- FindStat

Your data matches 111 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000675
(load all 26 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of centered multitunnels of a Dyck path. This is the number of factorisations

$D = A B C$ of a Dyck path, such that

$B$ is a Dyck path and

$A$ and

$B$ have the same length.

Matching statistic: St000011
(load all 35 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000056
(load all 166 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The decomposition (or block) number of a permutation. For

$\pi \in \mathcal{S}_n$ , this is given by

$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].

Matching statistic: St001461
(load all 70 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation

$\pi\in\mathfrak S_n$ is obtained by placing labels

$1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from

$i$ to

$\pi(i)$ for every label

$i$ . This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation

$\pi\in\mathfrak S_n$ stabilizes an interval

$I=\{a,a+1,\dots,b\}$ if

$\pi(I)=I$ . It is stabilized-interval-free, if the only interval

$\pi$ stablizes is

$\{1,\dots,n\}$ . Thus, this statistic is

$1$ if

$\pi$ is stabilized-interval-free.

Matching statistic: St000234
(load all 166 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of global ascents of a permutation. The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$ Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$ For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$ or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].

Matching statistic: St000439
(load all 9 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The position of the first down step of a Dyck path.

Matching statistic: St001226
(load all 16 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001226: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The 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. That is the number of i such that

$Ext_A^1(J,e_i J)=0$ .

Matching statistic: St000007
(load all 21 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Matching statistic: St000025
(load all 9 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St000084
(load all 6 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of subtrees.

The following 101 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000843The decomposition number of a perfect matching. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000237The number of small exceedances. St000546The number of global descents of a permutation. St000203The number of external nodes of a binary tree. St000678The number of up steps after the last double rise of a Dyck path. St000738The first entry in the last row of a standard tableau. St000883The number of longest increasing subsequences of a permutation. St000925The number of topologically connected components of a set partition. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000374The number of exclusive right-to-left minima of a permutation. St000382The first part of an integer composition. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000553The number of blocks of a graph. St000740The last entry of a permutation. St000822The Hadwiger number of the graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001029The size of the core of a graph. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

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

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000214The number of adjacencies of a permutation. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

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

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

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

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000061The number of nodes on the left branch of a binary tree. St000717The number of ordinal summands of a poset. St000504The cardinality of the first block of a set partition. St000654The first descent of a permutation. St000906The length of the shortest maximal chain in a poset. St000914The sum of the values of the Möbius function of a poset. St000990The first ascent of a permutation. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000648The number of 2-excedences of a permutation. St000924The number of topologically connected components of a perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000241The number of cyclical small excedances. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001545The second Elser number of a connected graph. St001889The size of the connectivity set of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions.