- FindStat

Your data matches 38 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001586

Mapping

St001586: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 0
[2]
 => 0
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 2
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 1
[2,1,1,1]
 => 3
[1,1,1,1,1]
 => 0
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 2
[3,3]
 => 0
[3,2,1]
 => 1
[3,1,1,1]
 => 0
[2,2,2]
 => 0
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 0
[7]
 => 0
[6,1]
 => 1
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 3
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 3
[2,1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => 0
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 2
[5,3]
 => 0
[5,2,1]
 => 1

Description

The number of odd parts smaller than the largest even part in an integer partition.

Matching statistic: St000771
(load all 2 compositions to match this statistic)

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%

Values

[1]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[2]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[1,1]
 => 11 => [2] => ([],2)
 => ? = 0 + 1
[3]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[2,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,1]
 => 111 => [3] => ([],3)
 => ? = 1 + 1
[4]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[3,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,2} + 1
[2,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,2} + 1
[2,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,2} + 1
[1,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,2} + 1
[5]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[3,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[3,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {1,1,3} + 1
[2,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {1,1,3} + 1
[1,1,1,1,1]
 => 11111 => [5] => ([],5)
 => ? ∊ {1,1,3} + 1
[6]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[5,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[4,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[3,3]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[3,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[2,2,2]
 => 000 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[2,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[2,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[1,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[7]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[5,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[5,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[4,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[3,3,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[3,2,2]
 => 100 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[3,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[2,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[2,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[1,1,1,1,1,1,1]
 => 1111111 => [7] => ([],7)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[8]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[7,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[6,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[5,3]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,4]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,2,2]
 => 000 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[3,3,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,3,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[3,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,1,1,1]
 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[2,2,2,2]
 => 0000 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[2,2,2,1,1]
 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[2,2,1,1,1,1]
 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[2,1,1,1,1,1,1]
 => 0111111 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[1,1,1,1,1,1,1,1]
 => 11111111 => [8] => ([],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[9]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[7,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[7,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[6,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[6,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[6,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[5,4]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[5,3,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[5,2,2]
 => 100 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[5,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[4,4,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,3,2]
 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[4,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,3,2,1]
 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,2,2,2,1]
 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[10]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[7,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[5,4,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[5,3,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,3,2,1]
 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,2,2,1]
 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[11]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[10,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[9,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[8,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[8,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[7,4]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[6,5]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[6,4,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[6,3,2]
 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[6,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[5,3,2,1]
 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1

Description

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$0$ , which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$2$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore statistic

$1$ .

Matching statistic: St000772
(load all 4 compositions to match this statistic)

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%

Values

[1]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[2]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[1,1]
 => 11 => [2] => ([],2)
 => ? = 0 + 1
[3]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[2,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,1,1]
 => 111 => [3] => ([],3)
 => ? = 1 + 1
[4]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[3,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,2} + 1
[2,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,2} + 1
[2,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,2} + 1
[1,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,2} + 1
[5]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[3,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[3,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {1,1,3} + 1
[2,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {1,1,3} + 1
[1,1,1,1,1]
 => 11111 => [5] => ([],5)
 => ? ∊ {1,1,3} + 1
[6]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[5,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[4,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[3,3]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[3,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[2,2,2]
 => 000 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[2,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[2,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[1,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,2,2,4} + 1
[7]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[5,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[5,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[4,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[3,3,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[3,2,2]
 => 100 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[3,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[2,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[2,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[1,1,1,1,1,1,1]
 => 1111111 => [7] => ([],7)
 => ? ∊ {0,0,1,1,1,2,3,3,5} + 1
[8]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[7,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[6,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[5,3]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,4]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,2,2]
 => 000 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[3,3,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[3,3,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[3,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,1,1,1]
 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[2,2,2,2]
 => 0000 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[2,2,2,1,1]
 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[2,2,1,1,1,1]
 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[2,1,1,1,1,1,1]
 => 0111111 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[1,1,1,1,1,1,1,1]
 => 11111111 => [8] => ([],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,2,3,4,4,6} + 1
[9]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[7,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[7,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[6,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[6,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[6,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[5,4]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[5,3,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[5,2,2]
 => 100 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[5,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,3,3,3,3,4,5,5,7} + 1
[4,4,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[4,3,2]
 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[4,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,3,2,1]
 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,2,2,2,1]
 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[10]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[7,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[5,4,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[5,3,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[5,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[4,3,2,1]
 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,2,2,1]
 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[11]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[10,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[9,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[8,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[8,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[7,4]
 => 10 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[6,5]
 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[6,4,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[6,3,2]
 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[6,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[5,3,2,1]
 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1

Description

The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$1$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic

$1$ . The graphs with statistic

$n-1$ ,

$n-2$ and

$n-3$ have been characterised, see [1].

Matching statistic: St001498
(load all 9 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 22%●distinct values known / distinct values provided: 7%

Values

[1]
 => [1,0,1,0]
 => [2,1] => [1,1,0,0]
 => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => ? = 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1}
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => ? ∊ {0,1}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,2}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,2}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,1,3}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,1,3}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,3}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,1,3}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,1,2,2,4}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,2,2,4}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,2,2,4}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,2,2,4}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,1,2,2,4}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 0
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,3,3,5}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,3,3,5}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,3,3,5}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,3,3,5}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,3,3,5}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,3,3,5}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,3,3,5}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 0
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,2,3,3,5}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 0
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 0
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,2,2,2,2,3,4,4,6}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [8,3,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [8,2,3,1,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [7,4,1,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [7,3,2,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7}
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => 0
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 0
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,6,1] => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 0
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,1,2] => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 0
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,6,1] => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 0
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => 0
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 0
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => 0
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 0
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [6,2,3,4,5,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 0
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,1,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 0
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0

Description

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.

Matching statistic: St001172

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 40%

Values

[1]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[2]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 0
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 0
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 2
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 3
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 0
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => 0
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 0
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => 3
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,1,1,2,3,4,6}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 4
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,1,1,2,3,4,6}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 2
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => 2
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,1,1,2,3,4,6}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 0
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,1,1,2,3,4,6}
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? ∊ {0,1,1,2,3,4,6}
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? ∊ {0,1,1,2,3,4,6}
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? ∊ {0,1,1,2,3,4,6}
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,2,2,3,4,5,5,7}
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[6,2,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[5,2,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[4,2,2,1,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[4,2,1,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[4,1,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[3,2,2,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[3,1,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[2,2,1,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[10,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[9,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[9,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[8,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[8,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[7,3,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[7,2,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[7,2,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[7,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}
[6,3,1,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,7,7,9}

Description

The number of 1-rises at odd height of a Dyck path.

Matching statistic: St000260

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 16%●distinct values known / distinct values provided: 7%

Values

[1]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? = 0 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,2} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,2} + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,2} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,3} + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {1,1,3} + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {1,1,3} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,1,2,2,4} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,1,2,2,4} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,2,2,4} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,1,2,2,4} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,0,1,2,2,4} + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,0,1,2,2,4} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,7] => ([(6,7)],8)
 => ? ∊ {0,1,1,1,1,2,3,3,5} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,7] => ([(6,7)],8)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,8] => ([(7,8)],9)
 => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 1
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000367
(load all 2 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000367: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%

Values

[1]
 => [1,0,1,0]
 => [2,1] => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => 0
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,3,4,2] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => [3,2,1] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [3,4,1,2] => 0
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,3,4,5,2] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [4,2,3,1] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,4,2,1] => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => 0
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [1,3,4,5,6,2] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [5,2,3,4,1] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [4,2,1,3] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [3,2,4,1] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [3,4,5,2,1] => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [3,4,5,6,1,2] => 0
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [1,3,4,5,6,7,2] => 0
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => [6,2,3,4,5,1] => 0
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [5,2,3,1,4] => 0
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [4,2,3,5,1] => 0
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [5,1,3,4,2] => 0
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [3,2,1,4] => 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [3,4,2,5,1] => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [4,5,1,3,2] => 0
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [3,4,1,5,2] => 0
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,1,6] => [3,4,5,6,2,1] => 4
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [3,4,5,6,7,1,2] => ? = 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? ∊ {0,1,1,5}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7] => [7,2,3,4,5,6,1] => ? ∊ {0,1,1,5}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,6,4] => [6,2,3,4,1,5] => 0
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5,6] => [5,2,3,4,6,1] => 0
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [5,2,1,4,3] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [4,2,3,1,5] => 0
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [3,2,4,5,1] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,1,3,5,2] => 0
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [4,5,2,1,3] => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [3,4,2,1,5] => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,3,4,1,5,6] => [3,4,5,2,6,1] => 3
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [3,5,1,4,2] => 0
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,1,5] => [3,4,5,1,6,2] => 0
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => [3,4,5,6,7,2,1] => ? ∊ {0,1,1,5}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [3,4,5,6,7,8,1,2] => ? ∊ {0,1,1,5}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [1,3,4,5,6,7,8,9,2] => ? ∊ {0,0,0,0,2,2,4,6}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6,8] => [8,2,3,4,5,6,7,1] => ? ∊ {0,0,0,0,2,2,4,6}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [6,1,2,3,4,7,5] => [7,2,3,4,5,1,6] => ? ∊ {0,0,0,0,2,2,4,6}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6,7] => [6,2,3,4,5,7,1] => ? ∊ {0,0,0,0,2,2,4,6}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,4] => [6,2,3,1,5,4] => 0
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,1,2,3,6,5] => [5,2,3,4,1,6] => 0
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,5,6] => [4,2,3,5,6,1] => 0
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [6,1,3,4,5,2] => 0
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,2,1,5,3] => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [4,2,5,1,3] => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [3,2,4,1,5] => 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,1,4,5,6] => [3,4,2,5,6,1] => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [4,5,2,3,1] => 0
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [3,1,4,5,2] => 0
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,5,2,1,4] => 3
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,3,4,5,1,6,7] => [3,4,5,6,2,7,1] => ? ∊ {0,0,0,0,2,2,4,6}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,4,5,7,1,6] => [3,4,5,6,1,7,2] => ? ∊ {0,0,0,0,2,2,4,6}
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1,8] => [3,4,5,6,7,8,2,1] => ? ∊ {0,0,0,0,2,2,4,6}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [3,4,5,6,7,8,9,1,2] => ? ∊ {0,0,0,0,2,2,4,6}
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,10,2] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7,9] => [9,2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,8,6] => [8,2,3,4,5,6,1,7] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,7,8] => [7,2,3,4,5,6,8,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [6,1,2,3,7,4,5] => [7,2,3,4,1,6,5] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,4,7,6] => [6,2,3,4,5,1,7] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [4,1,2,3,5,6,7] => [5,2,3,4,6,7,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,4,1,5,6,7] => [3,4,5,2,6,7,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,3,4,5,1,7,6] => [3,4,5,6,2,1,7] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2,3,4,5,6,1,7,8] => [3,4,5,6,7,2,8,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,6,7,1,5] => [3,4,5,7,1,6,2] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,8,1,7] => [3,4,5,6,7,1,8,2] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1,9] => [3,4,5,6,7,8,9,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [3,4,5,6,7,8,9,10,1,2] => ? ∊ {0,0,0,0,0,1,1,1,1,3,3,5,5,7}
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [1,3,4,5,6,7,8,9,10,11,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8,10] => [10,2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => [9,2,3,4,5,6,7,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6,8,9] => [8,2,3,4,5,6,7,9,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [7,1,2,3,4,8,5,6] => [8,2,3,4,5,1,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5,8,7] => [7,2,3,4,5,6,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6,7,8] => [6,2,3,4,5,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[6,4]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [6,1,2,7,3,4,5] => [7,2,3,1,5,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [5,1,2,3,7,4,6] => [6,2,3,4,1,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [5,1,2,3,6,7,4] => [6,2,3,4,7,1,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [4,1,2,3,5,7,6] => [5,2,3,4,6,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [3,1,2,4,5,6,7] => [4,2,3,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,7,1,2,3,4,5] => [7,1,3,4,5,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,1,4,5,6,7] => [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,3,4,1,5,7,6] => [3,4,5,2,6,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[4,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [2,3,4,5,1,6,7,8] => [3,4,5,6,2,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,3,4,7,1,5,6] => [3,4,5,1,6,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,3,4,6,1,7,5] => [3,4,5,7,2,1,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[3,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,3,4,5,6,1,8,7] => [3,4,5,6,7,2,1,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1,8,9] => [3,4,5,6,7,8,2,9,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [4,5,6,7,1,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,5,6,7,1,4] => [3,4,6,7,1,5,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}
[2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,7,8,1,6] => [3,4,5,6,8,1,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,4,4,5,6,6,8}

Description

The number of simsun double descents of a permutation. The restriction of a permutation

$\pi$ to

$[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of

$\pi$ of letters in

$[k]$ . A simsun double descent of a permutation

$\pi$ is a double descent of any restriction of

$\pi$ to

$[1,\ldots,k]$ for some

$k$ . (Note here that the same double descent can appear in multiple restrictions!)

Matching statistic: St001645

Mapping

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%

Values

[1]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[2]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[1,1]
 => 11 => [2] => ([],2)
 => ? = 0 + 1
[3]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[2,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,1]
 => 111 => [3] => ([],3)
 => ? = 0 + 1
[4]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[3,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,2} + 1
[2,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,2} + 1
[2,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,2} + 1
[1,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,2} + 1
[5]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[3,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[3,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0} + 1
[2,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[2,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0} + 1
[1,1,1,1,1]
 => 11111 => [5] => ([],5)
 => ? ∊ {0,0,0} + 1
[6]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[5,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[4,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[3,3]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[3,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[2,2,2]
 => 000 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[2,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[2,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[1,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,1,2,4} + 1
[7]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[5,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[5,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[4,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[3,3,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[3,2,2]
 => 100 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[3,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[2,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[2,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[2,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[1,1,1,1,1,1,1]
 => 1111111 => [7] => ([],7)
 => ? ∊ {0,0,0,0,0,0,1,2,3,5} + 1
[8]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[7,1]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[6,2]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[5,3]
 => 11 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[4,4]
 => 00 => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[4,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[4,2,2]
 => 000 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[3,3,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[3,3,1,1]
 => 1111 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[3,2,2,1]
 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[3,2,1,1,1]
 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[2,2,2,2]
 => 0000 => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[2,2,2,1,1]
 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[2,2,1,1,1,1]
 => 001111 => [2,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[2,1,1,1,1,1,1]
 => 0111111 => [1,6] => ([(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[1,1,1,1,1,1,1,1]
 => 11111111 => [8] => ([],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,4,4,6} + 1
[9]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[7,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[7,1,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,5,5,7} + 1
[6,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[6,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[6,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,5,5,7} + 1
[5,4]
 => 10 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[5,3,1]
 => 111 => [3] => ([],3)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,5,5,7} + 1
[4,4,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[4,3,2]
 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[10]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[7,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[5,4,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[5,3,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[4,3,2,1]
 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[11]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1
[10,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[9,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[8,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[8,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[7,4]
 => 10 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[6,5]
 => 01 => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[6,4,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[6,3,2]
 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[4,4,3]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[12]
 => 0 => [1] => ([],1)
 => 1 = 0 + 1
[9,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[7,4,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[7,3,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[6,3,2,1]
 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[5,5,2]
 => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 4 = 3 + 1
[5,4,3]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[13]
 => 1 => [1] => ([],1)
 => 1 = 0 + 1

Description

The pebbling number of a connected graph.

Matching statistic: St000456

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%

Values

[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,0} + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0} + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,1} + 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,1} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,0,2} + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,2} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,2} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {0,0,1,1,3} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,1,1,3} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,1,1,3} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,1,1,3} + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {0,0,1,1,3} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => ([(5,6)],7)
 => ? ∊ {0,0,0,0,1,2,2,4} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,1,2,2,4} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,1,2,2,4} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,1,2,2,4} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,1,2,2,4} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,1,2,2,4} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,1,2,2,4} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => ? ∊ {0,0,0,0,1,2,2,4} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? ∊ {0,0,0,0,0,1,1,2,3,3,5} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,3,4,4,6} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,10,9] => ([(8,9)],10)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,4,5,5,7} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,3,3,4,5,5,7} + 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 4 = 3 + 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 4 = 3 + 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [5,1,6,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 2 + 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [4,6,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 3 = 2 + 1
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,6,1,7,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 3 = 2 + 1
[3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 2 + 1
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,7,4] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [4,1,6,2,3,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 2 = 1 + 1
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [3,6,1,2,4,7,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 2 = 1 + 1
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,4,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
[6,3,1,1,1]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1

Description

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

Matching statistic: St000058

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 10%●distinct values known / distinct values provided: 7%

Values

[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 2 = 0 + 2
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 2 = 0 + 2
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 2 = 0 + 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => ? ∊ {0,1} + 2
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 2 = 0 + 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => ? ∊ {0,1} + 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => ? ∊ {0,0,0,0,2} + 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [6,3,2,5,4,1,8,7] => ? ∊ {0,0,0,0,2} + 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => ? ∊ {0,0,0,0,2} + 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,8,5,4,7,6,3] => ? ∊ {0,0,0,0,2} + 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => ? ∊ {0,0,0,0,2} + 2
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
 => [10,9,8,7,6,5,4,3,2,1,12,11] => 2 = 0 + 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [8,7,4,3,6,5,2,1,10,9] => ? ∊ {0,0,1,1,3} + 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => ? ∊ {0,0,1,1,3} + 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => ? ∊ {0,0,1,1,3} + 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => ? ∊ {0,0,1,1,3} + 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,10,9,6,5,8,7,4,3] => ? ∊ {0,0,1,1,3} + 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,12,11,10,9,8,7,6,5,4,3] => 2 = 0 + 2
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
 => [12,11,10,9,8,7,6,5,4,3,2,1,14,13] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
 => [10,9,8,5,4,7,6,3,2,1,12,11] => 2 = 0 + 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [8,5,4,3,2,7,6,1,10,9] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [8,3,2,7,6,5,4,1,10,9] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,10,7,6,5,4,9,8,3] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,10,5,4,9,8,7,6,3] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [2,1,12,11,10,7,6,9,8,5,4,3] => 2 = 0 + 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {0,0,0,0,0,1,2,2,4} + 2
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
 => [14,13,12,11,10,9,8,7,6,5,4,3,2,1,16,15] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
 => [12,11,10,9,6,5,8,7,4,3,2,1,14,13] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
 => [10,9,6,5,4,3,8,7,2,1,12,11] => 2 = 0 + 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
 => [10,9,4,3,8,7,6,5,2,1,12,11] => 2 = 0 + 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [6,5,4,3,2,1,8,7,10,9] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [8,3,2,5,4,7,6,1,10,9] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,8,7,6,5,4,3,10,9] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [6,3,2,5,4,1,10,9,8,7] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [4,3,2,1,10,7,6,9,8,5] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,10,5,4,7,6,9,8,3] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
 => [2,1,12,11,8,7,6,5,10,9,4,3] => 2 = 0 + 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,10,9,8,7,6,5] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
 => [2,1,12,11,6,5,10,9,8,7,4,3] => 2 = 0 + 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,14,13,12,11,8,7,10,9,6,5,4,3] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
 => [2,1,16,15,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {0,0,0,1,1,1,1,2,3,3,5} + 2
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18)]
 => [16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,18,17] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
 => [14,13,12,11,10,7,6,9,8,5,4,3,2,1,16,15] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
 => [12,11,10,7,6,5,4,9,8,3,2,1,14,13] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
 => [12,11,10,5,4,9,8,7,6,3,2,1,14,13] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]
 => [10,7,6,5,4,3,2,9,8,1,12,11] => 2 = 0 + 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
 => [10,9,4,3,6,5,8,7,2,1,12,11] => 2 = 0 + 2
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
 => [10,3,2,9,8,7,6,5,4,1,12,11] => 2 = 0 + 2
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
 => [8,7,6,5,4,3,2,1,12,11,10,9] => 2 = 0 + 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [6,3,2,5,4,1,8,7,10,9] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,8,5,4,7,6,3,10,9] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
 => [2,1,12,9,8,7,6,5,4,11,10,3] => 2 = 0 + 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,10,7,6,9,8,5] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)]
 => [2,1,12,11,6,5,8,7,10,9,4,3] => 2 = 0 + 2
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,14,13,12,9,8,7,6,11,10,5,4,3] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
 => [4,3,2,1,12,11,10,9,8,7,6,5] => 2 = 0 + 2
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,12,5,4,11,10,9,8,7,6,3] => 2 = 0 + 2
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
 => [2,1,14,13,12,7,6,11,10,9,8,5,4,3] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
 => [2,1,16,15,14,13,12,9,8,11,10,7,6,5,4,3] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
 => [2,1,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3] => ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,6} + 2
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,20)]
 => [18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,20,19] => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 2
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10),(17,18)]
 => [16,15,14,13,12,11,8,7,10,9,6,5,4,3,2,1,18,17] => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 2
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
 => [14,13,12,11,8,7,6,5,10,9,4,3,2,1,16,15] => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 2
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
 => [14,13,12,11,6,5,10,9,8,7,4,3,2,1,16,15] => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,5,5,7} + 2
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
 => [8,7,6,5,4,3,2,1,10,9,12,11] => 2 = 0 + 2
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9),(11,12)]
 => [10,7,4,3,6,5,2,9,8,1,12,11] => 2 = 0 + 2
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)]
 => [10,5,4,3,2,9,8,7,6,1,12,11] => 2 = 0 + 2
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
 => [10,3,2,9,6,5,8,7,4,1,12,11] => 2 = 0 + 2
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
 => [2,1,10,9,8,7,6,5,4,3,12,11] => 2 = 0 + 2
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,12),(10,11)]
 => [8,7,4,3,6,5,2,1,12,11,10,9] => 2 = 0 + 2
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
 => [2,1,12,9,6,5,8,7,4,11,10,3] => 2 = 0 + 2
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
 => [6,5,4,3,2,1,12,11,10,9,8,7] => 2 = 0 + 2
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]
 => [2,1,12,7,6,5,4,11,10,9,8,3] => 2 = 0 + 2
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
 => [4,3,2,1,12,11,8,7,10,9,6,5] => 2 = 0 + 2
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,12,5,4,11,8,7,10,9,6,3] => 2 = 0 + 2
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,12,11,10,9,8,7,6,5] => 2 = 0 + 2
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10),(11,12)]
 => [8,7,4,3,6,5,2,1,10,9,12,11] => 2 = 0 + 2
[5,3,2]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9),(11,12)]
 => [10,5,4,3,2,7,6,9,8,1,12,11] => 2 = 0 + 2
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
 => [10,3,2,7,6,5,4,9,8,1,12,11] => 2 = 0 + 2
[5,2,2,1]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
 => [10,3,2,5,4,9,8,7,6,1,12,11] => 2 = 0 + 2
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
 => [2,1,10,9,6,5,8,7,4,3,12,11] => 2 = 0 + 2
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,12),(10,11)]
 => [8,5,4,3,2,7,6,1,12,11,10,9] => 2 = 0 + 2
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
 => [8,3,2,7,6,5,4,1,12,11,10,9] => 2 = 0 + 2
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [(1,6),(2,5),(3,4),(7,12),(8,9),(10,11)]
 => [6,5,4,3,2,1,12,9,8,11,10,7] => 2 = 0 + 2
[4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,12,7,6,5,4,9,8,11,10,3] => 2 = 0 + 2
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
 => [4,3,2,1,12,9,8,7,6,11,10,5] => 2 = 0 + 2
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,12,5,4,9,8,7,6,11,10,3] => 2 = 0 + 2
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
 => [6,3,2,5,4,1,12,11,10,9,8,7] => 2 = 0 + 2
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
 => [4,3,2,1,12,7,6,11,10,9,8,5] => 2 = 0 + 2
[3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,12,5,4,7,6,11,10,9,8,3] => 2 = 0 + 2
[3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,12,11,8,7,10,9,6,5] => 2 = 0 + 2
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10),(11,12)]
 => [8,5,4,3,2,7,6,1,10,9,12,11] => 2 = 0 + 2
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
 => [8,3,2,7,6,5,4,1,10,9,12,11] => 2 = 0 + 2
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
 => [6,5,4,3,2,1,10,9,8,7,12,11] => 2 = 0 + 2

Description

The order of a permutation.

$\operatorname{ord}(\pi)$ is given by the minimial

$k$ for which

$\pi^k$ is the identity permutation.

The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000383The last part of an integer composition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000356The number of occurrences of the pattern 13-2. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001964The interval resolution global dimension of a poset. St000741The Colin de Verdière graph invariant. St000366The number of double descents of a permutation. St001130The number of two successive successions in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001820The size of the image of the pop stack sorting operator. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001557The number of inversions of the second entry of a permutation. St001330The hat guessing number of a graph. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000454The largest eigenvalue of a graph if it is integral. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice.