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Your data matches 95 different statistics following compositions of up to 3 maps.
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Matching statistic: St000381
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1
([],2)
=> [2] => 2
([(0,1)],2)
=> [1,1] => 1
([],3)
=> [3] => 3
([(1,2)],3)
=> [2,1] => 2
([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([],4)
=> [4] => 4
([(2,3)],4)
=> [3,1] => 3
([(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([],5)
=> [5] => 5
([(3,4)],5)
=> [4,1] => 4
([(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,3)],5)
=> [3,2] => 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000382
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ 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)
=> [2] => 2
([(0,1)],2)
=> [1,1] => 1
([],3)
=> [3] => 3
([(1,2)],3)
=> [2,1] => 2
([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([],4)
=> [4] => 4
([(2,3)],4)
=> [3,1] => 3
([(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3
([(0,3),(1,2)],4)
=> [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([],5)
=> [5] => 5
([(3,4)],5)
=> [4,1] => 4
([(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 4
([(1,4),(2,3)],5)
=> [3,2] => 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
Description
The first part of an integer composition.
Matching statistic: St000013
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 1
([],2)
=> [2] => [1,1,0,0]
=> 2
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 3
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(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
([(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
([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000025
Mp00324: Graphs —chromatic difference sequence⟶ 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)
=> [2] => [1,1,0,0]
=> 2
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 3
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(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
([(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
([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 2
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
Mp00324: Graphs —chromatic difference sequence⟶ 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)
=> [2] => [1,1,0,0]
=> 2
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 3
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(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
([(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
([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 2
Description
The position of the first return of a Dyck path.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1]
=> 1
([],2)
=> [2] => [2]
=> 2
([(0,1)],2)
=> [1,1] => [1,1]
=> 1
([],3)
=> [3] => [3]
=> 3
([(1,2)],3)
=> [2,1] => [2,1]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [2,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1]
=> 1
([],4)
=> [4] => [4]
=> 4
([(2,3)],4)
=> [3,1] => [3,1]
=> 3
([(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [3,1]
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [2,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1]
=> 1
([],5)
=> [5] => [5]
=> 5
([(3,4)],5)
=> [4,1] => [4,1]
=> 4
([(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [4,1]
=> 4
([(1,4),(2,3)],5)
=> [3,2] => [3,2]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [3,2]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [3,2]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [3,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [3,2]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [2,1,1,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000383
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ 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)
=> [2] => [2] => 2
([(0,1)],2)
=> [1,1] => [1,1] => 1
([],3)
=> [3] => [3] => 3
([(1,2)],3)
=> [2,1] => [1,2] => 2
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,1,1] => 1
([],4)
=> [4] => [4] => 4
([(2,3)],4)
=> [3,1] => [1,3] => 3
([(1,3),(2,3)],4)
=> [3,1] => [1,3] => 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,3] => 3
([(0,3),(1,2)],4)
=> [2,2] => [2,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,2] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,1,1,1] => 1
([],5)
=> [5] => [5] => 5
([(3,4)],5)
=> [4,1] => [1,4] => 4
([(2,4),(3,4)],5)
=> [4,1] => [1,4] => 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,4] => 4
([(1,4),(2,3)],5)
=> [3,2] => [2,3] => 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [2,3] => 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [2,3] => 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [2,3] => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [2,3] => 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,1,2] => 2
([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => 2
Description
The last part of an integer composition.
Matching statistic: St000723
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => ([],1)
=> 1
([],2)
=> [2] => ([],2)
=> 2
([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([],3)
=> [3] => ([],3)
=> 3
([(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> [4] => ([],4)
=> 4
([(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],5)
=> [5] => ([],5)
=> 5
([(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The maximal cardinality of a set of vertices with the same neighbourhood in a graph.
The set of so called mating graphs, for which this statistic equals $1$, is enumerated by [1].
Matching statistic: St001809
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> 1
([],2)
=> [2] => [1,1,0,0]
=> 2
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 3
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
([(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
([(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
([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 2
Description
The index of the step at the first peak of maximal height in a Dyck path.
Matching statistic: St000439
Mp00324: Graphs —chromatic difference sequence⟶ 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 = 1 + 1
([],2)
=> [2] => [1,1,0,0]
=> 3 = 2 + 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> 2 = 1 + 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
([(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
([(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
([(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
([(1,4),(2,3)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 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]
=> 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
([(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 = 2 + 1
([(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 = 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]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 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]
=> 3 = 2 + 1
Description
The position of the first down step of a Dyck path.
The following 85 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000273The domination number of a graph. St000297The number of leading ones in a binary word. St000363The number of minimal vertex covers of a graph. St000392The length of the longest run of ones in a binary word. St000544The cop number of a graph. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001415The length of the longest palindromic prefix of a binary word. St001829The common independence number of a graph. 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 $c_{n-1}$. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001322The size of a minimal independent dominating set in a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001316The domatic number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000741The Colin de Verdière graph invariant. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001656The monophonic position number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000482The (zero)-forcing number of a graph. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001286The annihilation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001530The depth of a Dyck path. St001654The monophonic hull number of a graph. St001949The rigidity index of a graph. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001655The general position number of a graph. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001330The hat guessing number of a graph. St000667The greatest common divisor of the parts of the partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001571The Cartan determinant of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000454The largest eigenvalue of a graph if it is integral. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a graph. St001651The Frankl number of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001890The maximum magnitude of the Möbius function of a poset.
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