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Your data matches 182 different statistics following compositions of up to 3 maps.
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Matching statistic: St000013
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
St000013: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000013: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
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: St000381
(load all 49 compositions to match this statistic)
(load all 49 compositions to match this statistic)
Mp00102: Dyck paths ārise compositionā¶ Integer compositions
Mp00041: Integer compositions āconjugateā¶ Integer compositions
St000381: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00041: Integer compositions āconjugateā¶ Integer compositions
St000381: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,1] => [2] => 2
[1,1,0,0]
=> [2] => [1,1] => 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => 3
[1,0,1,1,0,0]
=> [1,2] => [1,2] => 2
[1,1,0,0,1,0]
=> [2,1] => [2,1] => 2
[1,1,0,1,0,0]
=> [2,1] => [2,1] => 2
[1,1,1,0,0,0]
=> [3] => [1,1,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 2
[1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => 2
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,2,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => [1,2,1,1] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,2,1,1] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [2,1,1,1] => 2
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [1,5] => 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,4] => 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [1,1,4] => 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [3,3] => 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [1,2,3] => 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => 3
Description
The largest part of an integer composition.
Matching statistic: St000451
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00087: Permutations āinverse first fundamental transformationā¶ Permutations
St000451: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00087: Permutations āinverse first fundamental transformationā¶ Permutations
St000451: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => 2
[1,1,0,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 3
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 4
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,3,2,1,5] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,1,2,3,6,5] => 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [4,1,2,3,5,6] => 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [3,1,2,6,4,5] => 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [3,1,2,5,4,6] => 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [3,1,2,4,6,5] => 3
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000141
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St000141: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St000141: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => 2 = 3 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000662
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St000662: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00031: Dyck paths āto 312-avoiding permutationā¶ Permutations
St000662: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => 2 = 3 - 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001090
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St001090: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St001090: Permutations ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => 2 = 3 - 1
Description
The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
Mp00060: Permutations āRobinson-Schensted tableau shapeā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
Mp00060: Permutations āRobinson-Schensted tableau shapeā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [1,1]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,2,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [2,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [2,2,1,1]
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [3,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => [2,2,2]
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => [3,2,1]
=> 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => [3,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
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: St000097
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00236: Permutations āClarke-Steingrimsson-Zeng inverseā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000097: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00236: Permutations āClarke-Steingrimsson-Zeng inverseā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000097: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
Mp00236: Permutations āClarke-Steingrimsson-Zeng inverseā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000098: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00236: Permutations āClarke-Steingrimsson-Zeng inverseā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000098: Graphs ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 3
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
The following 172 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000147The largest part of an integer partition. St000527The width of the poset. St000982The length of the longest constant subword. St001330The hat guessing number of a graph. St000028The number of stack-sorts needed to sort a permutation. St000245The number of ascents of a permutation. St000392The length of the longest run of ones in a binary word. St000439The position of the first down step of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001058The breadth of the ordered tree. St001809The index of the step at the first peak of maximal height in a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000686The finitistic dominant dimension of a Dyck path. St000444The length of the maximal rise of a Dyck path. St001062The maximal size of a block of a set partition. 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:
St000306The bounce count of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000025The number of initial rises of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000172The Grundy number of a graph. St000528The height of a poset. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001268The size of the largest ordinal summand in the poset. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001399The distinguishing number of a poset. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001883The mutual visibility number of a graph. St000053The number of valleys of the Dyck path. St000171The degree of the graph. St000272The treewidth of a graph. St000503The maximal difference between two elements in a common block. St000536The pathwidth of a graph. St000651The maximal size of a rise in a permutation. St000730The maximal arc length of a set partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001644The dimension of a graph. St001826The maximal number of leaves on a vertex of a graph. St000308The height of the tree associated to a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001343The dimension of the reduced incidence algebra of a poset. St000209Maximum difference of elements in cycles. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001652The length of a longest interval of consecutive numbers. St001717The largest size of an interval in a poset. St001963The tree-depth of a graph. St000956The maximal displacement of a permutation. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001530The depth of a Dyck path. 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. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000094The depth of an ordered tree. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000015The number of peaks of a Dyck path. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000542The number of left-to-right-minima of a permutation. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000021The number of descents of a permutation. St000080The rank of the poset. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001117The game chromatic index of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001118The acyclic chromatic index of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000652The maximal difference between successive positions of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000454The largest eigenvalue of a graph if it is integral. St000663The number of right floats of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001589The nesting number of a perfect matching. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001590The crossing number of a perfect matching. St000317The cycle descent number of a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001864The number of excedances of a signed permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001624The breadth of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000983The length of the longest alternating subword. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St000035The number of left outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001875The number of simple modules with projective dimension at most 1. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000834The number of right outer peaks of a permutation. St000352The Elizalde-Pak rank of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000153The number of adjacent cycles of a permutation. St000237The number of small exceedances. St001621The number of atoms of a lattice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000672The number of minimal elements in Bruhat order not less than the permutation. St001083The number of boxed occurrences of 132 in a permutation. St001435The number of missing boxes in the first row. St000264The girth of a graph, which is not a tree. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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 $eA$. 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 $eA$. St001060The distinguishing index of a graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001867The number of alignments of type EN of a signed permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000201The number of leaf nodes in a binary tree. St000214The number of adjacencies of a permutation. St000390The number of runs of ones in a binary word. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000291The number of descents of a binary word. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. 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 $eA$. St000071The number of maximal chains in a poset. St000850The number of 1/2-balanced pairs in a poset.
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