searching the database
Your data matches 929 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St001232
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00118: Dyck paths āswap returns and last descentā¶ Dyck paths
St001232: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00118: Dyck paths āswap returns and last descentā¶ Dyck paths
St001232: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[]
=> []
=> [1,0]
=> [1,0]
=> 0
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001502
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001502: Dyck paths ā¶ ā¤Result quality: 80% āvalues known / values provided: 97%ādistinct values known / distinct values provided: 80%
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St001502: Dyck paths ā¶ ā¤Result quality: 80% āvalues known / values provided: 97%ādistinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 6 = 7 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 6 = 7 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6 = 7 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 8 = 9 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> 6 = 7 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> 6 = 7 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> 8 = 9 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> 6 = 7 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> 6 = 7 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> 8 = 9 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 6 = 7 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> 8 = 9 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[]
=> []
=> []
=> [1,0]
=> ? = 0 - 1
Description
The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras.
We use the code below to translate them to Dyck paths.
The algebras where the statistic returns 0 are exactly the higher Auslander algebras and are of special interest. It seems like they are counted by the number of divisors function.
Matching statistic: St001766
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
Mp00089: Permutations āInverse Kreweras complementā¶ Permutations
St001766: Permutations ā¶ ā¤Result quality: 80% āvalues known / values provided: 97%ādistinct values known / distinct values provided: 80%
Mp00024: Dyck paths āto 321-avoiding permutationā¶ Permutations
Mp00089: Permutations āInverse Kreweras complementā¶ Permutations
St001766: Permutations ā¶ ā¤Result quality: 80% āvalues known / values provided: 97%ādistinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 6 = 7 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 6 = 7 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 4 = 5 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => 6 = 7 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 8 = 9 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 6 = 7 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 6 = 7 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 8 = 9 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => [5,6,1,3,4,2] => 4 = 5 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => [5,6,1,3,4,2] => 4 = 5 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 6 = 7 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => [5,6,1,3,4,2] => 4 = 5 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 6 = 7 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 8 = 9 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => [5,6,1,3,4,2] => 4 = 5 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 6 = 7 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 8 = 9 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => 0 = 1 - 1
[]
=> []
=> [] => [] => ? = 0 - 1
Description
The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation.
Precisely the 132-avoiding permutations have only one associated reduced pipe dream.
Matching statistic: St000976
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
St000976: Dyck paths ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Mp00032: Dyck paths āinverse zeta mapā¶ Dyck paths
St000976: Dyck paths ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 7
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 7
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 9
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 9
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 5
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 5
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 5
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 9
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 5
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 9
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[]
=> []
=> []
=> ? = 0
Description
The sum of the positions of double up-steps of a Dyck path.
This is part of MacMahon's equal index of a word, see [1, p. 135].
Matching statistic: St000795
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00101: Dyck paths ādecomposition reverseā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St000795: Permutations ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Mp00101: Dyck paths ādecomposition reverseā¶ Dyck paths
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
St000795: Permutations ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 7
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 7
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 5
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 7
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => 9
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 7
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 7
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 7
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => 9
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => 5
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => 5
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => 5
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 7
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => 9
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => 5
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,1,4,5,2,6] => 7
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => 9
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
[]
=> []
=> []
=> [] => ? = 0
Description
The mad of a permutation.
According to [1], this is the sum of twice the number of occurrences of the vincular pattern of $(2\underline{31})$ plus the number of occurrences of the vincular patterns $(\underline{31}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000874
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00028: Dyck paths āreverseā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St000874: Dyck paths ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Mp00028: Dyck paths āreverseā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St000874: Dyck paths ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 7
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 7
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 5
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[]
=> []
=> []
=> []
=> ? = 0
Description
The position of the last double rise in a Dyck path.
If the Dyck path has no double rises, this statistic is $0$.
Matching statistic: St000830
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00073: Permutations āmajor-index to inversion-number bijectionā¶ Permutations
St000830: Permutations ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00073: Permutations āmajor-index to inversion-number bijectionā¶ Permutations
St000830: Permutations ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 1 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,1,3,2] => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,1,3,2] => 6 = 5 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => 8 = 7 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => 6 = 5 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => 8 = 7 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [6,1,2,3,5,4] => 10 = 9 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8 = 7 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [6,1,2,3,5,4] => 10 = 9 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,5,1,2,3] => [1,2,6,3,5,4] => 6 = 5 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,5,1,2,3] => [1,2,6,3,5,4] => 6 = 5 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8 = 7 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,5,1,2,3] => [1,2,6,3,5,4] => 6 = 5 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [6,1,2,3,5,4] => 10 = 9 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,5,1,2,3] => [1,2,6,3,5,4] => 6 = 5 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,6,2,3,5,4] => 8 = 7 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [6,1,2,3,5,4] => 10 = 9 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 2 = 1 + 1
[]
=> []
=> [] => [] => ? = 0 + 1
Description
The total displacement of a permutation.
This is, for a permutation $\pi$ of $n$, given by $\sum_{i = 1}^n | \pi(i) - i |.$
This is twice the statistic [[St000029]] and can be found in [3, Problem 5.1.1.28] and also in [1, 2].
Matching statistic: St000866
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000866: Permutations ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Mp00120: Dyck paths āLalanne-Kreweras involutionā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St000866: Permutations ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => ? = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 6 = 7 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 6 = 7 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4 = 5 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 6 = 7 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 8 = 9 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 6 = 7 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 6 = 7 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 8 = 9 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => 4 = 5 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => 4 = 5 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 6 = 7 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => 4 = 5 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 6 = 7 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 8 = 9 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => 4 = 5 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 6 = 7 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 8 = 9 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => 0 = 1 - 1
[]
=> []
=> []
=> [] => ? = 0 - 1
Description
The number of admissible inversions of a permutation in the sense of Shareshian-Wachs.
An admissible inversion of a permutation $\sigma$ is a pair $(\sigma_i,\sigma_j)$ such that
1. $i < j$ and $\sigma_i > \sigma_j$ and 2. either $\sigma_j < \sigma_{j+1}$ or there exists a $i < k < j$ with $\sigma_k < \sigma_j$.
This version was introduced by John Shareshian and Michelle L. Wachs in [1], for a closely related version, see [[St000463]].
Matching statistic: St001365
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St001365: Binary words ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Mp00023: Dyck paths āto non-crossing permutationā¶ Permutations
Mp00109: Permutations ādescent wordā¶ Binary words
St001365: Binary words ā¶ ā¤Result quality: 80% āvalues known / values provided: 93%ādistinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 1 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0 => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 6 = 5 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 8 = 7 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 8 = 7 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 6 = 5 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 8 = 7 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 01110 => 10 = 9 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 10110 => 8 = 7 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 10110 => 8 = 7 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 10110 => 8 = 7 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 01110 => 10 = 9 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 11010 => 6 = 5 + 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 11010 => 6 = 5 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 10110 => 8 = 7 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 11010 => 6 = 5 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 10110 => 8 = 7 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 01110 => 10 = 9 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 11010 => 6 = 5 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 10110 => 8 = 7 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 01110 => 10 = 9 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 11110 => 2 = 1 + 1
[]
=> []
=> [] => ? => ? = 0 + 1
Description
The number of lattice paths of the same length weakly above the path given by a binary word.
In particular, there are $2^n$ lattice paths weakly above the the length $n$ binary word $0\dots 0$, there is a unique path weakly above $1\dots 1$, and there are $\binom{2n}{n}$ paths weakly above the length $2n$ binary word $10\dots 10$.
Matching statistic: St000435
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St000435: Permutations ā¶ ā¤Result quality: 60% āvalues known / values provided: 83%ādistinct values known / distinct values provided: 60%
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St000435: Permutations ā¶ ā¤Result quality: 60% āvalues known / values provided: 83%ādistinct values known / distinct values provided: 60%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => 6 = 7 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => 6 = 7 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => 4 = 5 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => 6 = 7 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 9 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,3,7,5,6] => 6 = 7 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,3,7,5,6] => 6 = 7 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,3,7,5,6] => 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 9 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,2,5,3,7,4,6] => 4 = 5 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,2,5,3,7,4,6] => 4 = 5 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,3,7,5,6] => 6 = 7 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,2,5,3,7,4,6] => 4 = 5 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,3,7,5,6] => 6 = 7 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 9 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,2,5,3,7,4,6] => 4 = 5 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,3,7,5,6] => 6 = 7 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,2,4,7,5,6] => ? = 9 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => 0 = 1 - 1
[]
=> []
=> [1,0]
=> [1] => ? = 0 - 1
Description
The number of occurrences of the pattern 213 or of the pattern 231 in a permutation.
The following 919 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000909The number of maximal chains of maximal size in a poset. St001268The size of the largest ordinal summand in the poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000910The number of maximal chains of minimal length in a poset. St000002The number of occurrences of the pattern 123 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000463The number of admissible inversions of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000849The number of 1/3-balanced pairs in a poset. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001485The modular major index of a binary word. St000068The number of minimal elements in a poset. St000307The number of rowmotion orbits of a poset. St000691The number of changes of a binary word. St000847The number of standard Young tableaux whose descent set is the binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001638The book thickness of a graph. St001768The number of reduced words of a signed permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000215The number of adjacencies of a permutation, zero appended. St000218The number of occurrences of the pattern 213 in a permutation. St000264The girth of a graph, which is not a tree. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000921The number of internal inversions of a binary word. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001545The second Elser number of a connected graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001817The number of flag weak exceedances of a signed permutation. St001964The interval resolution global dimension of a poset. St000100The number of linear extensions of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001896The number of right descents of a signed permutations. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000422The energy of a graph, if it is integral. St000741The Colin de VerdiĆØre graph invariant. St001624The breadth of a lattice. St001870The number of positive entries followed by a negative entry in a signed permutation. St000090The variation of a composition. St001413Half the length of the longest even length palindromic prefix of a binary word. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St000383The last part of an integer composition. St000001The number of reduced words for a permutation. St000007The number of saliances of the permutation. St000022The number of fixed points of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000054The first entry of the permutation. St000091The descent variation of a composition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000535The rank-width of a graph. St000546The number of global descents of a permutation. St000635The number of strictly order preserving maps of a poset into itself. St000662The staircase size of the code of a permutation. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000696The number of cycles in the breakpoint graph of a permutation. St000703The number of deficiencies of a permutation. St000729The minimal arc length of a set partition. St000730The maximal arc length of a set partition. St000742The number of big ascents of a permutation after prepending zero. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000839The largest opener of a set partition. St000862The number of parts of the shifted shape of a permutation. St000872The number of very big descents of a permutation. St000884The number of isolated descents of a permutation. St000908The length of the shortest maximal antichain in a poset. St000925The number of topologically connected components of a set partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St001414Half the length of the longest odd length palindromic prefix 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. St001423The number of distinct cubes in a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001712The number of natural descents of a standard Young tableau. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001730The number of times the path corresponding to a binary word crosses the base line. St001743The discrepancy of a graph. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001821The sorting index of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001890The maximum magnitude of the Mƶbius function of a poset. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000043The number of crossings plus two-nestings of a perfect matching. St000052The number of valleys of a Dyck path not on the x-axis. St000119The number of occurrences of the pattern 321 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St000258The burning number of a graph. St000273The domination number of a graph. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000417The size of the automorphism group of the ordered tree. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000490The intertwining number of a set partition. St000491The number of inversions of a set partition. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000534The number of 2-rises of a permutation. St000544The cop number of a graph. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000630The length of the shortest palindromic decomposition of a binary word. St000648The number of 2-excedences of a permutation. St000657The smallest part of an integer composition. St000665The number of rafts of a permutation. St000676The number of odd rises of a Dyck path. St000679The pruning number of an ordered tree. St000731The number of double exceedences of a permutation. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000758The length of the longest staircase fitting into an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000842The breadth of a permutation. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000891The number of distinct diagonal sums of a permutation matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000905The number of different multiplicities of parts of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St000916The packing number of a graph. St000942The number of critical left to right maxima of the parking functions. St000983The length of the longest alternating subword. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001058The breadth of the ordered tree. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001115The number of even descents of a permutation. St001261The Castelnuovo-Mumford regularity of a graph. St001267The length of the Lyndon factorization of the binary word. St001298The number of repeated entries in the Lehmer code of a permutation. St001301The first Betti number of the order complex associated with the poset. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001363The Euler characteristic of a graph according to Knill. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001462The number of factors of a standard tableaux under concatenation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001625The Mƶbius invariant of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001720The minimal length of a chain of small intervals in a lattice. St001732The number of peaks visible from the left. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001829The common independence number of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000381The largest part of an integer composition. St000808The number of up steps of the associated bargraph. St000893The number of distinct diagonal sums of an alternating sign matrix. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000210Minimum over maximum difference of elements in cycles. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St001081The number of minimal length factorizations of a permutation into star transpositions. St001162The minimum jump of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001461The number of topologically connected components of the chord diagram of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001928The number of non-overlapping descents in a permutation. St000039The number of crossings of a permutation. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000089The absolute variation of a composition. St000105The number of blocks in the set partition. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000338The number of pixed points of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000487The length of the shortest cycle of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of a permutation. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000037The sign of a permutation. St000023The number of inner peaks of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000060The greater neighbor of the maximum. St000062The length of the longest increasing subsequence of the permutation. St000069The number of maximal elements of a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000115The single entry in the last row. St000133The "bounce" of a permutation. St000136The dinv of a parking function. St000155The number of exceedances (also excedences) of a permutation. St000174The flush statistic of a semistandard tableau. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000214The number of adjacencies of a permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000238The number of indices that are not small weak excedances. St000239The number of small weak excedances. St000245The number of ascents of a permutation. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000304The load of a permutation. St000308The height of the tree associated to a permutation. St000310The minimal degree of a vertex of a graph. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000363The number of minimal vertex covers of a graph. St000454The largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St000472The sum of the ascent bottoms of a permutation. St000488The number of cycles of a permutation of length at most 2. St000501The size of the first part in the decomposition of a permutation. St000502The number of successions of a set partitions. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000536The pathwidth of a graph. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000619The number of cyclic descents of a permutation. St000632The jump number of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000652The maximal difference between successive positions of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000717The number of ordinal summands of a poset. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000834The number of right outer peaks of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000868The aid statistic in the sense of Shareshian-Wachs. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000883The number of longest increasing subsequences of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000889The number of alternating sign matrices with the same antidiagonal sums. St000914The sum of the values of the Mƶbius function of a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000948The chromatic discriminant of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000959The number of strong Bruhat factorizations of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001061The number of indices that are both descents and recoils of a permutation. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001119The length of a shortest maximal path in a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001151The number of blocks with odd minimum. St001153The number of blocks with even minimum in a set partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001220The width of a permutation. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001246The maximal difference between two consecutive entries of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001260The permanent of an alternating sign matrix. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001271The competition number of a graph. St001272The number of graphs with the same degree sequence. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001277The degeneracy of a graph. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001316The domatic number of a graph. St001323The independence gap of a graph. St001346The number of parking functions that give the same permutation. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001375The pancake length of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001405The number of bonds in a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001427The number of descents of a signed permutation. St001463The number of distinct columns in the nullspace of a graph. St001468The smallest fixpoint of a permutation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001489The maximum of the number of descents and the number of inverse descents. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001546The number of monomials in the Tutte polynomial of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001565The number of arithmetic progressions of length 2 in a permutation. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001592The maximal number of simple paths between any two different vertices of a graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001597The Frobenius rank of a skew partition. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001735The number of permutations with the same set of runs. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001765The number of connected components of the friends and strangers graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001807The lower middle entry of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001842The major index of a set partition. St001850The number of Hecke atoms of a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001862The number of crossings of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001949The rigidity index of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St001958The degree of the polynomial interpolating the values of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000058The order of a permutation. St000061The number of nodes on the left branch of a binary tree. St000064The number of one-box pattern of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000077The number of boxed and circled entries. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000166The depth minus 1 of an ordered tree. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000213The number of weak exceedances (also weak excedences) of a permutation. St000222The number of alignments in the permutation. St000226The convexity of a permutation. St000244The cardinality of the automorphism group of a graph. St000252The number of nodes of degree 3 of a binary tree. St000268The number of strongly connected orientations of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000341The non-inversion sum of a permutation. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000364The exponent of the automorphism group of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000386The number of factors DDU in a Dyck path. St000401The size of the symmetry class of a permutation. St000402Half the size of the symmetry class of a permutation. St000403The Szeged index minus the Wiener index of a graph. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000469The distinguishing number of a graph. St000471The sum of the ascent tops of a permutation. St000482The (zero)-forcing number of a graph. St000489The number of cycles of a permutation of length at most 3. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St000538The number of even inversions of a permutation. St000552The number of cut vertices of a graph. St000624The normalized sum of the minimal distances to a greater element. St000636The hull number of a graph. St000637The length of the longest cycle in a graph. St000638The number of up-down runs of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000674The number of hills of a Dyck path. St000680The Grundy value for Hackendot on posets. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000699The toughness times the least common multiple of 1,. St000702The number of weak deficiencies of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000873The aix statistic of a permutation. St000894The trace of an alternating sign matrix. St000895The number of ones on the main diagonal of an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000931The number of occurrences of the pattern UUU in a Dyck path. St000943The number of spots the most unlucky car had to go further in a parking function. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. 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 $c_{n-1}$. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. 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 $eA$. St001204Call 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. 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 $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001281The normalized isoperimetric number of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001374The Padmakar-Ivan index of a graph. St001377The major index minus the number of inversions of a permutation. St001388The number of non-attacking neighbors of a permutation. St001399The distinguishing number of a poset. St001411The number of patterns 321 or 3412 in a permutation. St001433The flag major index of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001488The number of corners of a skew partition. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001510The number of self-evacuating linear extensions of a finite poset. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001516The number of cyclic bonds of a permutation. St001517The length of a longest pair of twins in a permutation. St001519The pinnacle sum of a permutation. St001530The depth of a Dyck path. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001535The number of cyclic alignments of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001580The acyclic chromatic number of a graph. St001596The number of two-by-two squares inside a skew partition. St001621The number of atoms of a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001644The dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001654The monophonic hull number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001691The number of kings in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001716The 1-improper chromatic number of a graph. St001718The number of non-empty open intervals in a poset. St001727The number of invisible inversions of a permutation. St001731The factorization defect of a permutation. St001734The lettericity of a graph. St001736The total number of cycles in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001819The flag Denert index of a signed permutation. St001822The number of alignments of a signed permutation. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001871The number of triconnected components of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001911A descent variant minus the number of inversions. St001927Sparre Andersen's number of positives of a signed permutation. St001947The number of ties in a parking function. St000094The depth of an ordered tree. St000187The determinant of an alternating sign matrix. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001060The distinguishing index of a graph. St001108The 2-dynamic chromatic number of a graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000981The length of the longest zigzag subpath. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001875The number of simple modules with projective dimension at most 1. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2.
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