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Your data matches 65 different statistics following compositions of up to 3 maps.
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Matching statistic: St000216
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
St000216: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 3
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 3
Description
The absolute length of a permutation.
The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.
Matching statistic: St000211
(load all 170 compositions to match this statistic)
(load all 170 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
[1,2] => {{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
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> 3
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: St000316
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 2
[3,1,2] => [3,2,1] => 2
[3,2,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,4,3,2] => 2
[1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,1,2,4] => 2
[2,3,4,1] => [4,1,2,3] => 3
[2,4,1,3] => [4,3,1,2] => 3
[2,4,3,1] => [3,4,1,2] => 2
[3,1,2,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,1,3] => 3
[3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => [2,4,1,3] => 2
[3,4,1,2] => [3,1,4,2] => 2
[3,4,2,1] => [4,1,3,2] => 3
[4,1,2,3] => [4,3,2,1] => 3
[4,1,3,2] => [3,4,2,1] => 2
[4,2,1,3] => [2,4,3,1] => 2
[4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => [4,2,3,1] => 3
[4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,3,4] => 2
[1,2,5,3,4] => [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,2,3,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => 3
[1,3,5,2,4] => [1,5,4,2,3] => 3
[1,3,5,4,2] => [1,4,5,2,3] => 2
[1,4,2,3,5] => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,5,3,2,4] => 3
[1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,3,5,2] => [1,3,5,2,4] => 2
[1,4,5,2,3] => [1,4,2,5,3] => 2
[1,4,5,3,2] => [1,5,2,4,3] => 3
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 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 3
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000155
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => 2
[3,1,2] => {{1,2,3}}
=> [2,3,1] => 2
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => 2
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 2
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => 2
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => 2
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => 3
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 3
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => 2
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 3
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000377
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [2]
=> 0
[2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,1,1]
=> [2,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> 1
[2,1,3] => [2,1]
=> [3]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> 2
[3,1,2] => [3]
=> [1,1,1]
=> 2
[3,2,1] => [2,1]
=> [3]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [3,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [2,2]
=> 1
[1,3,2,4] => [2,1,1]
=> [2,2]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [2,2]
=> 1
[2,1,3,4] => [2,1,1]
=> [2,2]
=> 1
[2,1,4,3] => [2,2]
=> [4]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 3
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 3
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 3
[3,2,1,4] => [2,1,1]
=> [2,2]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> [4]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 3
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 3
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [2,2]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 3
[4,3,2,1] => [2,2]
=> [4]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [3,2]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [4,1]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [4,1]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [4,1]
=> 2
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [4,1]
=> 2
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [2,2,1]
=> 2
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> 3
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001176
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [2]
=> 0
[2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,1,1]
=> [3]
=> 0
[1,3,2] => [2,1]
=> [2,1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> 2
[3,1,2] => [3]
=> [1,1,1]
=> 2
[3,2,1] => [2,1]
=> [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 3
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 3
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 3
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> [2,2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 3
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 3
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 3
[4,3,2,1] => [2,2]
=> [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 2
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 2
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> 2
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> 3
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000443
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
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%
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%
Values
[1,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 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: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[2,1] => [2]
=> [[1,2]]
=> 2 = 1 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[3,1,2] => [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St001007
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
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%
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%
Values
[1,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000053The number of valleys of the Dyck path. St000074The number of special entries. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000157The number of descents of a standard tableau. St000168The number of internal nodes of an ordered tree. St000228The size of a partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. 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. St000502The number of successions of a set partitions. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000728The dimension of a set partition. St000829The Ulam distance of a permutation to the identity permutation. St001169Number of simple modules with projective dimension at least two 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. 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. St001480The number of simple summands of the module J^2/J^3. St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. 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. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. 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. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. 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. St001864The number of excedances of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset.
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