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Your data matches 111 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
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(load all 3 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1 = 2 - 1
([],2)
=> [2] => 2 = 3 - 1
([(0,1)],2)
=> [1,1] => 1 = 2 - 1
([],3)
=> [3] => 3 = 4 - 1
([(1,2)],3)
=> [1,2] => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2 = 3 - 1
([],4)
=> [4] => 4 = 5 - 1
([(2,3)],4)
=> [1,3] => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 3 = 4 - 1
([],5)
=> [5] => 5 = 6 - 1
([(3,4)],5)
=> [1,4] => 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2 = 3 - 1
Description
The first part of an integer composition.
Matching statistic: St000439
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 2
([],2)
=> [2] => [1,1,0,0]
=> 3
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 2
([],3)
=> [3] => [1,1,1,0,0,0]
=> 4
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 3
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 5
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
Description
The position of the first down step of a Dyck path.
Matching statistic: St000025
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 1 = 2 - 1
([],2)
=> [2] => [1,1,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1 = 2 - 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 1 = 2 - 1
([],2)
=> [2] => [1,1,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1 = 2 - 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000383
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => 1 = 2 - 1
([],2)
=> [2] => [2] => 2 = 3 - 1
([(0,1)],2)
=> [1,1] => [1,1] => 1 = 2 - 1
([],3)
=> [3] => [3] => 3 = 4 - 1
([(1,2)],3)
=> [1,2] => [2,1] => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2 = 3 - 1
([],4)
=> [4] => [4] => 4 = 5 - 1
([(2,3)],4)
=> [1,3] => [3,1] => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [2,1,1] => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,2,1] => 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [2,2] => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,2,1] => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,3] => 3 = 4 - 1
([],5)
=> [5] => [5] => 5 = 6 - 1
([(3,4)],5)
=> [1,4] => [4,1] => 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,3,1] => 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => [3,2] => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [3,2] => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [2,1,1,1] => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,2,1,1] => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,2,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [2,1,2] => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,1,2,1] => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,1,1,1,1] => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2 = 3 - 1
Description
The last part of an integer composition.
Matching statistic: St000011
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> [1,0]
=> 1 = 2 - 1
([],2)
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [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]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [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]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [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]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [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]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [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]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
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: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => 1 => 1 = 2 - 1
([],2)
=> [2] => [1,1] => 11 => 2 = 3 - 1
([(0,1)],2)
=> [1,1] => [2] => 10 => 1 = 2 - 1
([],3)
=> [3] => [1,1,1] => 111 => 3 = 4 - 1
([(1,2)],3)
=> [1,2] => [2,1] => 101 => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [3] => 100 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,2] => 110 => 2 = 3 - 1
([],4)
=> [4] => [1,1,1,1] => 1111 => 4 = 5 - 1
([(2,3)],4)
=> [1,3] => [2,1,1] => 1011 => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [3,1] => 1001 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,2] => 1010 => 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => 1101 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [4] => 1000 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => 1101 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => 1000 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,2] => 1010 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,3] => 1100 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,2] => 1110 => 3 = 4 - 1
([],5)
=> [5] => [1,1,1,1,1] => 11111 => 5 = 6 - 1
([(3,4)],5)
=> [1,4] => [2,1,1,1] => 10111 => 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [3,1,1] => 10011 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [2,2,1] => 10101 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [2,1,2] => 10110 => 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,2,1,1] => 11011 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [4,1] => 10001 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [4,1] => 10001 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,2,1,1] => 11011 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [4,1] => 10001 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [3,2] => 10010 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [2,2,1] => 10101 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => 11001 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [3,2] => 10010 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => 11001 => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,3] => 10100 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => 11010 => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,2,1] => 11101 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [2,3] => 10100 => 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,4] => 11000 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 11010 => 2 = 3 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000505
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> {{1}}
=> 1 = 2 - 1
([],2)
=> [2] => [1,1,0,0]
=> {{1,2}}
=> 2 = 3 - 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1 = 2 - 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 4 - 1
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 3 - 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 4 - 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 6 - 1
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2 = 3 - 1
Description
The biggest entry in the block containing the 1.
Matching statistic: St000971
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> {{1}}
=> 1 = 2 - 1
([],2)
=> [2] => [1,1,0,0]
=> {{1,2}}
=> 2 = 3 - 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1 = 2 - 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 4 - 1
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 3 - 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 4 - 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 6 - 1
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2 = 3 - 1
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St001135
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => [1,0]
=> 1 = 2 - 1
([],2)
=> [2] => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [1,1] => [2] => [1,1,0,0]
=> 1 = 2 - 1
([],3)
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
([(1,2)],3)
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
([],4)
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(2,3)],4)
=> [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
([],5)
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
([(3,4)],5)
=> [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
The following 101 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000504The cardinality of the first block of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000823The number of unsplittable factors of the set partition. St000273The domination number of a graph. St000363The number of minimal vertex covers of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001316The domatic number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000617The number of global maxima of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000054The first entry of the permutation. St001118The acyclic chromatic index of a graph. St001060The distinguishing index of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another 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. St001330The hat guessing number of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000983The length of the longest alternating subword. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. 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. St001267The length of the Lyndon factorization of the binary word. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001471The magnitude of a Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000015The number of peaks of a Dyck path. St000144The pyramid weight of the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001530The depth of a Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000738The first entry in the last row of a standard tableau. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000740The last entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000553The number of blocks of a graph. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000989The number of final rises of a permutation. St000061The number of nodes on the left branch of a binary tree. St000454The largest eigenvalue of a graph if it is integral. St001568The smallest positive integer that does not appear twice in the partition. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000668The least common multiple of the parts of the partition. 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. St000933The number of multipartitions of sizes given by an integer partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000654The first descent of a permutation. St000338The number of pixed points of a permutation. St001621The number of atoms of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000642The size of the smallest orbit of antichains under Panyushev complementation.
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