- FindStat

Your data matches 161 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001290
(load all 19 compositions to match this statistic)

Mapping

St001290: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.

Matching statistic: St000013
(load all 4 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St001039
(load all 2 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

Matching statistic: St000314
(load all 22 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left-to-right-maxima of a permutation. An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a '''left-to-right-maximum''' if there does not exist a

$j < i$ such that

$\sigma_j > \sigma_i$ . This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.

Matching statistic: St000442
(load all 4 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St000542
(load all 10 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left-to-right-minima of a permutation. An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a left-to-right-minimum if there does not exist a j < i such that

$\sigma_j < \sigma_i$ .

Matching statistic: St000991
(load all 34 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].

Matching statistic: St001829

Mapping

Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The common independence number of a graph. The common independence number of a graph

$G$ is the greatest integer

$r$ such that every vertex of

$G$ belongs to some independent set

$X$ of vertices of cardinality at least

$r$ .

Matching statistic: St001203

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

We 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: In the list

$L$ delete the first entry

$c_0$ and substract from all other entries

$n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.

Matching statistic: St000007
(load all 7 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

The following 151 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000906The length of the shortest maximal chain in a poset. St000527The width of the poset. St000306The bounce count of a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000806The semiperimeter of the associated bargraph. St000259The diameter of a connected graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000451The length of the longest pattern of the form k 1 2. St000528The height of a poset. St000662The staircase size of the code of a permutation. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000308The height of the tree associated to a permutation. St000028The number of stack-sorts needed to sort a permutation. St000366The number of double descents of a permutation. St000141The maximum drop size of a permutation. St000703The number of deficiencies of a permutation. St001875The number of simple modules with projective dimension at most 1. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000717The number of ordinal summands of a poset. St000765The number of weak records in an integer composition. St001060The distinguishing index of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. 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. St001530The depth of a Dyck path. St000080The rank of the poset. St000094The depth of an ordered tree. St000352The Elizalde-Pak rank of a permutation. 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

$eA$ . 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001948The number of augmented double ascents of a permutation. St001782The order of rowmotion on the set of order ideals of a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001200The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ . St000680The Grundy value for Hackendot on posets. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000474Dyson's crank of a partition. St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St001933The largest multiplicity of a part in an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000383The last part of an integer composition. St000237The number of small exceedances. St000741The Colin de Verdière graph invariant. St000455The second largest eigenvalue of a graph if it is integral. St000731The number of double exceedences of a permutation. St000392The length of the longest run of ones in a binary word. St001862The number of crossings of a signed permutation. St000365The number of double ascents of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000873The aix statistic of a permutation. St001618The cardinality of the Frattini sublattice of a lattice. St000153The number of adjacent cycles of a permutation. St000877The depth of the binary word interpreted as a path. St001115The number of even descents of a permutation. St000983The length of the longest alternating subword. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001330The hat guessing number of a graph. St000441The number of successions of a permutation. St000381The largest part of an integer composition. St001566The length of the longest arithmetic progression in a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001730The number of times the path corresponding to a binary word crosses the base line. St000864The number of circled entries of the shifted recording tableau of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St000241The number of cyclical small excedances. St000317The cycle descent number of a permutation. St000654The first descent of a permutation. St000732The number of double deficiencies of a permutation. St000899The maximal number of repetitions of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000648The number of 2-excedences of a permutation. St000649The number of 3-excedences of a permutation. St000894The trace of an alternating sign matrix. St000932The number of occurrences of the pattern UDU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000834The number of right outer peaks of a permutation. St001096The size of the overlap set of a permutation. St000534The number of 2-rises of a permutation.