- FindStat

Your data matches 63 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001737
(load all 34 compositions to match this statistic)

Mapping

St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of type 2 in a permutation. A position

$i\in[1,n-1]$ is a descent of type 2 of a permutation

$\pi$ of

$n$ letters, if it is a descent and if

$\pi(j) < \pi(i)$ for all

$j < i$ .

Matching statistic: St001280
(load all 4 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St000374
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [[1]]
 => [1] => 0
[1,2] => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 0
[2,1] => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 1
[3,1,2] => [1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [3,2,1] => 1
[3,2,1] => [1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [3,2,1] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [3,2,1,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [3,2,1,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 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: St000386

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of factors DDU in a Dyck path.

Matching statistic: St000670

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The reversal length of a permutation. A reversal in a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form

$\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for

$1 \leq i < j \leq n$ . This statistic is then given by the minimal number of reversals needed to sort a permutation. The reversal distance between two permutations plays an important role in studying DNA structures.

Matching statistic: St000884
(load all 7 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of isolated descents of a permutation. A descent

$i$ is isolated if neither

$i+1$ nor

$i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].

Matching statistic: St000994
(load all 6 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation

$\pi$ is an index

$i$ such that

$\pi^{-1}(i) < i > \pi(i)$ . Analogously, a '''cycle valley''' is an index

$i$ such that

$\pi^{-1}(i) > i < \pi(i)$ . Clearly, every cycle of

$\pi$ contains as many peaks as valleys.

Matching statistic: St000996
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Matching statistic: St001269
(load all 6 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.

Matching statistic: St001489

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].

The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000010The length of the partition. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000251The number of nonsingleton blocks of a set partition. St000659The number of rises of length at least 2 of a Dyck path. St000354The number of recoils of a permutation. St000390The number of runs of ones in a binary word. St000829The Ulam distance of a permutation to the identity permutation. St000919The number of maximal left branches of a binary tree. St000703The number of deficiencies of a permutation. St000035The number of left outer peaks of a permutation. St000834The number of right outer peaks of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001874Lusztig's a-function for the symmetric group. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000702The number of weak deficiencies of a permutation. 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

$eA$ . St000455The second largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St000092The number of outer peaks of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. 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. St000850The number of 1/2-balanced pairs in a poset. St001935The number of ascents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice.