- FindStat

Your data matches 85 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000120
(load all 11 compositions to match this statistic)

Mapping

St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b

Matching statistic: St000316
(load all 12 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

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

$\sigma$ is a **non-left-to-right-maximum** if there exists a

$j < i$ such that

$\sigma_j > \sigma_i$ .

Matching statistic: St000337
(load all 7 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$ where

$\textrm{inv} \, \tau_{i}$ is the number of inversions of

$\tau_{i}$ .

Matching statistic: St000374
(load all 12 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000703
(load all 37 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of deficiencies of a permutation. This is defined as

$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$ The number of exceedances is [[St000155]].

Matching statistic: St000021
(load all 28 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Matching statistic: St000024
(load all 7 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: 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] => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0]
 => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 1

Description

The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

Matching statistic: St000155
(load all 37 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exceedances (also excedences) of a permutation. This is defined as

$exc(\sigma) = \#\{ i : \sigma(i) > i \}$ . It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic

$(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here,

$den$ is the Denert index of a permutation, see [[St000156]].

Matching statistic: St000329
(load all 9 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

Matching statistic: St000996
(load all 13 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

The following 75 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000676The number of odd rises of a Dyck path. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000012The area of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000161The sum of the sizes of the right subtrees of a binary tree. St000168The number of internal nodes of an ordered tree. St000204The number of internal nodes of a binary tree. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000246The number of non-inversions of a permutation. St000332The positive inversions of an alternating sign matrix. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000007The number of saliances of the permutation. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St000740The last entry of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000288The number of ones in a binary word. St000354The number of recoils of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St000984The number of boxes below precisely one peak. St001480The number of simple summands of the module J^2/J^3. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001346The number of parking functions that give the same permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001427The number of descents of a signed permutation. St000619The number of cyclic descents of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.