- FindStat

Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000093

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000786

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes,

$[2,2,2]$ and

$[3,2,1]$ . Therefore, the statistic on this graph is

$3$ .

Matching statistic: St001337

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The upper domination number of a graph. This is the maximum cardinality of a minimal dominating set of

$G$ . The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of

$K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].

Matching statistic: St001338

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The upper irredundance number of a graph. A set

$S$ of vertices is irredundant, if there is no vertex in

$S$ , whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of

$S$ . The upper irredundance number is the largest size of a maximal irredundant set. The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of

$K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].

Matching statistic: St000393

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1] => [1] =>  => ? = 1 - 1
[1,0,1,0]
 => [2,1] => [2,1] => 1 => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [1,2] => 0 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => 10 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => 11 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => 10 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,3,2] => [1,3,2] => 01 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 00 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 101 => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => 110 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 100 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => 110 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,3,2,1] => 111 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [4,3,1,2] => 101 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => 110 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => 100 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => 010 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,4,3,2] => 011 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => 100 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => 010 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [1,2,4,3] => 001 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 000 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => 1100 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => 1101 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [5,2,1,4,3] => 1110 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => 1010 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => 1100 => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => 1101 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => 1110 => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => 1100 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [4,3,2,1,5] => 1110 => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [4,3,1,2,5] => 1010 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [4,1,3,2,5] => 1100 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => 1011 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [5,4,1,3,2] => 1101 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,4,3,2] => 1110 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => 1010 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [5,1,3,2,4] => 1100 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => 1001 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => 0110 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => 1000 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => 0110 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [5,1,2,4,3] => 1010 => 3 = 4 - 1

Description

The number of strictly increasing runs in a binary word.

Matching statistic: St000691

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1] =>  =>  => ? = 1 - 2
[1,0,1,0]
 => [2,1] => 1 => 1 => 0 = 2 - 2
[1,1,0,0]
 => [1,2] => 0 => 0 => 0 = 2 - 2
[1,0,1,0,1,0]
 => [2,1,3] => 10 => 01 => 1 = 3 - 2
[1,0,1,1,0,0]
 => [2,3,1] => 10 => 01 => 1 = 3 - 2
[1,1,0,0,1,0]
 => [3,1,2] => 10 => 01 => 1 = 3 - 2
[1,1,0,1,0,0]
 => [1,3,2] => 01 => 00 => 0 = 2 - 2
[1,1,1,0,0,0]
 => [1,2,3] => 00 => 10 => 1 = 3 - 2
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 101 => 001 => 1 = 3 - 2
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 100 => 101 => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 100 => 101 => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 100 => 101 => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 100 => 101 => 2 = 4 - 2
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 110 => 011 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 100 => 101 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => 100 => 101 => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 010 => 100 => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 010 => 100 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 100 => 101 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 010 => 100 => 1 = 3 - 2
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => 001 => 110 => 1 = 3 - 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 000 => 010 => 2 = 4 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 1010 => 1001 => 2 = 4 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 1000 => 0101 => 3 = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 1010 => 1001 => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => 1010 => 1001 => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 1000 => 0101 => 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 1010 => 1001 => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 1000 => 0101 => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => 1001 => 1101 => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 1001 => 1101 => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 1000 => 0101 => 3 = 5 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1000 => 0101 => 3 = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => 1000 => 0101 => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => 1000 => 0101 => 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1000 => 0101 => 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 1100 => 1011 => 2 = 4 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 1000 => 0101 => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 1100 => 1011 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 1100 => 1011 => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1000 => 0101 => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => 1100 => 1011 => 2 = 4 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => 1000 => 0101 => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => 1001 => 1101 => 2 = 4 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => 0101 => 1100 => 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0100 => 0100 => 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => 1000 => 0101 => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => 0100 => 0100 => 2 = 4 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => 0100 => 0100 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => 0100 => 0100 => 2 = 4 - 2
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 1100 => 1011 => 2 = 4 - 2

Description

The number of changes of a binary word. This is the number of indices

$i$ such that

$w_i \neq w_{i+1}$ .

Matching statistic: St000619
(load all 2 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%

Values

[1,0]
 => [1] => [.,.]
 => [1] => ? = 1 - 1
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,2] => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [[.,.],[.,.]]
 => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
 => [2,1,5,7,6,4,3] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [2,1,4,7,6,5,3] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [3,2,1,7,6,5,4] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,4,1,3,6,7,5] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
 => [2,1,6,5,7,4,3] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,6,3,7,5] => [[.,[.,.]],[[.,.],[[.,.],.]]]
 => [2,1,4,6,7,5,3] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,4,6,1,3,7,5] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
 => [3,2,1,6,7,5,4] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,4,6,7,3,5] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,5,4,3,7,6,2] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,4,1,6,7,3,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
 => [2,1,5,4,7,6,3] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,6,1,7,3,5] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
 => [3,2,1,5,7,6,4] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => [4,3,2,1,7,6,5] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,4,1,3,7,5,6] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
 => [2,1,5,7,6,4,3] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,4,1,7,3,5,6] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [2,1,4,7,6,5,3] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,4,7,1,3,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [3,2,1,7,6,5,4] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,4,1,3,5,7,6] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
 => [2,1,6,7,5,4,3] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,1,5,3,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
 => [2,1,4,6,7,5,3] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,4,5,1,3,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
 => [3,2,1,6,7,5,4] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,4,5,7,3,6] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,5,4,3,7,6,2] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,7,3,6] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
 => [2,1,5,4,7,6,3] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,7,3,6] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
 => [3,2,1,5,7,6,4] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,4,5,7,1,3,6] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => [4,3,2,1,7,6,5] => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
 => [2,1,7,6,5,4,3] => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,3,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [2,1,4,7,6,5,3] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,4,5,1,3,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [3,2,1,7,6,5,4] => ? = 7 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,1,4,5,6,3,7] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,5,4,3,7,6,2] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,4,1,5,6,3,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
 => [2,1,5,4,7,6,3] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,4,5,1,6,3,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
 => [3,2,1,5,7,6,4] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,4,5,6,1,3,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => [4,3,2,1,7,6,5] => ? = 7 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,4,5,6,7,3] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
 => [1,6,5,4,3,7,2] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,4,1,5,6,7,3] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
 => [2,1,6,5,4,7,3] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,4,5,1,6,7,3] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
 => [3,2,1,6,5,7,4] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,4,5,6,1,7,3] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
 => [4,3,2,1,6,7,5] => ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,5,6,7,1,3] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [5,4,3,2,1,7,6] => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
 => [2,1,5,7,6,4,3] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,5,1,6,3,4,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [2,1,4,7,6,5,3] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,6,1,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [3,2,1,7,6,5,4] => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,6,7,4] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
 => [2,1,6,5,7,4,3] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,5,1,6,3,7,4] => [[.,[.,.]],[[.,.],[[.,.],.]]]
 => [2,1,4,6,7,5,3] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,5,6,1,3,7,4] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
 => [3,2,1,6,7,5,4] => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,6,7,3,4] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,5,4,3,7,6,2] => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,5,1,6,7,3,4] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
 => [2,1,5,4,7,6,3] => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,5,6,1,7,3,4] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
 => [3,2,1,5,7,6,4] => ? = 6 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,5,6,7,1,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => [4,3,2,1,7,6,5] => ? = 7 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,7,4,6] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
 => [2,1,5,7,6,4,3] => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,5,1,7,3,4,6] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [2,1,4,7,6,5,3] => ? = 6 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [2,5,7,1,3,4,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [3,2,1,7,6,5,4] => ? = 7 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [2,5,1,3,4,7,6] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
 => [2,1,6,7,5,4,3] => ? = 6 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
 => [1,4,6,7,5,3,2] => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
 => [2,1,4,6,7,5,3] => ? = 5 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
 => [3,2,1,6,7,5,4] => ? = 6 - 1

Description

The number of cyclic descents of a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is given by the number of indices

$1 \leq i \leq n$ such that

$\pi(i) > \pi(i+1)$ where we set

$\pi(n+1) = \pi(1)$ .

Matching statistic: St001237
(load all 7 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5 = 4 + 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]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => ? = 5 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => ? = 5 + 1

Description

The number of simple modules with injective dimension at most one or dominant dimension at least one.

Matching statistic: St001211

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%

Values

[1,0]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
 => ? = 7 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 7 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 7 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 7 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => ? = 7 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 7 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => ? = 5 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
 => ? = 6 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 5 + 1

Description

The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.

Matching statistic: St001773

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001773: Signed permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 57%

Values

[1,0]
 => [1] => [1] => [-1] => 0 = 1 - 1
[1,0,1,0]
 => [2,1] => [2,1] => [1,-2] => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [1,2] => [2,-1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => [1,3,-2] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => [1,2,-3] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => [3,1,-2] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,3,2] => [1,3,2] => [3,2,-1] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [2,3,-1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => ? = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => ? = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => ? = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => ? = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => ? = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => ? = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => ? = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => ? = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => ? = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [4,1,5,2,3] => [4,5,1,3,-2] => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 5 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [4,1,2,5,3] => [3,5,1,4,-2] => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => ? = 4 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [4,1,2,3,5] => [3,4,1,5,-2] => ? = 5 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => ? = 4 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => ? = 4 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,1,-2] => ? = 5 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => ? = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => ? = 4 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 5 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,4,3,6,5,-2] => ? = 4 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5] => [2,4,1,3,6,5] => [1,4,2,6,5,-3] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,6,3,5] => [2,1,4,6,3,5] => [1,5,3,6,4,-2] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5] => [2,4,1,6,3,5] => [1,5,2,6,4,-3] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => [2,4,6,1,3,5] => [1,5,2,6,3,-4] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,4,3,5,6,-2] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,4,1,3,5,6] => [2,4,1,3,5,6] => [1,4,2,5,6,-3] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,1,4,5,3,6] => [2,1,4,5,3,6] => [1,5,3,4,6,-2] => ? = 5 - 1

Description

The number of minimal elements in Bruhat order not less than the signed permutation. The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent. This is the size of the essential set of the signed permutation, see [1].