searching the database
Your data matches 46 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: St001738
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
([],1)
=> 2
([],2)
=> 2
([(0,1)],2)
=> 2
([],3)
=> 2
([(1,2)],3)
=> 3
([(0,2),(1,2)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> 2
([(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 3
([(0,3),(1,2),(2,3)],4)
=> 3
([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> 2
([(3,4)],5)
=> 3
([(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
Description
The minimal order of a graph which is not an induced subgraph of the given graph.
For example, the graph with two isolated vertices is not an induced subgraph of the complete graph on three vertices.
By contrast, the minimal number of vertices of a graph which is not a subgraph of a graph is one plus the clique number [[St000097]].
Matching statistic: St000099
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> [1] => 1 = 2 - 1
([],2)
=> [2] => [1,1,0,0]
=> [1,2] => 1 = 2 - 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> [2,1] => 1 = 2 - 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 1 = 2 - 1
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 2 = 3 - 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 2 - 1
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2 = 3 - 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 2 - 1
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 3 = 4 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 2 = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 3 = 4 - 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000023
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> [1] => 0 = 2 - 2
([],2)
=> [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> [2,1] => 0 = 2 - 2
([],3)
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0 = 2 - 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1 = 3 - 2
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 3 - 2
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 2 - 2
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 3 - 2
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 3 - 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 3 - 2
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 3 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 3 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 3 - 2
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 2 - 2
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0 = 2 - 2
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1 = 3 - 2
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1 = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1 = 3 - 2
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 3 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2 = 4 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1 = 3 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2 = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 4 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 4 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1 = 3 - 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 4 - 2
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St001792
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],2)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
Description
The arboricity of a graph.
This is the minimum number of forests that covers all edges of the graph.
Matching statistic: St000624
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> [1] => ? = 2 - 2
([],2)
=> [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> [2,1] => 0 = 2 - 2
([],3)
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0 = 2 - 2
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1 = 3 - 2
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 3 - 2
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 2 - 2
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 3 - 2
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 3 - 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 3 - 2
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 3 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 3 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 3 - 2
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 2 - 2
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0 = 2 - 2
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1 = 3 - 2
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1 = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1 = 3 - 2
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 3 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2 = 4 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1 = 3 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2 = 4 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 4 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 4 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1 = 3 - 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1 = 3 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 4 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 3 - 2
Description
The normalized sum of the minimal distances to a greater element.
Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is
$$
\sum_{i=1}^n \min_d(\pi_{i-1-d}>\pi_i\text{ or }\pi_{i+1+d}>\pi_i)
$$
A closely related statistic appears in [1].
The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently satisfies the functional equation
$$
(x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x^3 = 0.
$$
Matching statistic: St001012
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001012: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001012: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> ? = 2
([],2)
=> []
=> []
=> ? = 2
([(0,1)],2)
=> [1]
=> [1,0]
=> 2
([],3)
=> []
=> []
=> ? = 2
([(1,2)],3)
=> [1]
=> [1,0]
=> 2
([(0,2),(1,2)],3)
=> [2]
=> [1,0,1,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([],4)
=> []
=> []
=> ? = 2
([(2,3)],4)
=> [1]
=> [1,0]
=> 2
([(1,3),(2,3)],4)
=> [2]
=> [1,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([],5)
=> []
=> []
=> ? ∊ {2,3,4,4,4}
([(3,4)],5)
=> [1]
=> [1,0]
=> 2
([(2,4),(3,4)],5)
=> [2]
=> [1,0,1,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> 4
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,4,4,4}
Description
Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001504
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> []
=> ? = 2
([],2)
=> []
=> []
=> []
=> ? = 2
([(0,1)],2)
=> [1]
=> [1]
=> [1,0]
=> 2
([],3)
=> []
=> []
=> []
=> ? = 2
([(1,2)],3)
=> [1]
=> [1]
=> [1,0]
=> 2
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([],4)
=> []
=> []
=> []
=> ? = 3
([(2,3)],4)
=> [1]
=> [1]
=> [1,0]
=> 2
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([],5)
=> []
=> []
=> []
=> ? ∊ {3,3,4,4,4}
([(3,4)],5)
=> [1]
=> [1]
=> [1,0]
=> 2
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 4
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4}
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001028
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> []
=> ? = 2
([],2)
=> []
=> []
=> []
=> ? = 2
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([],3)
=> []
=> []
=> []
=> ? = 2
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
([],4)
=> []
=> []
=> []
=> ? = 2
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 3
([],5)
=> []
=> []
=> []
=> ? ∊ {3,3,3,3,3,4,4,4,4}
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([(2,4),(3,4)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,4,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,4,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,4,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,4,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,4,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,4,4,4,4}
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001240
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001240: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001240: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> []
=> ? = 2
([],2)
=> []
=> []
=> []
=> ? = 2
([(0,1)],2)
=> [1]
=> [1]
=> [1,0]
=> 2
([],3)
=> []
=> []
=> []
=> ? = 2
([(1,2)],3)
=> [1]
=> [1]
=> [1,0]
=> 2
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([],4)
=> []
=> []
=> []
=> ? = 2
([(2,3)],4)
=> [1]
=> [1]
=> [1,0]
=> 2
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([],5)
=> []
=> []
=> []
=> ? ∊ {2,3,3,3,3,3,4,4,4}
([(3,4)],5)
=> [1]
=> [1]
=> [1,0]
=> 2
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 4
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,3,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,3,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,3,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,3,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,3,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,3,4,4,4}
Description
The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
Matching statistic: St001500
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001500: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001500: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> []
=> ? = 2
([],2)
=> []
=> []
=> []
=> ? = 2
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([],3)
=> []
=> []
=> []
=> ? = 3
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([],4)
=> []
=> []
=> []
=> ? = 3
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 3
([],5)
=> []
=> []
=> []
=> ? ∊ {3,3,3,3,3,3,4,4,4}
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {3,3,3,3,3,3,4,4,4}
Description
The global dimension of magnitude 1 Nakayama algebras.
We use the code below to translate them to Dyck paths.
The following 36 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000144The pyramid weight of the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000260The radius of a connected graph. St001469The holeyness of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001624The breadth of a lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St001323The independence gap of a graph. St001871The number of triconnected components of a graph. 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. St000454The largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph.
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!