2,1,1,1,2] => [2,1,1,1,2,1,1,1] => 1011110111 => 0100001000 => ? = 1 + 1
[1,1,6,2] => [2,6,1,1] => 1010000011 => 0101111100 => ? = 1 + 1
[1,7,2] => [2,7,1] => 1010000001 => 0101111110 => ? = 1 + 1
[1,8,1] => [1,8,1] => 1100000001 => 0011111110 => ? = 2 + 1
[1,9] => [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
[4,6] => [6,4] => 1000001000 => 0111110111 => ? = 1 + 1
[5,5] => [5,5] => 1000010000 => 0111101111 => ? = 1 + 1
[6,4] => [4,6] => 1000100000 => 0111011111 => ? = 1 + 1
[9,1] => [1,9] => 1100000000 => 0011111111 => ? = 2 + 1
[1,10] => [10,1] => 10000000001 => 01111111110 => ? = 1 + 1
[10,1] => [1,10] => 11000000000 => 00111111111 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 111111111111 => 000000000000 => ? = 12 + 1
[1,2,2,2,2,2,1] => [1,2,2,2,2,2,1] => 110101010101 => 001010101010 => ? = 2 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 89%●distinct values known / distinct values provided: 82%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 89%●distinct values known / distinct values provided: 82%
Values
[1] => [1] => 1 => 1
[1,1] => [1,1] => 11 => 2
[2] => [2] => 10 => 1
[1,1,1] => [1,1,1] => 111 => 3
[1,2] => [2,1] => 101 => 1
[2,1] => [1,2] => 110 => 2
[3] => [3] => 100 => 1
[1,1,1,1] => [1,1,1,1] => 1111 => 4
[1,1,2] => [2,1,1] => 1011 => 1
[1,2,1] => [1,2,1] => 1101 => 2
[1,3] => [3,1] => 1001 => 1
[2,1,1] => [1,1,2] => 1110 => 3
[2,2] => [2,2] => 1010 => 1
[3,1] => [1,3] => 1100 => 2
[4] => [4] => 1000 => 1
[1,1,1,1,1] => [1,1,1,1,1] => 11111 => 5
[1,1,1,2] => [2,1,1,1] => 10111 => 1
[1,1,2,1] => [1,2,1,1] => 11011 => 2
[1,1,3] => [3,1,1] => 10011 => 1
[1,2,1,1] => [1,1,2,1] => 11101 => 3
[1,2,2] => [2,2,1] => 10101 => 1
[1,3,1] => [1,3,1] => 11001 => 2
[1,4] => [4,1] => 10001 => 1
[2,1,1,1] => [1,1,1,2] => 11110 => 4
[2,1,2] => [2,1,2] => 10110 => 1
[2,2,1] => [1,2,2] => 11010 => 2
[2,3] => [3,2] => 10010 => 1
[3,1,1] => [1,1,3] => 11100 => 3
[3,2] => [2,3] => 10100 => 1
[4,1] => [1,4] => 11000 => 2
[5] => [5] => 10000 => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 111111 => 6
[1,1,1,1,2] => [2,1,1,1,1] => 101111 => 1
[1,1,1,2,1] => [1,2,1,1,1] => 110111 => 2
[1,1,1,3] => [3,1,1,1] => 100111 => 1
[1,1,2,1,1] => [1,1,2,1,1] => 111011 => 3
[1,1,2,2] => [2,2,1,1] => 101011 => 1
[1,1,3,1] => [1,3,1,1] => 110011 => 2
[1,1,4] => [4,1,1] => 100011 => 1
[1,2,1,1,1] => [1,1,1,2,1] => 111101 => 4
[1,2,1,2] => [2,1,2,1] => 101101 => 1
[1,2,2,1] => [1,2,2,1] => 110101 => 2
[1,2,3] => [3,2,1] => 100101 => 1
[1,3,1,1] => [1,1,3,1] => 111001 => 3
[1,3,2] => [2,3,1] => 101001 => 1
[1,4,1] => [1,4,1] => 110001 => 2
[1,5] => [5,1] => 100001 => 1
[2,1,1,1,1] => [1,1,1,1,2] => 111110 => 5
[2,1,1,2] => [2,1,1,2] => 101110 => 1
[2,1,2,1] => [1,2,1,2] => 110110 => 2
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 2
[1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1,1] => 1110111111 => ? = 3
[1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => 1010111111 => ? = 1
[1,1,1,1,1,1,3,1] => [1,3,1,1,1,1,1,1] => 1100111111 => ? = 2
[1,1,1,1,1,1,4] => [4,1,1,1,1,1,1] => 1000111111 => ? = 1
[1,1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1,1] => 1111011111 => ? = 4
[1,1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 5
[1,1,1,2,1,1,1,1,1] => [1,1,1,1,1,2,1,1,1] => 1111110111 => ? = 6
[1,1,1,2,1,1,1,2] => [2,1,1,1,2,1,1,1] => 1011110111 => ? = 1
[1,1,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1,1] => 1111111011 => ? = 7
[1,1,6,2] => [2,6,1,1] => 1010000011 => ? = 1
[1,2,2,2,2,1] => [1,2,2,2,2,1] => 1101010101 => ? = 2
[1,7,2] => [2,7,1] => 1010000001 => ? = 1
[1,8,1] => [1,8,1] => 1100000001 => ? = 2
[1,9] => [9,1] => 1000000001 => ? = 1
[2,6,2] => [2,6,2] => 1010000010 => ? = 1
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,3] => 1111111100 => ? = 8
[4,1,1,1,1,1,1] => [1,1,1,1,1,1,4] => 1111111000 => ? = 7
[4,6] => [6,4] => 1000001000 => ? = 1
[5,1,1,1,1,1] => [1,1,1,1,1,5] => 1111110000 => ? = 6
[5,5] => [5,5] => 1000010000 => ? = 1
[6,4] => [4,6] => 1000100000 => ? = 1
[7,1,1,1] => [1,1,1,7] => 1111000000 => ? = 4
[7,3] => [3,7] => 1001000000 => ? = 1
[8,1,1] => [1,1,8] => 1110000000 => ? = 3
[8,2] => [2,8] => 1010000000 => ? = 1
[9,1] => [1,9] => 1100000000 => ? = 2
[1,10] => [10,1] => 10000000001 => ? = 1
[10,1] => [1,10] => 11000000000 => ? = 2
[1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 111111111111 => ? = 12
[1,2,2,2,2,2,1] => [1,2,2,2,2,2,1] => 110101010101 => ? = 2
[1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
Description
The number of leading ones in a binary word.
Matching statistic: St000745
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 91%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 91%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 1
[1,1] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2
[2] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 2
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [[1,2],[3],[4],[5],[6]]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [[1,3],[2],[4],[5],[6]]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [[1,2,3],[4],[5],[6]]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [[1,2],[3,4],[5],[6]]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [[1,3,4],[2],[5],[6]]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [[1,2,3,4],[5],[6]]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [[1,5],[2],[3],[4],[6]]
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [[1,2],[3,5],[4],[6]]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [[1,2,3],[4,5],[6]]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [[1,4,5],[2],[3],[6]]
=> 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [[1,2,5],[3,4],[6]]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [[1,3,4,5],[2],[6]]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,2,3,4,5],[6]]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [[1,2],[3,6],[4],[5]]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [[1,3],[2,6],[4],[5]]
=> 2
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,2,1] => ?
=> ? = 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 1
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,2,1] => ?
=> ? = 1
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,2,3,1] => ?
=> ? = 1
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,2,3,1] => ?
=> ? = 1
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,2,3,4,1] => ?
=> ? = 1
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,5,1,2] => ?
=> ? = 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,1,2,3] => ?
=> ? = 2
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,1,2,3] => ?
=> ? = 3
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [9,7,8,6,5,4,3,2,1] => [[1,3],[2],[4],[5],[6],[7],[8],[9]]
=> ? = 2
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [9,8,6,7,5,4,3,2,1] => ?
=> ? = 3
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [9,6,7,8,5,4,3,2,1] => [[1,3,4],[2],[5],[6],[7],[8],[9]]
=> ? = 2
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,6,4,3,2,1] => [[1,5],[2],[3],[4],[6],[7],[8],[9]]
=> ? = 4
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [9,8,5,6,7,4,3,2,1] => ?
=> ? = 3
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,5,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9]]
=> ? = 5
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [9,8,7,4,5,6,3,2,1] => [[1,5,6],[2],[3],[4],[7],[8],[9]]
=> ? = 4
[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,3,4,2,1] => [[1,7],[2],[3],[4],[5],[6],[8],[9]]
=> ? = 6
[1,1,2,1,1,3] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [7,8,9,6,5,3,4,2,1] => ?
=> ? = 1
[1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,3,4,5,2,1] => ?
=> ? = 5
[1,1,5,2] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [8,9,3,4,5,6,7,2,1] => ?
=> ? = 1
[1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,2,3,1] => [[1,8],[2],[3],[4],[5],[6],[7],[9]]
=> ? = 7
[1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,2,3,1] => [[1,2],[3,8],[4],[5],[6],[7],[9]]
=> ? = 1
[1,2,1,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,8,9,6,5,4,2,3,1] => ?
=> ? = 1
[1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,2,3,4,1] => [[1,7,8],[2],[3],[4],[5],[6],[9]]
=> ? = 6
[1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [8,9,2,3,4,5,6,7,1] => [[1,2,5,6,7,8],[3,4],[9]]
=> ? = 1
[2,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,3,1,2] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? = 1
[2,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [9,7,8,6,5,4,3,1,2] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
=> ? = 2
[2,1,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,8,9,6,5,4,3,1,2] => ?
=> ? = 1
[2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [6,7,8,9,5,4,3,1,2] => ?
=> ? = 1
[2,1,1,5] => [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [5,6,7,8,9,4,3,1,2] => ?
=> ? = 1
[2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5,6,7,8,9,3,1,2] => [[1,2,3,4,5,6],[7,9],[8]]
=> ? = 1
[2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [5,6,7,8,9,3,4,1,2] => ?
=> ? = 1
[2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [8,9,3,4,5,6,7,1,2] => [[1,2,5,6,7],[3,4],[8,9]]
=> ? = 1
[2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,8,1,2] => [[1,3,4,5,6,7],[2,9],[8]]
=> ? = 2
[3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,8,9,6,5,4,1,2,3] => ?
=> ? = 1
[3,4,2] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [8,9,4,5,6,7,1,2,3] => ?
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [10,8,9,7,6,5,4,3,2,1] => [[1,3],[2],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 2
[1,1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [10,9,7,8,6,5,4,3,2,1] => ?
=> ? = 3
[1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [9,10,7,8,6,5,4,3,2,1] => ?
=> ? = 1
[1,1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [10,7,8,9,6,5,4,3,2,1] => ?
=> ? = 2
[1,1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [10,9,8,6,7,5,4,3,2,1] => [[1,5],[2],[3],[4],[6],[7],[8],[9],[10]]
=> ? = 4
[1,1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
=> ? = 5
[1,1,1,2,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,6,4,5,3,2,1] => ?
=> ? = 6
[1,1,1,2,1,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [9,10,8,7,6,4,5,3,2,1] => ?
=> ? = 1
[1,1,2,1,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,6,5,3,4,2,1] => ?
=> ? = 7
[1,1,6,2] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [9,10,3,4,5,6,7,8,2,1] => ?
=> ? = 1
[1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,6,5,4,2,3,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8],[10]]
=> ? = 8
[1,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [10,8,9,6,7,4,5,2,3,1] => [[1,3],[2,5],[4,7],[6,9],[8],[10]]
=> ? = 2
[1,7,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [9,10,2,3,4,5,6,7,8,1] => ?
=> ? = 1
[1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,5,6,7,8,9,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000439
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 91%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 91%
Values
[1] => [1,0]
=> [1,0]
=> 2 = 1 + 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 4 = 3 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 6 = 5 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 4 + 1
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 4 + 1
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 5 + 1
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4 + 1
[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 6 + 1
[1,1,2,1,1,3] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 5 + 1
[1,1,5,2] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 1
[1,2,1,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6 + 1
[2,1,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,1,1,4] => [1,1,0,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[2,1,1,5] => [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1 + 1
[2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 + 1
[3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[3,2,1,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 5 + 1
[3,4,2] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
[4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 1
[4,2,1,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4 + 1
[4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 5 + 1
[5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 1
[5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
[6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000383
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 82%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 82%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [2] => 2
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3] => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,1] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,1,1,1,1,1,1] => [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]
=> [8] => ? = 8
[1,1,1,1,1,2,1] => [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]
=> [6,2] => ? = 2
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,2] => ? = 2
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [5,3] => ? = 3
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ? = 5
[1,1,5,1] => [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]
=> [6,2] => ? = 2
[1,2,1,1,1,1,1] => [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]
=> [2,6] => ? = 6
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,5,1] => ? = 1
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,5,1] => ? = 1
[1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,5,1] => ? = 1
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,5,1] => ? = 1
[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,5] => ? = 5
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [6,2] => ? = 2
[2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,3,4] => ? = 4
[2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,5] => ? = 5
[2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4] => ? = 4
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,6] => ? = 6
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,5,1] => ? = 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,5,1] => ? = 1
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,5,1] => ? = 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,2,4] => ? = 4
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,5,1] => ? = 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,5] => ? = 5
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [4,2,2] => ? = 2
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [5,3] => ? = 3
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,2] => ? = 2
[1,1,1,1,1,1,1,1,1] => [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]
=> [9] => ? = 9
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,2] => ? = 2
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [6,3] => ? = 3
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [7,2] => ? = 2
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [5,4] => ? = 4
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]
=> [6,3] => ? = 3
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5] => ? = 5
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,4] => ? = 4
[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,6] => ? = 6
[1,1,2,1,1,3] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,5,1] => ? = 1
[1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5] => ? = 5
[1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,7] => ? = 7
[1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,6,1] => ? = 1
[1,2,1,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? => ? = 1
[1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,6] => ? = 6
[1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [7,2] => ? = 2
[2,2,1,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,6] => ? = 6
[2,2,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? => ? = 1
[2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? => ? = 1
[3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,7] => ? = 7
[3,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [2,6,1] => ? = 1
[3,2,1,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? => ? = 5
[3,4,2] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000678
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 73%
St000678: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 73%
Values
[1] => [1,0]
=> ? = 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 3
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,2,1,1,3] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,5,2] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,2,1,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000617
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 82%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 82%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,1] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 2
[2] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 2
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 3
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,2,2,1] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[1,1,2,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,3,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,1,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,1,2,1] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2
[1,2,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[1,2,2,1,1] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[1,2,3,1] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[1,3,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[2,1,1,3] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,2,2] => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[2,1,3,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[3,1,2,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[3,1,3] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
[3,2,1,1] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3
[4,1,2] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,1,1,1,1,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,2,1,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[1,1,1,1,2,2] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,3,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,3,2] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,2,1,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,2,2,1,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3
[1,1,2,4] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,3,1,2] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 1
[1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[1,1,4,2] => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,5,1] => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,1,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,2,1,1,2,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[1,2,3,2] => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 1
[1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 1
[1,3,1,1,2] => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 1
[1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[1,3,2,2] => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 1
[1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 1
[2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,2,2] => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,4] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The number of global maxima of a Dyck path.
The following 154 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000700The protection number of an ordered tree. St000010The length of the partition. St001176The size of a partition minus its first part. St000993The multiplicity of the largest part of an integer partition. St000759The smallest missing part in an integer partition. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000971The smallest closer of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000883The number of longest increasing subsequences of a permutation. St000505The biggest entry in the block containing the 1. St000733The row containing the largest entry of a standard tableau. St001050The number of terminal closers of a set partition. St000054The first entry of the permutation. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001733The number of weak left to right maxima of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001691The number of kings in a graph. St001829The common independence number of a graph. St000237The number of small exceedances. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000069The number of maximal elements of a poset. St000908The length of the shortest maximal antichain in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St000234The number of global ascents of a permutation. St000363The number of minimal vertex covers of a graph. St000287The number of connected components of a graph. St000553The number of blocks of a graph. St000917The open packing number of a graph. St001672The restrained domination number of a graph. St000068The number of minimal elements in a poset. St001316The domatic number of a graph. St001828The Euler characteristic of a graph. St000914The sum of the values of the Möbius function of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000286The number of connected components of the complement of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000501The size of the first part in the decomposition of a permutation. St000989The number of final rises of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000542The number of left-to-right-minima of a permutation. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000181The number of connected components of the Hasse diagram for the poset. St001461The number of topologically connected components of the chord diagram of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000843The decomposition number of a perfect matching. St000738The first entry in the last row of a standard tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000455The second largest eigenvalue of a graph if it is integral. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000740The last entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000335The difference of lower and upper interactions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000699The toughness times the least common multiple of 1,. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000258The burning number of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000061The number of nodes on the left branch of a binary tree. St001340The cardinality of a minimal non-edge isolating set of a graph. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St000133The "bounce" of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St001530The depth of a Dyck path. St000264The girth of a graph, which is not a tree. St000756The sum of the positions of the left to right maxima of a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
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