- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000691
(load all 5 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => 0 = 2 - 2
[1,2] => [2] => 10 => 1 = 3 - 2
[2,1] => [1,1] => 11 => 0 = 2 - 2
[1,2,3] => [3] => 100 => 1 = 3 - 2
[1,3,2] => [2,1] => 101 => 2 = 4 - 2
[2,1,3] => [1,2] => 110 => 1 = 3 - 2
[2,3,1] => [2,1] => 101 => 2 = 4 - 2
[3,1,2] => [1,2] => 110 => 1 = 3 - 2
[3,2,1] => [1,1,1] => 111 => 0 = 2 - 2
[1,2,3,4] => [4] => 1000 => 1 = 3 - 2
[1,2,4,3] => [3,1] => 1001 => 2 = 4 - 2
[1,3,2,4] => [2,2] => 1010 => 3 = 5 - 2
[1,3,4,2] => [3,1] => 1001 => 2 = 4 - 2
[1,4,2,3] => [2,2] => 1010 => 3 = 5 - 2
[1,4,3,2] => [2,1,1] => 1011 => 2 = 4 - 2
[2,1,3,4] => [1,3] => 1100 => 1 = 3 - 2
[2,1,4,3] => [1,2,1] => 1101 => 2 = 4 - 2
[2,3,1,4] => [2,2] => 1010 => 3 = 5 - 2
[2,3,4,1] => [3,1] => 1001 => 2 = 4 - 2
[2,4,1,3] => [2,2] => 1010 => 3 = 5 - 2
[2,4,3,1] => [2,1,1] => 1011 => 2 = 4 - 2
[3,1,2,4] => [1,3] => 1100 => 1 = 3 - 2
[3,1,4,2] => [1,2,1] => 1101 => 2 = 4 - 2
[3,2,1,4] => [1,1,2] => 1110 => 1 = 3 - 2
[3,2,4,1] => [1,2,1] => 1101 => 2 = 4 - 2
[3,4,1,2] => [2,2] => 1010 => 3 = 5 - 2
[3,4,2,1] => [2,1,1] => 1011 => 2 = 4 - 2
[4,1,2,3] => [1,3] => 1100 => 1 = 3 - 2
[4,1,3,2] => [1,2,1] => 1101 => 2 = 4 - 2
[4,2,1,3] => [1,1,2] => 1110 => 1 = 3 - 2
[4,2,3,1] => [1,2,1] => 1101 => 2 = 4 - 2
[4,3,1,2] => [1,1,2] => 1110 => 1 = 3 - 2
[4,3,2,1] => [1,1,1,1] => 1111 => 0 = 2 - 2
[1,2,3,4,5] => [5] => 10000 => 1 = 3 - 2
[1,2,3,5,4] => [4,1] => 10001 => 2 = 4 - 2
[1,2,4,3,5] => [3,2] => 10010 => 3 = 5 - 2
[1,2,4,5,3] => [4,1] => 10001 => 2 = 4 - 2
[1,2,5,3,4] => [3,2] => 10010 => 3 = 5 - 2
[1,2,5,4,3] => [3,1,1] => 10011 => 2 = 4 - 2
[1,3,2,4,5] => [2,3] => 10100 => 3 = 5 - 2
[1,3,2,5,4] => [2,2,1] => 10101 => 4 = 6 - 2
[1,3,4,2,5] => [3,2] => 10010 => 3 = 5 - 2
[1,3,4,5,2] => [4,1] => 10001 => 2 = 4 - 2
[1,3,5,2,4] => [3,2] => 10010 => 3 = 5 - 2
[1,3,5,4,2] => [3,1,1] => 10011 => 2 = 4 - 2
[1,4,2,3,5] => [2,3] => 10100 => 3 = 5 - 2
[1,4,2,5,3] => [2,2,1] => 10101 => 4 = 6 - 2
[1,4,3,2,5] => [2,1,2] => 10110 => 3 = 5 - 2
[1,4,3,5,2] => [2,2,1] => 10101 => 4 = 6 - 2
[1,4,5,2,3] => [3,2] => 10010 => 3 = 5 - 2

Description

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

$i$ such that

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

Matching statistic: St001486

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => [1,1] => 2
[1,2] => [2] => 10 => [1,2] => 3
[2,1] => [1,1] => 11 => [1,1,1] => 2
[1,2,3] => [3] => 100 => [1,3] => 3
[1,3,2] => [2,1] => 101 => [1,2,1] => 4
[2,1,3] => [1,2] => 110 => [1,1,2] => 3
[2,3,1] => [2,1] => 101 => [1,2,1] => 4
[3,1,2] => [1,2] => 110 => [1,1,2] => 3
[3,2,1] => [1,1,1] => 111 => [1,1,1,1] => 2
[1,2,3,4] => [4] => 1000 => [1,4] => 3
[1,2,4,3] => [3,1] => 1001 => [1,3,1] => 4
[1,3,2,4] => [2,2] => 1010 => [1,2,2] => 5
[1,3,4,2] => [3,1] => 1001 => [1,3,1] => 4
[1,4,2,3] => [2,2] => 1010 => [1,2,2] => 5
[1,4,3,2] => [2,1,1] => 1011 => [1,2,1,1] => 4
[2,1,3,4] => [1,3] => 1100 => [1,1,3] => 3
[2,1,4,3] => [1,2,1] => 1101 => [1,1,2,1] => 4
[2,3,1,4] => [2,2] => 1010 => [1,2,2] => 5
[2,3,4,1] => [3,1] => 1001 => [1,3,1] => 4
[2,4,1,3] => [2,2] => 1010 => [1,2,2] => 5
[2,4,3,1] => [2,1,1] => 1011 => [1,2,1,1] => 4
[3,1,2,4] => [1,3] => 1100 => [1,1,3] => 3
[3,1,4,2] => [1,2,1] => 1101 => [1,1,2,1] => 4
[3,2,1,4] => [1,1,2] => 1110 => [1,1,1,2] => 3
[3,2,4,1] => [1,2,1] => 1101 => [1,1,2,1] => 4
[3,4,1,2] => [2,2] => 1010 => [1,2,2] => 5
[3,4,2,1] => [2,1,1] => 1011 => [1,2,1,1] => 4
[4,1,2,3] => [1,3] => 1100 => [1,1,3] => 3
[4,1,3,2] => [1,2,1] => 1101 => [1,1,2,1] => 4
[4,2,1,3] => [1,1,2] => 1110 => [1,1,1,2] => 3
[4,2,3,1] => [1,2,1] => 1101 => [1,1,2,1] => 4
[4,3,1,2] => [1,1,2] => 1110 => [1,1,1,2] => 3
[4,3,2,1] => [1,1,1,1] => 1111 => [1,1,1,1,1] => 2
[1,2,3,4,5] => [5] => 10000 => [1,5] => 3
[1,2,3,5,4] => [4,1] => 10001 => [1,4,1] => 4
[1,2,4,3,5] => [3,2] => 10010 => [1,3,2] => 5
[1,2,4,5,3] => [4,1] => 10001 => [1,4,1] => 4
[1,2,5,3,4] => [3,2] => 10010 => [1,3,2] => 5
[1,2,5,4,3] => [3,1,1] => 10011 => [1,3,1,1] => 4
[1,3,2,4,5] => [2,3] => 10100 => [1,2,3] => 5
[1,3,2,5,4] => [2,2,1] => 10101 => [1,2,2,1] => 6
[1,3,4,2,5] => [3,2] => 10010 => [1,3,2] => 5
[1,3,4,5,2] => [4,1] => 10001 => [1,4,1] => 4
[1,3,5,2,4] => [3,2] => 10010 => [1,3,2] => 5
[1,3,5,4,2] => [3,1,1] => 10011 => [1,3,1,1] => 4
[1,4,2,3,5] => [2,3] => 10100 => [1,2,3] => 5
[1,4,2,5,3] => [2,2,1] => 10101 => [1,2,2,1] => 6
[1,4,3,2,5] => [2,1,2] => 10110 => [1,2,1,2] => 5
[1,4,3,5,2] => [2,2,1] => 10101 => [1,2,2,1] => 6
[1,4,5,2,3] => [3,2] => 10010 => [1,3,2] => 5

Description

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

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

$c_i$ cells in the

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

Matching statistic: St000340
(load all 4 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 89%

Values

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

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

Matching statistic: St000453

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 78%

Values

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

Description

The number of distinct Laplacian eigenvalues of a graph.

Matching statistic: St000777
(load all 2 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 78%

Values

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

Description

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

Matching statistic: St000638
(load all 15 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
St000638: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of up-down runs of a permutation. An '''up-down run''' of a permutation

$\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or

$\pi_{1}$ if 1 is a descent of

$\pi$ . For example, the up-down runs of

$\pi=85712643$ are

$8$ ,

$85$ ,

$57$ ,

$71$ ,

$126$ , and

$643$ .