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Your data matches 733 different statistics following compositions of up to 3 maps.
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Matching statistic: St000645
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
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
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 3
[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
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 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: St000018
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000018: 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
[1,0,1,1,0,0]
=> [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
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[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
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000019
(load all 166 compositions to match this statistic)
(load all 166 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000019: 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
[1,0,1,1,0,0]
=> [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
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[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
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
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: St000228
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000228: Integer partitions ⟶ 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
[1,1,0,0]
=> []
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 2
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 0
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000384
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000384: Integer partitions ⟶ 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
[1,1,0,0]
=> []
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 2
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 0
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000459
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000459: Integer partitions ⟶ 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
[1,1,0,0]
=> []
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 2
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 0
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000784
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000784: Integer partitions ⟶ 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
[1,1,0,0]
=> []
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 2
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> 0
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St001090
(load all 68 compositions to match this statistic)
(load all 68 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001090: 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
[1,0,1,1,0,0]
=> [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
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[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
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 3
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0
Description
The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
Matching statistic: St001300
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> 0
[1,0,1,0]
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> ([],2)
=> 0
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,0,1,0,0]
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> ([],3)
=> 0
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(2,5),(3,5),(4,5)],6)
=> 3
[1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5)],6)
=> 4
[1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(2,3),(2,4),(2,5)],6)
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(3,4),(3,5)],6)
=> 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(4,5)],6)
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> ([],6)
=> 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ([(3,6),(4,6),(5,6)],7)
=> 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ([(4,6),(5,6)],7)
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> 6
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ([(2,3),(2,4),(2,5),(2,6)],7)
=> 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ([(3,4),(3,5),(3,6)],7)
=> 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ([(4,5),(4,6)],7)
=> 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(5,6)],7)
=> 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([],7)
=> 0
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000058
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [2,1] => 2 = 1 + 1
[1,1,0,0]
=> [1,2] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => 3 = 2 + 1
[1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 4 = 3 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 5 = 4 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 6 = 5 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 5 = 4 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => 4 = 3 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 6 = 5 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 5 = 4 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => 4 = 3 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,2,4,5,6] => 3 = 2 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 7 = 6 + 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => 6 = 5 + 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => 5 = 4 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => 4 = 3 + 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 7 = 6 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => 6 = 5 + 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => 4 = 3 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => 3 = 2 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => 2 = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 1 = 0 + 1
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
The following 723 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000110The number of permutations less than or equal to a permutation in left weak order. St000532The total number of rook placements on a Ferrers board. St001400The total number of Littlewood-Richardson tableaux of given shape. St000013The height of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000081The number of edges of a graph. St000141The maximum drop size of a permutation. St000171The degree of the graph. St000211The rank of the set partition. St000237The number of small exceedances. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St001094The depth index of a set partition. St001161The major index north count of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001479The number of bridges of a graph. St001759The Rajchgot index of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000147The largest part of an integer partition. St000381The largest part of an integer composition. St000383The last part of an integer composition. St000468The Hosoya index of a graph. St000505The biggest entry in the block containing the 1. St000808The number of up steps of the associated bargraph. St000971The smallest closer of a set partition. St001389The number of partitions of the same length below the given integer partition. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000008The major index of the composition. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000153The number of adjacent cycles of a permutation. St000169The cocharge of a standard tableau. St000225Difference between largest and smallest parts in a partition. St000234The number of global ascents of a permutation. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000330The (standard) major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000369The dinv deficit of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000391The sum of the positions of the ones in a binary word. St000445The number of rises of length 1 of a Dyck path. St000537The cutwidth of a graph. St000651The maximal size of a rise in a permutation. St000676The number of odd rises of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000703The number of deficiencies of a permutation. St000766The number of inversions of an integer composition. St000877The depth of the binary word interpreted as a path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001697The shifted natural comajor index of a standard Young tableau. St001721The degree of a binary word. St001777The number of weak descents in an integer composition. St001910The height of the middle non-run of a Dyck path. St000003The number of standard Young tableaux of the partition. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000047The number of standard immaculate tableaux of a given shape. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000273The domination number of a graph. St000277The number of ribbon shaped standard tableaux. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000507The number of ascents of a standard tableau. St000544The cop number of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000839The largest opener of a set partition. St000916The packing number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001128The exponens consonantiae of a partition. St001313The number of Dyck paths above the lattice path given by a binary word. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001462The number of factors of a standard tableaux under concatenation. St001523The degree of symmetry of a Dyck path. St001780The order of promotion on the set of standard tableaux of given shape. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001809The index of the step at the first peak of maximal height in a Dyck path. St001829The common independence number of a graph. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000439The position of the first down step of a Dyck path. 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. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420The number of Dyck paths that are weakly above 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. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000693The modular (standard) major index of a standard tableau. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001671Haglund's hag of a permutation. St000444The length of the maximal rise of a Dyck path. St000740The last entry of a permutation. St000925The number of topologically connected components of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001463The number of distinct columns in the nullspace of a graph. St001733The number of weak left to right maxima of a Dyck path. St000209Maximum difference of elements in cycles. St000446The disorder of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St000030The sum of the descent differences of a permutations. St000653The last descent of a permutation. St000794The mak of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000501The size of the first part in the decomposition of a permutation. St000702The number of weak deficiencies of a permutation. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. 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. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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. 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. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000809The reduced reflection length of the permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000100The number of linear extensions of a poset. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000833The comajor index of a permutation. St001268The size of the largest ordinal summand in the poset. St001779The order of promotion on the set of linear extensions of a poset. St000539The number of odd inversions of a permutation. St000795The mad of a permutation. St000359The number of occurrences of the pattern 23-1. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000831The number of indices that are either descents or recoils. St001497The position of the largest weak excedence of a permutation. St000463The number of admissible inversions of a permutation. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000002The number of occurrences of the pattern 123 in a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000155The number of exceedances (also excedences) of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000356The number of occurrences of the pattern 13-2. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000572The dimension exponent of a set partition. 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. St000144The pyramid weight of the Dyck path. St000443The number of long tunnels of a Dyck path. St000863The length of the first row of the shifted shape of a 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. 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. 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. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001118The acyclic chromatic index of a graph. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001658The total number of rook placements on a Ferrers board. St001955The number of natural descents for set-valued two row standard Young tableaux. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000050The depth or height of a binary tree. St000189The number of elements in the poset. St000203The number of external nodes of a binary tree. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000236The number of cyclical small weak excedances. St000308The height of the tree associated to a permutation. St000336The leg major index of a standard tableau. St000528The height of a poset. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000628The balance of a binary word. St000657The smallest part of an integer composition. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000983The length of the longest alternating subword. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001250The number of parts of a partition that are not congruent 0 modulo 3. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001343The dimension of the reduced incidence algebra of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001554The number of distinct nonempty subtrees of a binary tree. St001566The length of the longest arithmetic progression in a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001717The largest size of an interval in a poset. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St000067The inversion number of the alternating sign matrix. St000070The number of antichains in a poset. St000080The rank of the poset. St000085The number of linear extensions of the tree. St000089The absolute variation of a composition. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000248The number of anti-singletons of a set partition. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000296The length of the symmetric border of a binary word. St000306The bounce count of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000521The number of distinct subtrees of an ordered tree. St000674The number of hills of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000819The propagating number of a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. 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 $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001067The number of simple modules of dominant dimension at least two 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. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001312Number of parabolic noncrossing partitions indexed by the composition. St001405The number of bonds in a permutation. St001415The length of the longest palindromic prefix of a binary word. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St000806The semiperimeter of the associated bargraph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000770The major index of an integer partition when read from bottom to top. St000878The number of ones minus the number of zeros of a binary word. St001688The sum of the squares of the heights of the peaks of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000530The number of permutations with the same descent word as the given permutation. St000924The number of topologically connected components of a perfect matching. St001214The aft of an integer partition. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000837The number of ascents of distance 2 of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001911A descent variant minus the number of inversions. St000335The difference of lower and upper interactions. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001397Number of pairs of incomparable elements in a finite poset. St001959The product of the heights of the peaks of a Dyck path. St000014The number of parking functions supported by a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001480The number of simple summands of the module J^2/J^3. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000015The number of peaks of a Dyck path. St000029The depth of a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000134The size of the orbit of an alternating sign matrix under gyration. St000166The depth minus 1 of an ordered tree. St000210Minimum over maximum difference of elements in cycles. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001077The prefix exchange distance of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000094The depth of an ordered tree. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000221The number of strong fixed points of a permutation. St000222The number of alignments in the permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000471The sum of the ascent tops of a permutation. St000487The length of the shortest cycle of a permutation. St000673The number of non-fixed points of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001061The number of indices that are both descents and recoils of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001468The smallest fixpoint of a permutation. St000747A variant of the major index of a set partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000553The number of blocks of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000796The stat' of a permutation. St000989The number of final rises of a permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St000004The major index of a permutation. St000057The Shynar inversion number of a standard tableau. St000156The Denert index of a permutation. St000224The sorting index of a permutation. St000305The inverse major index of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St001117The game chromatic index of a graph. St001428The number of B-inversions of a signed permutation. St001869The maximum cut size of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000005The bounce statistic of a Dyck path. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000091The descent variation of a composition. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000204The number of internal nodes of a binary tree. St000220The number of occurrences of the pattern 132 in a permutation. St000271The chromatic index of a graph. St000304The load of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000448The number of pairs of vertices of a graph with distance 2. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001311The cyclomatic number of a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001341The number of edges in the center of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001584The area statistic between a Dyck path and its bounce path. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001841The number of inversions of a set partition. St000078The number of alternating sign matrices whose left key is the permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001587Half of the largest even part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St001081The number of minimal length factorizations of a permutation into star transpositions. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000064The number of one-box pattern of a permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000294The number of distinct factors of a binary word. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000472The sum of the ascent bottoms of a permutation. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000719The number of alignments in a perfect matching. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000906The length of the shortest maximal chain in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001074The number of inversions of the cyclic embedding of a permutation. 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. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001589The nesting number of a perfect matching. St001684The reduced word complexity of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000044The number of vertices of the unicellular map given by a perfect matching. St000060The greater neighbor of the maximum. St000197The number of entries equal to positive one in the alternating sign matrix. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000385The number of vertices with out-degree 1 in a binary tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000520The number of patterns in a permutation. St000627The exponent of a binary word. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000780The size of the orbit under rotation of a perfect matching. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000945The number of matchings in the dihedral orbit of a perfect matching. St001052The length of the exterior of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001346The number of parking functions that give the same permutation. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St000355The number of occurrences of the pattern 21-3. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000327The number of cover relations in a poset. St000868The aid statistic in the sense of Shareshian-Wachs. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000462The major index minus the number of excedences of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000001The number of reduced words for a permutation. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000961The shifted major index of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000255The number of reduced Kogan faces with the permutation as type. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001555The order of a signed permutation. St000039The number of crossings of a permutation. St000095The number of triangles of a graph. St000217The number of occurrences of the pattern 312 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000873The aix statistic of a permutation. 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. St001082The number of boxed occurrences of 123 in a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001377The major index minus the number of inversions of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001727The number of invisible inversions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. 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. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000450The number of edges minus the number of vertices plus 2 of a graph. St001220The width of a permutation. St000235The number of indices that are not cyclical small weak excedances. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000045The number of linear extensions of a binary tree. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001330The hat guessing number of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000075The orbit size of a standard tableau under promotion. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001933The largest multiplicity of a part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000477The weight of a partition according to Alladi. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000438The position of the last up step in a Dyck path. St000475The number of parts equal to 1 in a partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000667The greatest common divisor of the parts of the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000733The row containing the largest entry of a standard tableau. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St000145The Dyson rank of a partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001060The distinguishing index of a graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St000494The number of inversions of distance at most 3 of a permutation. St001394The genus of a permutation. St001596The number of two-by-two squares inside a skew partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000993The multiplicity of the largest part of an integer partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000441The number of successions of a permutation. St000466The Gutman (or modified Schultz) index of a connected graph. St000741The Colin de Verdière graph invariant. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001433The flag major index of a signed permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000937The number of positive values of the symmetric group character corresponding to the partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000662The staircase size of the code of a permutation. St001115The number of even descents of a permutation. St000490The intertwining number of a set partition. St000670The reversal length of a permutation. St000317The cycle descent number of a permutation. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. 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. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001535The number of cyclic alignments of a permutation. St000570The Edelman-Greene number of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000073The number of boxed entries. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000353The number of inner valleys of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000447The number of pairs of vertices of a graph with distance 3. St000495The number of inversions of distance at most 2 of a permutation. St000497The lcb statistic of a set partition. St000538The number of even inversions of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000677The standardized bi-alternating inversion number of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001403The number of vertical separators in a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001569The maximal modular displacement of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000092The number of outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000307The number of rowmotion orbits of a poset. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000903The number of different parts of an integer composition. St001349The number of different graphs obtained from the given graph by removing an edge. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St000456The monochromatic index of a connected graph. St001651The Frankl number of a lattice. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. 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 $eA$. St001618The cardinality of the Frattini sublattice of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St000879The number of long braid edges in the graph of braid moves of a permutation. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition.
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