- FindStat

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000337

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ 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]
 => [1,2] => [1] => [1] => 0
[1,1,0,0]
 => [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [2,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2] => [3,1,2] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,3,2] => [3,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3] => [2,3,1] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,3,4,2] => [3,1,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,3,4,2] => [3,1,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,3,4] => [2,3,4,1] => 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

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ 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]
 => [1,2] => [1] => [1] => 0
[1,1,0,0]
 => [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [2,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2] => [3,1,2] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,3,2] => [3,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3] => [2,3,1] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,3,4,2] => [3,1,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,3,4,2] => [3,1,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,3,4] => [2,3,4,1] => 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

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ 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]
 => [1,2] => [1] => [1] => 0
[1,1,0,0]
 => [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [2,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2] => [3,1,2] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,3,2] => [3,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3] => [2,3,1] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,3,4,2] => [3,1,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,3,4,2] => [3,1,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,3,4] => [2,3,4,1] => 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: St000316

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%

Values

[1,0]
 => [1] => [] => [] => ? = 0
[1,0,1,0]
 => [1,2] => [1] => [1] => 0
[1,1,0,0]
 => [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [2,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2] => [3,1,2] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,3,2] => [3,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3] => [2,3,1] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,3,4,2] => [3,1,2,4] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,3,4,2] => [3,1,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,1] => [2,1,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,3,4] => [2,3,4,1] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,1,2,4] => [1,3,4,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,5,8,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,7,8,5,6] => [1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 5
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,6,7,4,8,5] => [1,2,3,6,7,4,5] => [6,7,1,2,3,4,5] => ? = 5
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,8,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [1,2,4,5,6,7,3] => [4,1,2,3,5,6,7] => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,2,4,6,7,8,3,5] => [1,2,4,6,7,3,5] => [4,6,1,2,3,5,7] => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,2,5,3,6,7,4,8] => [1,2,5,3,6,7,4] => [5,1,2,6,3,4,7] => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,2,5,3,6,7,8,4] => [1,2,5,3,6,7,4] => [5,1,2,6,3,4,7] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,2,5,6,3,7,8,4] => [1,2,5,6,3,7,4] => [5,1,6,2,3,4,7] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,2,5,6,7,3,8,4] => [1,2,5,6,7,3,4] => [5,6,1,2,3,4,7] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,8,3,4] => [1,2,5,6,7,3,4] => [5,6,1,2,3,4,7] => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,2,6,3,4,7,8,5] => [1,2,6,3,4,7,5] => [6,1,2,7,3,4,5] => ? = 5
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,2,6,3,7,4,5,8] => [1,2,6,3,7,4,5] => [1,6,2,7,3,4,5] => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,2] => [3,1,2,4,5,6,7] => ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,3,4,6,7,2,8,5] => [1,3,4,6,7,2,5] => [3,6,1,2,4,5,7] => ? = 4
[1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,5,2,3,4,7,6,8] => [1,5,2,3,4,7,6] => [5,1,2,3,7,4,6] => ? = 5
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,5,2,3,6,7,8,4] => [1,5,2,3,6,7,4] => [5,1,2,3,6,4,7] => ? = 4
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,5,6,7,8,2,3,4] => [1,5,6,7,2,3,4] => [5,6,7,1,2,3,4] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,6,2,3,4,5,7,8] => [1,6,2,3,4,5,7] => [1,2,3,4,6,5,7] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,6,2,3,4,7,5,8] => [1,6,2,3,4,7,5] => [6,1,2,3,7,4,5] => ? = 5
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,6,2,3,4,8,5,7] => [1,6,2,3,4,5,7] => [1,2,3,4,6,5,7] => ? = 1
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,6,2,8,3,4,5,7] => [1,6,2,3,4,5,7] => [1,2,3,4,6,5,7] => ? = 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,6,8,2,3,4,5,7] => [1,6,2,3,4,5,7] => [1,2,3,4,6,5,7] => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 1
[1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,8,5,6,7] => [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,8,4,5,6,7] => [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,6,3,4,5,8,7] => [2,1,6,3,4,5,7] => [2,3,4,6,1,7,5] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [2,1,6,3,4,8,5,7] => [2,1,6,3,4,5,7] => [2,3,4,6,1,7,5] => ? = 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,7,8,3,4,5,6] => [2,1,7,3,4,5,6] => [2,3,4,5,1,7,6] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,8,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,1,4,5,7,8,6] => [2,3,1,4,5,7,6] => [2,1,3,4,7,5,6] => ? = 3
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,8,5,6,7] => [2,3,1,4,5,6,7] => [2,3,4,5,6,1,7] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [2,3,4,1,5,6,7,8] => [2,3,4,1,5,6,7] => [2,3,4,5,1,6,7] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5,6,8,7] => [2,3,4,1,5,6,7] => [2,3,4,5,1,6,7] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [2,3,4,1,5,7,6,8] => [2,3,4,1,5,7,6] => [2,1,3,7,4,5,6] => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [2,3,4,1,5,7,8,6] => [2,3,4,1,5,7,6] => [2,1,3,7,4,5,6] => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [2,3,4,1,5,8,6,7] => [2,3,4,1,5,6,7] => [2,3,4,5,1,6,7] => ? = 1
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [2,3,4,1,8,5,6,7] => [2,3,4,1,5,6,7] => [2,3,4,5,1,6,7] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,4,5,1,6,7,8] => [2,3,4,5,1,6,7] => [2,3,4,1,5,6,7] => ? = 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

$\sigma_j > \sigma_i$ .