Your data matches 472 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000645
(load all 21 compositions to match this statistic)
Mapping
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 0
[1,0,1,0]
 => 1
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => 0
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by $$ \sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a) $$
Matching statistic: St000030
(load all 78 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 0
[1,0,1,0]
 => [2,1] => 1
[1,1,0,0]
 => [1,2] => 0
[1,0,1,0,1,0]
 => [3,2,1] => 2
[1,0,1,1,0,0]
 => [2,3,1] => 2
[1,1,0,0,1,0]
 => [3,1,2] => 2
[1,1,0,1,0,0]
 => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 4
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 4
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 3
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
Description
The sum of the descent differences of a permutations. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$ See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000029
(load all 63 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,2] => [2,1] => 1
[1,0,1,0,1,0]
 => [2,3,1] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [2,1,3] => [1,3,2] => 1
[1,1,0,0,1,0]
 => [1,3,2] => [3,2,1] => 2
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => 2
[1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [1,4,3,2] => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [1,4,2,3] => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [4,2,3,1] => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [3,2,4,1] => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,1,3,2] => 3
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,1,2,3] => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,4,2] => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,4,3,1] => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,4,2,1] => 4
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [3,4,1,2] => 4
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [2,3,4,1] => 3
Description
The depth of a permutation. This is given by $$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$. Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Matching statistic: St000222
(load all 12 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [2,1] => 0
[1,0,1,0]
 => [3,1,2] => [3,2,1] => 1
[1,1,0,0]
 => [2,3,1] => [3,1,2] => 0
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,3,2,1] => 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,1,3] => 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,3,1,2] => 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,3,1] => 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,4,3,2,1] => 4
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,3,2,1,4] => 3
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,4,2,1,3] => 2
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [5,3,4,2,1] => 3
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,2,1,3,4] => 3
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,4,3,1,2] => 3
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,3,1,2,4] => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,4,2,3,1] => 3
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,2,5,3,1] => 3
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,3,1,4] => 4
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,3,4,1,2] => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,4,1] => 3
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 0
Description
The number of alignments in the permutation.
Matching statistic: St000246
(load all 9 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => 1
[1,1,0,0]
 => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => 2
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => 4
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 0
Description
The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 1 => 0
[1,0,1,0]
 => [1,1] => 11 => 0
[1,1,0,0]
 => [2] => 10 => 1
[1,0,1,0,1,0]
 => [1,1,1] => 111 => 0
[1,0,1,1,0,0]
 => [1,2] => 110 => 2
[1,1,0,0,1,0]
 => [2,1] => 101 => 1
[1,1,0,1,0,0]
 => [3] => 100 => 2
[1,1,1,0,0,0]
 => [3] => 100 => 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 2
[1,0,1,1,0,1,0,0]
 => [1,3] => 1100 => 4
[1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 4
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 1
[1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 3
[1,1,0,1,0,0,1,0]
 => [3,1] => 1001 => 2
[1,1,0,1,0,1,0,0]
 => [4] => 1000 => 3
[1,1,0,1,1,0,0,0]
 => [4] => 1000 => 3
[1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 2
[1,1,1,0,0,1,0,0]
 => [4] => 1000 => 3
[1,1,1,0,1,0,0,0]
 => [4] => 1000 => 3
[1,1,1,1,0,0,0,0]
 => [4] => 1000 => 3
Description
The number of inversions of a binary word.
Matching statistic: St000305
(load all 6 compositions to match this statistic)
Mapping
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [1] => 0
[1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 0
[1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 2
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 2
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 3
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 4
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 3
Description
The inverse major index of a permutation. This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000394
(load all 15 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000799
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000799: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,2] => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 3
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 4
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 3
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 3
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
Description
The number of occurrences of the vincular pattern |213 in a permutation. This is the number of occurrences of the pattern $(2,1,3)$, such that the letter matched by $2$ is the first entry of the permutation.
Matching statistic: St000800
(load all 5 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000800: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2,1] => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 3
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 3
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 3
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 3
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 4
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 4
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 3
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
Description
The number of occurrences of the vincular pattern |231 in a permutation. This is the number of occurrences of the pattern $(2,3,1)$, such that the letter matched by $2$ is the first entry of the permutation.
The following 462 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000957The number of Bruhat lower covers of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001671Haglund's hag of a permutation. St001759The Rajchgot index of a permutation. St001861The number of Bruhat lower covers of a permutation. St000004The major index of a permutation. St000018The number of inversions of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000081The number of edges of a graph. St000136The dinv of a parking function. St000156The Denert index of a permutation. St000171The degree of the graph. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000220The number of occurrences of the pattern 132 in a permutation. St000224The sorting index of a permutation. St000290The major index of a binary word. St000304The load of a permutation. St000327The number of cover relations in a poset. St000330The (standard) major index of a standard tableau. St000462The major index minus the number of excedences of a permutation. St000572The dimension exponent of a set partition. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000728The dimension of a set partition. St000868The aid statistic in the sense of Shareshian-Wachs. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001511The minimal number of transpositions needed to sort a permutation in either direction. 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. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001822The number of alignments of a signed permutation. St001841The number of inversions of a set partition. St001869The maximum cut size of a graph. St001894The depth of a signed permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001725The harmonious chromatic number of a graph. St000797The stat`` of a permutation. St000798The makl of a permutation. St001077The prefix exchange distance of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000391The sum of the positions of the ones in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St001721The degree of a binary word. 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. St001330The hat guessing number of a graph. St000050The depth or height of a binary tree. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001554The number of distinct nonempty subtrees of a binary tree. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001555The order of a signed permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000356The number of occurrences of the pattern 13-2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000662The staircase size of the code of a permutation. St001060The distinguishing index of a graph. St001090The number of pop-stack-sorts needed to sort a permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000259The diameter of a connected graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000532The total number of rook placements on a Ferrers board. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the 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. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001645The pebbling number of a connected graph. St001769The reflection length of a signed permutation. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000670The reversal length of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000422The energy of a graph, if it is integral. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000451The length of the longest pattern of the form k 1 2. St000624The normalized sum of the minimal distances to a greater element. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000155The number of exceedances (also excedences) of a permutation. St000264The girth of a graph, which is not a tree. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000358The number of occurrences of the pattern 31-2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000491The number of inversions of a set partition. St000516The number of stretching pairs of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000682The Grundy value of Welter's game on a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000837The number of ascents of distance 2 of a permutation. St000956The maximal displacement of a permutation. St000993The multiplicity of the largest part of an integer partition. St001115The number of even descents of a permutation. St001118The acyclic chromatic index of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001469The holeyness of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001520The number of strict 3-descents. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001569The maximal modular displacement of a permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000092The number of outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000619The number of cyclic descents of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001557The number of inversions of the second entry of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001778The largest greatest common divisor of an element and its image in a permutation. St001792The arboricity of a graph. St001863The number of weak excedances of a signed permutation. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000735The last entry on the main diagonal of a standard tableau. St000438The position of the last up step in a Dyck path. St000632The jump number of the poset. St000981The length of the longest zigzag subpath. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001638The book thickness of a graph. St000652The maximal difference between successive positions of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001623The number of doubly irreducible elements of a lattice. St000939The number of characters of the symmetric group whose value on the partition is positive. St000307The number of rowmotion orbits of a poset. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000893The number of distinct diagonal sums of an alternating sign matrix. St001875The number of simple modules with projective dimension at most 1. St000260The radius of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001846The number of elements which do not have a complement in the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000527The width of the poset. St000617The number of global maxima of a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000909The number of maximal chains of maximal size in a poset. St001820The size of the image of the pop stack sorting operator. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001948The number of augmented double ascents of a permutation. St000736The last entry in the first row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000023The number of inner peaks of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000232The number of crossings of a set partition. St000338The number of pixed points of a permutation. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000365The number of double ascents of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000456The monochromatic index of a connected graph. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000497The lcb statistic of a set partition. St000534The number of 2-rises of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000668The least common multiple of the parts of the partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000770The major index of an integer partition when read from bottom to top. St000779The tier of a permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000871The number of very big ascents of a permutation. St000884The number of isolated descents of a permutation. St000933The number of multipartitions of sizes given by an integer partition. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001153The number of blocks with even minimum in a set partition. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001377The major index minus the number of inversions of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001470The cyclic holeyness of a permutation. St001535The number of cyclic alignments of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001684The reduced word complexity of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001712The number of natural descents of a standard Young tableau. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001911A descent variant minus the number of inversions. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000062The length of the longest increasing subsequence of the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000174The flush statistic of a semistandard tableau. St000239The number of small weak excedances. St000253The crossing number of a set partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000315The number of isolated vertices of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000354The number of recoils of a permutation. St000366The number of double descents of a permutation. St000502The number of successions of a set partitions. St000570The Edelman-Greene number of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000834The number of right outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000988The orbit size of a permutation under Foata's bijection. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001162The minimum jump of a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001394The genus of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001487The number of inner corners of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001728The number of invisible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001937The size of the center of a parking function. St001946The number of descents in a parking function. St000090The variation of a composition. St000166The depth minus 1 of an ordered tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000254The nesting number of a set partition. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000335The difference of lower and upper interactions. St000352The Elizalde-Pak rank of a permutation. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000654The first descent of a permutation. St000739The first entry in the last row of a semistandard tableau. St000824The sum of the number of descents and the number of recoils of a permutation. St000836The number of descents of distance 2 of a permutation. St000842The breadth of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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 $eA$. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001637The number of (upper) dissectors of a poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001864The number of excedances of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000094The depth of an ordered tree. St000383The last part of an integer composition. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000542The number of left-to-right-minima of a permutation. St000839The largest opener of a set partition. St000922The minimal number such that all substrings of this length are unique. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).