searching the database
Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000259
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 0
[1,2] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001488
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 43%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 43%
Values
[1] => [1,0]
=> []
=> [[],[]]
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [[1],[]]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ? = 3
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 3
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ? = 4
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 3
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 3
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 2
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 3
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> ? = 3
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> ? = 2
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> ? = 3
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[4],[]]
=> 2
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[4,3,3,2,1],[]]
=> ? = 5
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ? = 3
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ? = 3
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ? = 4
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ? = 4
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> ? = 4
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> ? = 4
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 3
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 3
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 3
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 3
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 3
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 3
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[4,4,3,1,1],[]]
=> ? = 4
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[4,3,3,1,1],[]]
=> ? = 4
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ? = 3
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ? = 3
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ? = 4
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ? = 4
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 3
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 3
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 3
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 3
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 3
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 3
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ? = 3
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 3
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 3
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ? = 3
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 3
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? = 3
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,4,2,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,4,3,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,4,5,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,5,2,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,5,3,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,5,3,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 2
[5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[5],[]]
=> 2
[5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[5],[]]
=> 2
[5,1,3,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[5],[]]
=> 2
[5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[5],[]]
=> 2
Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
Matching statistic: St001023
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001023: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 71%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001023: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 71%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3 = 0 + 3
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 4 = 1 + 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5 = 2 + 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 6 = 3 + 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 6 = 3 + 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 5 = 2 + 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 7 = 4 + 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 6 = 3 + 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 6 = 3 + 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 6 = 3 + 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 6 = 3 + 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 6 = 3 + 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5 = 2 + 3
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 6 = 3 + 3
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 6 = 3 + 3
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 5 = 2 + 3
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 7 = 4 + 3
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 6 = 3 + 3
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 6 = 3 + 3
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 5 = 2 + 3
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 6 = 3 + 3
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 6 = 3 + 3
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 5 = 2 + 3
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 6 = 3 + 3
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 2 + 3
[1,2,3,5,6,4] => [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]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 5 + 3
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 3
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 3
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
[1,2,5,6,3,4] => [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]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
[1,2,5,6,4,3] => [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]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 3
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 3
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 3
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 3
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 3
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 3
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 4 + 3
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 3
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 3
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 3
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 3
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 3
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 3
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 3
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 3
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 3
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 3
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 3
[1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,2,6,3,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,2,6,5,3] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,3,6,5,2] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,6,3,2,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,6,3,5,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 3
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
[1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 3
Description
Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000454
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 43%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => [.,.]
=> ([],1)
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[1,5,2,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,5,2,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,5,3,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,5,3,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,5,4,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,5,4,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,3,1,4,5] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4
[2,3,1,5,4] => [2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[3,1,2,4,5] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[3,1,2,5,4] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,1,4,2,5] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[3,2,1,4,5] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[3,2,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[3,2,4,1,5] => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,1,2,3,5] => [4,1,5,3,2] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,1,3,2,5] => [4,1,5,3,2] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,2,1,3,5] => [4,2,1,5,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[4,2,3,1,5] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[4,3,1,2,5] => [4,3,1,5,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[4,3,2,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 5
[1,2,3,6,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[1,2,3,6,5,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[1,2,5,3,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 4
[1,2,5,4,6,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 4
[1,2,5,6,3,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 4
[1,2,5,6,4,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 4
[1,2,6,3,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[1,2,6,3,5,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[1,2,6,4,3,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[1,2,6,4,5,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3
[6,3,1,2,4,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,1,2,5,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,1,4,2,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,1,4,5,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,1,5,2,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,1,5,4,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,2,1,4,5,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,2,1,5,4,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,2,4,1,5,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,2,4,5,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,2,5,1,4,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,2,5,4,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,4,1,2,5,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,4,1,5,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,4,2,1,5,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,4,2,5,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,4,5,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,4,5,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,5,1,2,4,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,5,1,4,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,5,2,1,4,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,5,2,4,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,5,4,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
[6,3,5,4,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!