- FindStat

Your data matches 96 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001185
(load all 40 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0

Description

The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.

Matching statistic: St001271
(load all 10 compositions to match this statistic)

Mapping

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => ([],1)
 => 0
[1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [2] => ([],2)
 => 0
[1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 0
[1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [3] => ([],3)
 => 0
[1,1,1,0,0,0]
 => [3] => ([],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => 0
[1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => 0
[1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => 0
[1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => 0
[1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => 0

Description

The competition number of a graph. The competition graph of a digraph

$D$ is a (simple undirected) graph which has the same vertex set as

$D$ and has an edge between

$x$ and

$y$ if and only if there exists a vertex

$v$ in

$D$ such that

$(x, v)$ and

$(y, v)$ are arcs of

$D$ . For any graph,

$G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number

$k(G)$ is the smallest number of such isolated vertices.

Matching statistic: St001640
(load all 31 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [2,1] => [1,2] => 1
[1,1,0,0]
 => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [2,1,3] => [3,1,2] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,3,2] => [2,3,1] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 0
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,4,3,1] => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => 0
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [5,2,4,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [5,2,1,4,3] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,5,1,4,3] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,5,4,3] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,2,5,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,2,1,5,3] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [5,4,2,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => 0

Description

The number of ascent tops in the permutation such that all smaller elements appear before.

Matching statistic: St000007
(load all 10 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => 1 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => 2 = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => 2 = 1 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 1 = 0 + 1

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000745
(load all 7 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [[1]]
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [[1],[2]]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [[1,2]]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [[1,2],[3]]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [[1,3],[2]]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [[1,2,3]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [[1,2],[3,4]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [[1,3],[2,4]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [[1,3,4],[2]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[1,3,4],[2]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [[1,2,4],[3]]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [[1,2,5],[3,4]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [[1,2,4],[3,5]]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [[1,3,5],[2,4]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [[1,2,5],[3,4]]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [[1,2,4,5],[3]]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [[1,3,5],[2,4]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [[1,3,4],[2,5]]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [[1,3,5],[2,4]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [[1,3,4],[2,5]]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [[1,2,3],[4,5]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [[1,2,3,5],[4]]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => 1 = 0 + 1

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St001184
(load all 10 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001184: 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 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1

Description

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

Matching statistic: St000133
(load all 2 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [1,0,1,0]
 => [2,1] => [1,2] => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [2,3,1] => [3,1,2] => 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => [3,2,1] => 0
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,3,2] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [2,1,3] => [3,2,1] => 0
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [3,4,1,2] => 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,4,2,1] => 0
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [3,2,1,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,4,2,1] => 0
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [3,2,4,1] => 0
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,4,3,1] => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,4,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [3,2,4,1] => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,3,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,4,3,1] => 0
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [3,2,4,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [3,4,5,2,1] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,4,2,1,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [3,4,5,2,1] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,4,2,5,1] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2,5,4,1] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,4,2,1,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2,5,4,1] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,4,2,5,1] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [3,2,4,1,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2,5,4,1] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,4,2,5,1] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [3,2,4,5,1] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,4,5,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,4,5,3,1] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,4,3,1,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,4,5,3,1] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,4,3,5,1] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [3,2,5,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2,5,4,1] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,4,3,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2,5,4,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,4,3,5,1] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [3,2,4,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [3,2,5,4,1] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,4,3,5,1] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [3,2,4,5,1] => 0

Description

The "bounce" of a permutation.

Matching statistic: St000234
(load all 4 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [2,1] => [1,2] => [1,2] => 1
[1,1,0,0]
 => [1,2] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [1,3,2] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
 => [2,1,3] => [3,2,1] => [3,2,1] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,1,2,3] => [4,1,3,2] => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,3,1,2] => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [3,4,1,2] => [3,4,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,4,3] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,4,2,1] => [3,4,2,1] => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [4,2,3,1] => [4,2,3,1] => 0
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [3,2,4,1] => [3,2,4,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,5,2,3,4] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,1,2,3] => [5,4,1,3,2] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [4,5,1,2,3] => [4,5,1,3,2] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,5,3,4] => [1,5,4,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,1,4,2,3] => [5,1,4,3,2] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,5,4,2,3] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,4,3,1,2] => [5,4,3,1,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [4,5,3,1,2] => [4,5,3,1,2] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,4,5,2,3] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,3,4,1,2] => [5,3,4,1,2] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [4,3,5,1,2] => [4,3,5,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => [3,5,4,1,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,5,4] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [5,1,2,4,3] => [5,1,4,3,2] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,5,2,4,3] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,4,1,3,2] => [5,4,1,3,2] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,2,5,4,3] => [1,5,4,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [5,1,4,3,2] => [5,1,4,3,2] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,5,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [5,3,4,2,1] => [5,3,4,2,1] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,3,5,2,1] => [4,3,5,2,1] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [3,4,5,2,1] => [3,5,4,2,1] => 0

Description

The number of global ascents of a permutation. The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$ Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$ For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$ or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].

Matching statistic: St000237

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [[1]]
 => [1] => 0
[1,0,1,0]
 => [2,1] => [[1],[2]]
 => [2,1] => 1
[1,1,0,0]
 => [1,2] => [[1,2]]
 => [1,2] => 0
[1,0,1,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => [2,1,3] => 1
[1,0,1,1,0,0]
 => [2,3,1] => [[1,2],[3]]
 => [3,1,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [[1,3],[2]]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => [3,1,2] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [[1,2,3]]
 => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => [2,4,1,3] => 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => [3,1,2,4] => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [[1,3],[2,4]]
 => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => [3,1,2,4] => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [[1,2,4],[3]]
 => [3,1,2,4] => 0
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => [4,1,2,3] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 0

Description

The number of small exceedances. This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Matching statistic: St000261

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => ([],1)
 => 0
[1,0,1,0]
 => [2,1] => [1,1] => ([(0,1)],2)
 => 1
[1,1,0,0]
 => [1,2] => [2] => ([],2)
 => 0
[1,0,1,0,1,0]
 => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [1,2] => ([(1,2)],3)
 => 0
[1,1,0,0,1,0]
 => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [1,2] => ([(1,2)],3)
 => 0
[1,1,1,0,0,0]
 => [1,2,3] => [3] => ([],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 0
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 0
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4] => ([],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => 0

Description

The edge connectivity of a graph. This is the minimum number of edges that has to be removed to make the graph disconnected.

The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000352The Elizalde-Pak rank of a permutation. St000475The number of parts equal to 1 in a partition. St000546The number of global descents of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000286The number of connected components of the complement of a graph. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000542The number of left-to-right-minima of a permutation. St000733The row containing the largest entry of a standard tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001316The domatic number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000439The position of the first down step of a Dyck path. 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

$c_{n-1}$ . St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001204Call 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 special CNakayama algebra. St000990The first ascent of a permutation. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000390The number of runs of ones in a binary word. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000678The number of up steps after the last double rise of a Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001498The normalised height of a Nakayama algebra with magnitude 1. St000461The rix statistic of a permutation. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001933The largest multiplicity of a part in an integer partition. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000260The radius of a connected graph. St000929The constant term of the character polynomial of an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001552The number of inversions between excedances and fixed points of a permutation. St000264The girth of a graph, which is not a tree. St000221The number of strong fixed points of a permutation. St000873The aix statistic of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000456The monochromatic index of a connected graph. St001948The number of augmented double ascents of a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001545The second Elser number of a connected graph. 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

$eA$ . St001372The length of a longest cyclic run of ones of a binary word. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001050The number of terminal closers of a set partition. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.