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Your data matches 192 different statistics following compositions of up to 3 maps.
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Matching statistic: St000316
(load all 46 compositions to match this statistic)
(load all 46 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 52 compositions to match this statistic)
(load all 52 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: St000211
(load all 126 compositions to match this statistic)
(load all 126 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 0
[1,2] => {{1},{2}}
=> 0
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => {{1,2,3}}
=> 2
[3,1,2] => {{1,2,3}}
=> 2
[3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> 3
[2,4,1,3] => {{1,2,3,4}}
=> 3
[2,4,3,1] => {{1,2,4},{3}}
=> 2
[3,1,2,4] => {{1,2,3},{4}}
=> 2
[3,1,4,2] => {{1,2,3,4}}
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,2,3,4}}
=> 3
[4,1,2,3] => {{1,2,3,4}}
=> 3
[4,1,3,2] => {{1,2,4},{3}}
=> 2
[4,2,1,3] => {{1,3,4},{2}}
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> 1
[4,3,1,2] => {{1,2,3,4}}
=> 3
[4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000443
(load all 52 compositions to match this statistic)
(load all 52 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: St000740
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => 2 = 1 + 1
[2,1] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => 3 = 2 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 3 = 2 + 1
[2,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[1,3,2,4] => [1,3,2,4] => 4 = 3 + 1
[1,3,4,2] => [1,2,4,3] => 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => 4 = 3 + 1
[2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[2,3,1,4] => [1,3,2,4] => 4 = 3 + 1
[2,3,4,1] => [1,2,4,3] => 3 = 2 + 1
[2,4,1,3] => [2,4,1,3] => 3 = 2 + 1
[2,4,3,1] => [1,4,3,2] => 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => 4 = 3 + 1
[3,1,4,2] => [2,1,4,3] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => 4 = 3 + 1
[3,2,4,1] => [2,1,4,3] => 3 = 2 + 1
[3,4,1,2] => [2,4,1,3] => 3 = 2 + 1
[3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[4,1,2,3] => [4,1,2,3] => 3 = 2 + 1
[4,1,3,2] => [4,1,3,2] => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => 3 = 2 + 1
[4,2,3,1] => [4,1,3,2] => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 5 = 4 + 1
[1,2,4,5,3] => [1,2,3,5,4] => 4 = 3 + 1
[1,2,5,3,4] => [1,2,5,3,4] => 4 = 3 + 1
[1,2,5,4,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 5 = 4 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 4 = 3 + 1
[1,3,4,2,5] => [1,2,4,3,5] => 5 = 4 + 1
[1,3,4,5,2] => [1,2,3,5,4] => 4 = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => 4 = 3 + 1
[1,3,5,4,2] => [1,2,5,4,3] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => 5 = 4 + 1
[1,4,2,5,3] => [1,3,2,5,4] => 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => 5 = 4 + 1
[1,4,3,5,2] => [1,3,2,5,4] => 4 = 3 + 1
[1,4,5,2,3] => [1,3,5,2,4] => 4 = 3 + 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001007
(load all 52 compositions to match this statistic)
(load all 52 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 52 compositions to match this statistic)
(load all 52 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 52 compositions to match this statistic)
(load all 52 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: St000019
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => 1
[2,1] => [[.,.],.]
=> [1,2] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 4
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 4
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 4
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 4
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 4
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 4
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 4
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 4
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
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].
The following 182 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000168The number of internal nodes of an ordered tree. 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. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001215Let X be the direct sum of all simple modules 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. 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. St000054The first entry of the permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. 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. St001497The position of the largest weak excedence of a permutation. St000010The length of the partition. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000067The inversion number of the alternating sign matrix. St000074The number of special entries. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000228The size of a partition. St000237The number of small exceedances. St000292The number of ascents of a binary word. St000293The number of inversions 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. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000369The dinv deficit of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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. St001726The number of visible inversions of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. 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. St000542The number of left-to-right-minima of a permutation. St000738The first entry in the last 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. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. 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. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000653The last descent of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000957The number of Bruhat lower covers of a permutation. St001480The number of simple summands of the module J^2/J^3. St000702The number of weak deficiencies of a permutation. St000288The number of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000728The dimension of a set partition. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St000956The maximal displacement of a permutation. St000984The number of boxes below precisely one peak. 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). St000444The length of the maximal rise of a Dyck path. St000925The number of topologically connected components of a set partition. St001812The biclique partition number of a graph. St000159The number of distinct parts of the integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000746The number of pairs with odd minimum in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. 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. St001875The number of simple modules with projective dimension at most 1. St000259The diameter of a connected graph. St001060The distinguishing index of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000242The number of indices that are not cyclical small weak excedances. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000619The number of cyclic descents of a permutation. St000327The number of cover relations in a poset. St001668The number of points of the poset minus the width of the poset. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. 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 $eA$. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001861The number of Bruhat lower covers of a permutation. St001769The reflection length of a signed permutation. St001645The pebbling number of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. 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 $K[x]/(x^n)$. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2.
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