- FindStat

Your data matches 121 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000013
(load all 24 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000381
(load all 14 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer composition.

Matching statistic: St000141
(load all 60 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000662
(load all 6 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The staircase size of the code of a permutation. The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ . The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ . This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

Matching statistic: St000010
(load all 4 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000011

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000097

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000098

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000147
(load all 5 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000451
(load all 5 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the longest pattern of the form k 1 2...(k-1).

The following 111 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001330The hat guessing number of a graph. St000245The number of ascents of a permutation. St000439The position of the first down step of a Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St000527The width of the poset. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001058The breadth of the ordered tree. St000392The length of the longest run of ones in a binary word. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000028The number of stack-sorts needed to sort a permutation. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000444The length of the maximal rise of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

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

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St000442The maximal area to the right of an up step of a Dyck path. St000025The number of initial rises of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000172The Grundy number of a graph. St000528The height of a poset. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000730The maximal arc length of a set partition. St000845The maximal number of elements covered by an element in a poset. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000308The height of the tree associated to a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000171The degree of the graph. St000651The maximal size of a rise in a permutation. St001343The dimension of the reduced incidence algebra of a poset. St001652The length of a longest interval of consecutive numbers. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001268The size of the largest ordinal summand in the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000306The bounce count of a Dyck path. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000209Maximum difference of elements in cycles. St000470The number of runs in a permutation. St001717The largest size of an interval in a poset. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000094The depth of an ordered tree. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000542The number of left-to-right-minima of a permutation. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000021The number of descents of a permutation. St000080The rank of the poset. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

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

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001589The nesting number of a perfect matching. St001590The crossing number of a perfect matching. St000317The cycle descent number of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000983The length of the longest alternating subword. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .