- FindStat

Your data matches 110 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000691

Mapping

Mp00109: Permutations —descent word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => 0
[2,1] => 1 => 0
[1,2,3] => 00 => 0
[1,3,2] => 01 => 1
[2,1,3] => 10 => 1
[2,3,1] => 01 => 1
[3,1,2] => 10 => 1
[3,2,1] => 11 => 0
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 1
[1,3,2,4] => 010 => 2
[1,3,4,2] => 001 => 1
[1,4,2,3] => 010 => 2
[1,4,3,2] => 011 => 1
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 2
[2,3,1,4] => 010 => 2
[2,3,4,1] => 001 => 1
[2,4,1,3] => 010 => 2
[2,4,3,1] => 011 => 1
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 2
[3,2,1,4] => 110 => 1
[3,2,4,1] => 101 => 2
[3,4,1,2] => 010 => 2
[3,4,2,1] => 011 => 1
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 2
[4,2,1,3] => 110 => 1
[4,2,3,1] => 101 => 2
[4,3,1,2] => 110 => 1
[4,3,2,1] => 111 => 0
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => 0010 => 2
[1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => 0010 => 2
[1,2,5,4,3] => 0011 => 1
[1,3,2,4,5] => 0100 => 2
[1,3,2,5,4] => 0101 => 3
[1,3,4,2,5] => 0010 => 2
[1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => 0010 => 2
[1,3,5,4,2] => 0011 => 1
[1,4,2,3,5] => 0100 => 2
[1,4,2,5,3] => 0101 => 3
[1,4,3,2,5] => 0110 => 2
[1,4,3,5,2] => 0101 => 3
[1,4,5,2,3] => 0010 => 2
[1,4,5,3,2] => 0011 => 1

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: St001486

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition

$c=(c_1,\dots,c_n)$ with

$c_i$ cells in the

$i$ -th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

Matching statistic: St001036
(load all 6 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000053

Mapping

Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => [1] => [1,0]
 => 0
[2,1] => 1 => [1] => [1,0]
 => 0
[1,2,3] => 00 => [2] => [1,1,0,0]
 => 0
[1,3,2] => 01 => [1,1] => [1,0,1,0]
 => 1
[2,1,3] => 10 => [1,1] => [1,0,1,0]
 => 1
[2,3,1] => 01 => [1,1] => [1,0,1,0]
 => 1
[3,1,2] => 10 => [1,1] => [1,0,1,0]
 => 1
[3,2,1] => 11 => [2] => [1,1,0,0]
 => 0
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1

Description

The number of valleys of the Dyck path.

Matching statistic: St000272
(load all 4 compositions to match this statistic)

Mapping

Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => [1] => ([],1)
 => 0
[2,1] => 1 => [1] => ([],1)
 => 0
[1,2,3] => 00 => [2] => ([],2)
 => 0
[1,3,2] => 01 => [1,1] => ([(0,1)],2)
 => 1
[2,1,3] => 10 => [1,1] => ([(0,1)],2)
 => 1
[2,3,1] => 01 => [1,1] => ([(0,1)],2)
 => 1
[3,1,2] => 10 => [1,1] => ([(0,1)],2)
 => 1
[3,2,1] => 11 => [2] => ([],2)
 => 0
[1,2,3,4] => 000 => [3] => ([],3)
 => 0
[1,2,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,3,2,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,4,2] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,4,2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4,3,2] => 011 => [1,2] => ([(1,2)],3)
 => 1
[2,1,3,4] => 100 => [1,2] => ([(1,2)],3)
 => 1
[2,1,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3,1,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,4,1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,4,3,1] => 011 => [1,2] => ([(1,2)],3)
 => 1
[3,1,2,4] => 100 => [1,2] => ([(1,2)],3)
 => 1
[3,1,4,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1,4] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,2,1] => 011 => [1,2] => ([(1,2)],3)
 => 1
[4,1,2,3] => 100 => [1,2] => ([(1,2)],3)
 => 1
[4,1,3,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,2,1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[4,2,3,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,3,1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[4,3,2,1] => 111 => [3] => ([],3)
 => 0
[1,2,3,4,5] => 0000 => [4] => ([],4)
 => 0
[1,2,3,5,4] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2,4,3,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2,5,3,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,3,2,4,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,3,2,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,4,5,2] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,5,2,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,5,4,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,4,2,3,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2,5] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,5,2,3] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,5,3,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1

Description

The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.

Matching statistic: St000306

Mapping

Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => [1] => [1,0]
 => 0
[2,1] => 1 => [1] => [1,0]
 => 0
[1,2,3] => 00 => [2] => [1,1,0,0]
 => 0
[1,3,2] => 01 => [1,1] => [1,0,1,0]
 => 1
[2,1,3] => 10 => [1,1] => [1,0,1,0]
 => 1
[2,3,1] => 01 => [1,1] => [1,0,1,0]
 => 1
[3,1,2] => 10 => [1,1] => [1,0,1,0]
 => 1
[3,2,1] => 11 => [2] => [1,1,0,0]
 => 0
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000362

Mapping

Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => [1] => ([],1)
 => 0
[2,1] => 1 => [1] => ([],1)
 => 0
[1,2,3] => 00 => [2] => ([],2)
 => 0
[1,3,2] => 01 => [1,1] => ([(0,1)],2)
 => 1
[2,1,3] => 10 => [1,1] => ([(0,1)],2)
 => 1
[2,3,1] => 01 => [1,1] => ([(0,1)],2)
 => 1
[3,1,2] => 10 => [1,1] => ([(0,1)],2)
 => 1
[3,2,1] => 11 => [2] => ([],2)
 => 0
[1,2,3,4] => 000 => [3] => ([],3)
 => 0
[1,2,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,3,2,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,4,2] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,4,2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4,3,2] => 011 => [1,2] => ([(1,2)],3)
 => 1
[2,1,3,4] => 100 => [1,2] => ([(1,2)],3)
 => 1
[2,1,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3,1,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,4,1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,4,3,1] => 011 => [1,2] => ([(1,2)],3)
 => 1
[3,1,2,4] => 100 => [1,2] => ([(1,2)],3)
 => 1
[3,1,4,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1,4] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,2,1] => 011 => [1,2] => ([(1,2)],3)
 => 1
[4,1,2,3] => 100 => [1,2] => ([(1,2)],3)
 => 1
[4,1,3,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,2,1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[4,2,3,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,3,1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[4,3,2,1] => 111 => [3] => ([],3)
 => 0
[1,2,3,4,5] => 0000 => [4] => ([],4)
 => 0
[1,2,3,5,4] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2,4,3,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2,5,3,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,3,2,4,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,3,2,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,4,5,2] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,5,2,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,5,4,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,4,2,3,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2,5] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,5,2,3] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,5,3,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1

Description

The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . Finding a minimal vertex cover is an NP-hard optimization problem.

Matching statistic: St000536
(load all 4 compositions to match this statistic)

Mapping

Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => [1] => ([],1)
 => 0
[2,1] => 1 => [1] => ([],1)
 => 0
[1,2,3] => 00 => [2] => ([],2)
 => 0
[1,3,2] => 01 => [1,1] => ([(0,1)],2)
 => 1
[2,1,3] => 10 => [1,1] => ([(0,1)],2)
 => 1
[2,3,1] => 01 => [1,1] => ([(0,1)],2)
 => 1
[3,1,2] => 10 => [1,1] => ([(0,1)],2)
 => 1
[3,2,1] => 11 => [2] => ([],2)
 => 0
[1,2,3,4] => 000 => [3] => ([],3)
 => 0
[1,2,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,3,2,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,4,2] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,4,2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4,3,2] => 011 => [1,2] => ([(1,2)],3)
 => 1
[2,1,3,4] => 100 => [1,2] => ([(1,2)],3)
 => 1
[2,1,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3,1,4] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[2,4,1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,4,3,1] => 011 => [1,2] => ([(1,2)],3)
 => 1
[3,1,2,4] => 100 => [1,2] => ([(1,2)],3)
 => 1
[3,1,4,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1,4] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,2,1] => 011 => [1,2] => ([(1,2)],3)
 => 1
[4,1,2,3] => 100 => [1,2] => ([(1,2)],3)
 => 1
[4,1,3,2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,2,1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[4,2,3,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,3,1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[4,3,2,1] => 111 => [3] => ([],3)
 => 0
[1,2,3,4,5] => 0000 => [4] => ([],4)
 => 0
[1,2,3,5,4] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2,4,3,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,2,5,3,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,5,4,3] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,3,2,4,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,3,2,5,4] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2,5] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,4,5,2] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,3,5,2,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,3,5,4,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,4,2,3,5] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,5,3] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2,5] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,5,2] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,5,2,3] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,5,3,2] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 1

Description

The pathwidth of a graph.

Matching statistic: St001035

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The convexity degree of the parallelogram polyomino associated with the Dyck path. A parallelogram polyomino is

$k$ -convex if

$k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino. For example, any rotation of a Ferrers shape has convexity degree at most one. The (bivariate) generating function is given in Theorem 2 of [1].

Matching statistic: St001169

Mapping

Mp00109: Permutations —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0 => [1] => [1,0]
 => 0
[2,1] => 1 => [1] => [1,0]
 => 0
[1,2,3] => 00 => [2] => [1,1,0,0]
 => 0
[1,3,2] => 01 => [1,1] => [1,0,1,0]
 => 1
[2,1,3] => 10 => [1,1] => [1,0,1,0]
 => 1
[2,3,1] => 01 => [1,1] => [1,0,1,0]
 => 1
[3,1,2] => 10 => [1,1] => [1,0,1,0]
 => 1
[3,2,1] => 11 => [2] => [1,1,0,0]
 => 0
[1,2,3,4] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,4,3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,3,2,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,3,4,2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,4,2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,4,3,2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3,4] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,4,3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,3,1,4] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,3,4,1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,4,1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,4,3,1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2,4] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,4,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,2,1,4] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[3,2,4,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,4,1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[3,4,2,1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[4,1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[4,1,3,2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[4,2,1,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[4,2,3,1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[4,3,1,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[4,3,2,1] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,3,4,5] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,5,4] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,2,4,3,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,2,4,5,3] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,2,5,3,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,2,5,4,3] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,5,4] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,4,2,5] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,3,4,5,2] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,3,5,2,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,3,5,4,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,4,2,3,5] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,4,2,5,3] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,4,3,2,5] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,4,3,5,2] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,4,5,2,3] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,4,5,3,2] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1

Description

Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.

The following 100 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000388The number of orbits of vertices of a graph under automorphisms. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001812The biclique partition number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001083The number of boxed occurrences of 132 in a permutation. St000422The energy of a graph, if it is integral. St001488The number of corners of a skew partition. St001537The number of cyclic crossings of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St000528The height of a poset. St000080The rank of the poset. 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

$eA$ . St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001668The number of points of the poset minus the width of the poset. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001792The arboricity of a graph. St001962The proper pathwidth of a graph. St000785The number of distinct colouring schemes of a graph. St001883The mutual visibility number of a graph. St001746The coalition number of a graph. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a graph. St001638The book thickness of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001822The number of alignments of a signed permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000640The rank of the largest boolean interval in a poset. St001734The lettericity of a graph. St001642The Prague dimension of a graph. St000077The number of boxed and circled entries. St001575The minimal number of edges to add or remove to make a graph edge transitive. St000299The number of nonisomorphic vertex-induced subtrees. St001271The competition number of a graph.