Your data matches 491 different statistics following compositions of up to 3 maps.
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Mp00201: Dyck paths RingelPermutations
St000487: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => 2
[1,0,1,0]
=> [3,1,2] => 3
[1,1,0,0]
=> [2,3,1] => 3
[1,0,1,0,1,0]
=> [4,1,2,3] => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => 4
[1,1,0,0,1,0]
=> [2,4,1,3] => 4
[1,1,0,1,0,0]
=> [4,3,1,2] => 4
[1,1,1,0,0,0]
=> [2,3,4,1] => 4
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 5
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 5
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 5
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 5
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 5
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 5
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 5
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 5
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 6
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 6
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 6
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 6
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 6
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 6
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 6
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 6
Description
The length of the shortest cycle of a permutation.
Mp00201: Dyck paths RingelPermutations
Mp00151: Permutations to cycle typeSet partitions
St001075: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => {{1,2}}
=> 2
[1,0,1,0]
=> [3,1,2] => {{1,2,3}}
=> 3
[1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[1,0,1,0,1,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> 4
[1,1,0,0,1,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> 4
[1,1,0,1,0,0]
=> [4,3,1,2] => {{1,2,3,4}}
=> 4
[1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => {{1,2,3,4,5}}
=> 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => {{1,2,3,4,5}}
=> 5
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> 5
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => {{1,2,3,4,5}}
=> 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> 5
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> 5
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> 5
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => {{1,2,3,4,5}}
=> 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,3,5},{2,4}}
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => {{1,2,3,4,5}}
=> 5
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> 5
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => {{1,2,3,4,5}}
=> 5
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => {{1,2,3,4,5}}
=> 5
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => {{1,2,3,4,5,6}}
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => {{1,2,3,4,5,6}}
=> 6
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => {{1,2,3,4,5,6}}
=> 6
Description
The minimal size of a block of a set partition.
Mp00103: Dyck paths peeling mapDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St000117: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
Description
The number of centered tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.
Mp00103: Dyck paths peeling mapDyck paths
Mp00296: Dyck paths Knuth-KrattenthalerDyck paths
St000335: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
Description
The difference of lower and upper interactions. An ''upper interaction'' in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
Mp00103: Dyck paths peeling mapDyck paths
Mp00118: Dyck paths swap returns and last descentDyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
Description
The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].
Mp00103: Dyck paths peeling mapDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
St000685: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
Description
The dominant dimension of the LNakayama algebra associated to a Dyck path. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Mp00103: Dyck paths peeling mapDyck paths
Mp00118: Dyck paths swap returns and last descentDyck paths
St001265: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0 = 2 - 2
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 3 - 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
Description
The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra.
Mp00201: Dyck paths RingelPermutations
Mp00151: Permutations to cycle typeSet partitions
Mp00128: Set partitions to compositionInteger compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => {{1,2}}
=> [2] => 2
[1,0,1,0]
=> [3,1,2] => {{1,2,3}}
=> [3] => 3
[1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> [3] => 3
[1,0,1,0,1,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> [4] => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> [4] => 4
[1,1,0,0,1,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> [4] => 4
[1,1,0,1,0,0]
=> [4,3,1,2] => {{1,2,3,4}}
=> [4] => 4
[1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> [4] => 4
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,3,5},{2,4}}
=> [3,2] => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
=> [4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
=> [4,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
=> [3,3] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
=> [4,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
=> [3,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => {{1,2,3,4,5,6}}
=> [6] => 6
Description
The last part of an integer composition.
Mp00201: Dyck paths RingelPermutations
Mp00151: Permutations to cycle typeSet partitions
Mp00128: Set partitions to compositionInteger compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => {{1,2}}
=> [2] => 2
[1,0,1,0]
=> [3,1,2] => {{1,2,3}}
=> [3] => 3
[1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> [3] => 3
[1,0,1,0,1,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> [4] => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> [4] => 4
[1,1,0,0,1,0]
=> [2,4,1,3] => {{1,2,3,4}}
=> [4] => 4
[1,1,0,1,0,0]
=> [4,3,1,2] => {{1,2,3,4}}
=> [4] => 4
[1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> [4] => 4
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => {{1,3,5},{2,4}}
=> [3,2] => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => {{1,2,3,4,5}}
=> [5] => 5
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> [5] => 5
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
=> [4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
=> [4,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
=> [3,3] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
=> [4,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => {{1,2,3,4,5,6}}
=> [6] => 6
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
=> [3,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => {{1,2,3,4,5,6}}
=> [6] => 6
Description
The smallest part of an integer composition.
Matching statistic: St000982
Mp00201: Dyck paths RingelPermutations
Mp00108: Permutations cycle typeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000982: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2]
=> 100 => 2
[1,0,1,0]
=> [3,1,2] => [3]
=> 1000 => 3
[1,1,0,0]
=> [2,3,1] => [3]
=> 1000 => 3
[1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> 10000 => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> 10000 => 4
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> 10000 => 4
[1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> 10000 => 4
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> 10000 => 4
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> 100000 => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> 100000 => 5
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> 100000 => 5
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> 100000 => 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> 100000 => 5
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> 100000 => 5
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> 100000 => 5
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> 100000 => 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> 10100 => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> 100000 => 5
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> 100000 => 5
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> 100000 => 5
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> 100000 => 5
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 100000 => 5
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> 1000000 => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> 1000000 => 6
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> 1000000 => 6
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> 1000000 => 6
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> 1000000 => 6
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> 1000000 => 6
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> 1000000 => 6
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> 1000000 => 6
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> 100100 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> 1000000 => 6
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> 1000000 => 6
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> 1000000 => 6
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> 1000000 => 6
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6]
=> 1000000 => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> 1000000 => 6
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> 1000000 => 6
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> 1000000 => 6
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> 1000000 => 6
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [6]
=> 1000000 => 6
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> 1000000 => 6
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> 1000000 => 6
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> 100100 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> 11000 => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> 100100 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> 1000000 => 6
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> 1000000 => 6
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> 11000 => 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> 1000000 => 6
Description
The length of the longest constant subword.
The following 481 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. 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. 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. St001910The height of the middle non-run of a Dyck path. St001933The largest multiplicity of a part in an integer partition. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000221The number of strong fixed points of a permutation. St000236The number of cyclical small weak excedances. St000338The number of pixed points of a permutation. St000617The number of global maxima of a Dyck path. St000765The number of weak records in an integer composition. St000899The maximal number of repetitions of an integer composition. St000974The length of the trunk of an ordered tree. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in 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. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in 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. 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 $g$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001523The degree of symmetry of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000237The number of small exceedances. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000488The number of cycles of a permutation of length at most 2. St000654The first descent of a permutation. St000674The number of hills of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001820The size of the image of the pop stack sorting operator. St001637The number of (upper) dissectors of a poset. St001644The dimension of a graph. St001668The number of points of the poset minus the width of the poset. St000528The height of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000912The number of maximal antichains in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001626The number of maximal proper sublattices of a lattice. St001623The number of doubly irreducible elements of a lattice. St000080The rank of the poset. St001812The biclique partition number of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000451The length of the longest pattern of the form k 1 2. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001645The pebbling number of a connected graph. St001782The order of rowmotion on the set of order ideals of a poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001959The product of the heights of the peaks of a Dyck path. 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. St001845The number of join irreducibles minus the rank of a lattice. St000013The height of a Dyck path. St000210Minimum over maximum difference of elements in cycles. St000693The modular (standard) major index of a standard tableau. St001485The modular major index of a binary word. St001884The number of borders of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001948The number of augmented double ascents of a permutation. St001875The number of simple modules with projective dimension at most 1. St001684The reduced word complexity of a permutation. St001555The order of a signed permutation. St000327The number of cover relations in a poset. St000031The number of cycles in the cycle decomposition of a permutation. St000245The number of ascents of a permutation. St000019The cardinality of the support of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001621The number of atoms of a lattice. St000288The number of ones in a binary word. St000035The number of left outer peaks of a permutation. St000568The hook number of a binary tree. St000676The number of odd rises of a Dyck path. St000884The number of isolated descents of a permutation. St000919The number of maximal left branches of a binary tree. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000527The width of the poset. St000845The maximal number of elements covered by an element in a poset. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000007The number of saliances of the permutation. St000018The number of inversions of a permutation. St000490The intertwining number of a set partition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the 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 $3$. St000502The number of successions of a set partitions. St000550The number of modular elements of a lattice. St000731The number of double exceedences of a permutation. St000011The number of touch points (or returns) of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000742The number of big ascents of a permutation after prepending zero. St001817The number of flag weak exceedances of a signed permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000356The number of occurrences of the pattern 13-2. St000647The number of big descents of a permutation. St001115The number of even descents of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000971The smallest closer of a set partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000271The chromatic index of a graph. St000651The maximal size of a rise in a permutation. St000670The reversal length of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001050The number of terminal closers of a set partition. St001343The dimension of the reduced incidence algebra of a poset. St001434The number of negative sum pairs of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000135The number of lucky cars of the parking function. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000004The major index of a permutation. St000021The number of descents of a permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000489The number of cycles of a permutation of length at most 3. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000653The last descent of a permutation. St000702The number of weak deficiencies of a permutation. St000794The mak of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000823The number of unsplittable factors of the set partition. St000829The Ulam distance of a permutation to the identity permutation. St000833The comajor index of a permutation. St000925The number of topologically connected components of a set partition. St000956The maximal displacement of a permutation. St000991The number of right-to-left minima of a permutation. St001062The maximal size of a block of a set partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. St001517The length of a longest pair of twins in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001769The reflection length of a signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000358The number of occurrences of the pattern 31-2. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000624The normalized sum of the minimal distances to a greater element. St000989The number of final rises of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000015The number of peaks of a Dyck path. St000058The order of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000144The pyramid weight of the Dyck path. St000172The Grundy number of a graph. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000259The diameter of a connected graph. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000443The number of long tunnels of a Dyck path. St000485The length of the longest cycle of a permutation. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000673The number of non-fixed points of a permutation. St000691The number of changes of a binary word. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. 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 $c_{n-1}$. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple 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. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001120The length of a longest path in a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001417The length of a longest palindromic subword of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001437The flex of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001494The Alon-Tarsi number of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001566The length of the longest arithmetic progression in a permutation. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001692The number of vertices with higher degree than the average degree in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001883The mutual visibility number of a graph. St001892The flag excedance statistic of a signed permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St001963The tree-depth of a graph. St001978The codimension of the alternating sign matrix variety. St000017The number of inversions of a standard tableau. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000053The number of valleys of the Dyck path. St000056The decomposition (or block) number of a permutation. St000075The orbit size of a standard tableau under promotion. St000083The number of left oriented leafs of a binary tree except the first one. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000156The Denert index of a permutation. St000171The degree of the graph. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000240The number of indices that are not small excedances. St000272The treewidth of a graph. St000305The inverse major index of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000446The disorder of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000536The pathwidth of a graph. St000542The number of left-to-right-minima of a permutation. St000552The number of cut vertices of a graph. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000756The sum of the positions of the left to right maxima of a permutation. St000785The number of distinct colouring schemes of a graph. St000820The number of compositions obtained by rotating the composition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000890The number of nonzero entries in an alternating sign matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000924The number of topologically connected components of a perfect matching. St000957The number of Bruhat lower covers of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000990The first ascent of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001093The detour number of a graph. St001112The 3-weak dynamic number of a graph. St001114The number of odd descents of a permutation. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001286The annihilation number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001358The largest degree of a regular subgraph of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001424The number of distinct squares in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001480The number of simple summands of the module J^2/J^3. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001642The Prague dimension of a graph. St001671Haglund's hag of a permutation. St001726The number of visible inversions of a permutation. St001737The number of descents of type 2 in a permutation. St001746The coalition number of a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001829The common independence number of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001894The depth of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St001974The rank of the alternating sign matrix. St000005The bounce statistic of a Dyck path. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000204The number of internal nodes of a binary tree. St000222The number of alignments in the permutation. St000234The number of global ascents of a permutation. St000317The cycle descent number of a permutation. St000331The number of upper interactions of a Dyck path. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000501The size of the first part in the decomposition of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000565The major index of a set partition. 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. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. 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. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000688The global dimension minus the 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. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000872The number of very big descents of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. 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. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive 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. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001307The number of induced stars on four vertices in a graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001411The number of patterns 321 or 3412 in a permutation. St001469The holeyness of a permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001520The number of strict 3-descents. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. 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. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001468The smallest fixpoint of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. 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 $eA$. 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 $eA$. 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 $eA$. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000873The aix statistic of a permutation. St001569The maximal modular displacement of a permutation. St000462The major index minus the number of excedences of a permutation. St001731The factorization defect of a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St000307The number of rowmotion orbits of a poset. St001638The book thickness of a graph.