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Your data matches 139 different statistics following compositions of up to 3 maps.
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Matching statistic: St000316
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
St000316: 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] => 2
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 3
[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] => 2
[1,2,5,4,3] => 2
[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] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
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: St000024
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
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%
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,2] => [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => [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
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
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: St000443
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
Description
The number of long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St001007
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001187
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001187: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001187: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
Description
The number of simple modules with grade at least one in the corresponding Nakayama algebra.
Matching statistic: St001224
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001224: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001224: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
Description
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the first Ext-group between X and the regular module.
Matching statistic: St000021
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
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: St000053
(load all 56 compositions to match this statistic)
(load all 56 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000120
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
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: St000168
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [[]]
=> 0
[1,2] => [1,0,1,0]
=> [[],[]]
=> 0
[2,1] => [1,1,0,0]
=> [[[]]]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [[[],[]]]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 3
[1,2,3,4,5] => [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
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 2
Description
The number of internal nodes of an ordered tree.
A node is internal if it is neither the root nor a leaf.
The following 129 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000332The positive inversions of an alternating sign matrix. 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000676The number of odd rises of a Dyck path. St000740The last entry of a permutation. St000991The number of right-to-left minima 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. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000052The number of valleys of a Dyck path not on the x-axis. St000067The inversion number of the alternating sign matrix. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000161The sum of the sizes of the right subtrees of a binary tree. St000214The number of adjacencies of a permutation. St000292The number of ascents of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001427The number of descents of a signed permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000007The number of saliances of the permutation. St000010The length of the partition. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000390The number of runs of ones in a binary word. St000507The number of ascents of a standard tableau. St000542The number of left-to-right-minima of a permutation. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000216The absolute length of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001480The number of simple summands of the module J^2/J^3. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000288The number of ones in a binary word. 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. St000653The last descent of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000925The number of topologically connected components of a set partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001812The biclique partition number of a graph. St000159The number of distinct parts of the integer partition. St000746The number of pairs with odd minimum in a perfect matching. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000242The number of indices that are not cyclical small weak excedances. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000619The number of cyclic descents of a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000327The number of cover relations in a poset. St001668The number of points of the poset minus the width of the poset. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001769The reflection length of a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001896The number of right descents of a signed permutations. St001712The number of natural descents of a standard Young tableau. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001877Number of indecomposable injective modules with projective dimension 2.
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