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Your data matches 344 different statistics following compositions of up to 3 maps.
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Matching statistic: St000395
(load all 48 compositions to match this statistic)
(load all 48 compositions to match this statistic)
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 4
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 5
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> 4
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001018
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
St001018: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 4
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0]
=> 5
[1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> 4
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001034
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> 6
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St001348
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,0,0,1,0]
=> 6
[1,0,1,1,0,1,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 4
Description
The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St000029
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000029: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => 1
[1,0,1,0]
=> [3,1,2] => 2
[1,1,0,0]
=> [2,3,1] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => 3
[1,0,1,1,0,0]
=> [3,1,4,2] => 3
[1,1,0,0,1,0]
=> [2,4,1,3] => 3
[1,1,0,1,0,0]
=> [4,3,1,2] => 4
[1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 4
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 4
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 6
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 5
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 4
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 5
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 6
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
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: St000189
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
Description
The number of elements in the poset.
Matching statistic: St000197
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000197: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 3
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 3
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 4
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 4
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 5
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 4
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 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]]
=> 5
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 6
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 5
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 5
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 6
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
Description
The number of entries equal to positive one in the alternating sign matrix.
Matching statistic: St000229
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> 1
[1,0,1,0]
=> {{1},{2}}
=> 2
[1,1,0,0]
=> {{1,2}}
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 3
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 3
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 4
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 4
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 5
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 4
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 4
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 4
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 5
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 6
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 5
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 4
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 6
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 5
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
Description
Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition.
This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is
$$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$
This statistic is called ''dimension index'' in [2]
Matching statistic: St000394
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001717
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> ([],1)
=> 1
[1,0,1,0]
=> ([(0,1)],2)
=> 2
[1,1,0,0]
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
Description
The largest size of an interval in a poset.
The following 334 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000018The number of inversions of a permutation. St000030The sum of the descent differences of a permutations. St000224The sorting index of a permutation. St000228The size of a partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000728The dimension of a set partition. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001622The number of join-irreducible elements of a lattice. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001726The number of visible inversions of a permutation. St001746The coalition number of a graph. St001869The maximum cut size of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St000004The major index of a permutation. St000067The inversion number of the alternating sign matrix. St000081The number of edges of a graph. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000223The number of nestings in the permutation. St000246The number of non-inversions of a permutation. St000332The positive inversions of an alternating sign matrix. St000548The number of different non-empty partial sums of an integer partition. St000794The mak of a permutation. St000795The mad of a permutation. St000874The position of the last double rise in a Dyck path. St000883The number of longest increasing subsequences of a permutation. 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. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001725The harmonious chromatic number of a graph. St001727The number of invisible inversions of a permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000070The number of antichains in a poset. St000108The number of partitions contained in the given partition. St000171The degree of the graph. St000438The position of the last up step in a Dyck path. St000625The sum of the minimal distances to a greater element. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000656The number of cuts of a poset. St001645The pebbling number of a connected graph. St000494The number of inversions of distance at most 3 of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001875The number of simple modules with projective dimension at most 1. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000670The reversal length of a permutation. St000680The Grundy value for Hackendot on posets. St001343The dimension of the reduced incidence algebra of a poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001554The number of distinct nonempty subtrees of a binary tree. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000703The number of deficiencies of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000259The diameter of a connected graph. St000624The normalized sum of the minimal distances to a greater element. St000060The greater neighbor of the maximum. St000062The length of the longest increasing subsequence of the permutation. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000308The height of the tree associated to a permutation. 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. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000702The number of weak deficiencies of a permutation. St000730The maximal arc length of a set partition. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001286The annihilation 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. St001405The number of bonds in a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001821The sorting index of a signed permutation. St000133The "bounce" of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000519The largest length of a factor maximising the subword complexity. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000837The number of ascents of distance 2 of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in 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 $K[x]/(x^n)$. St001298The number of repeated entries in the Lehmer code of a permutation. St001323The independence gap of a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001565The number of arithmetic progressions of length 2 in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St001077The prefix exchange distance of a permutation. St001416The length of a longest palindromic factor of a binary word. St000806The semiperimeter of the associated bargraph. St000264The girth of a graph, which is not a tree. St000080The rank of the poset. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000632The jump number of the poset. 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. St001820The size of the image of the pop stack sorting operator. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000527The width of the poset. St000528The height of a poset. St000845The maximal number of elements covered by an element in a poset. St000906The length of the shortest maximal chain in a poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St000462The major index minus the number of excedences of a permutation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000307The number of rowmotion orbits of a poset. St001644The dimension of a graph. St001812The biclique partition number of a graph. St001638The book thickness of a graph. St001555The order of a signed permutation. St000306The bounce count of a Dyck path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000116The major index of a semistandard tableau obtained by standardizing. St000168The number of internal nodes of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000422The energy of a graph, if it is integral. 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. St000736The last entry in the first row of a semistandard tableau. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St000989The number of final rises 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. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001487The number of inner corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001570The minimal number of edges to add to make a graph Hamiltonian. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001596The number of two-by-two squares inside a skew partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001684The reduced word complexity of 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. St001768The number of reduced words of a signed permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001928The number of non-overlapping descents in a permutation. St001946The number of descents in a parking function. St000002The number of occurrences of the pattern 123 in a permutation. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000075The orbit size of a standard tableau under promotion. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000105The number of blocks in the set partition. St000141The maximum drop size of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000210Minimum over maximum difference of elements in cycles. St000211The rank of the set partition. St000251The number of nonsingleton blocks of a set partition. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000357The number of occurrences of the pattern 12-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000443The number of long tunnels of a Dyck path. St000461The rix statistic of a permutation. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000662The staircase size of the code of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000798The makl of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. 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. St000925The number of topologically connected components of a set partition. St000956The maximal displacement of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001114The number of odd descents of a permutation. St001117The game chromatic index of a graph. 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 $K[x]/(x^n)$. St001187The number of simple modules with grade at least one 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 $eA$. 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. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001497The position of the largest weak excedence of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001517The length of a longest pair of twins in a permutation. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001642The Prague dimension of a graph. St001649The length of a longest trail in a graph. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001874Lusztig's a-function for the symmetric group. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001935The number of ascents in a parking function. St001964The interval resolution global dimension of a poset. St000094The depth of an ordered tree. St000135The number of lucky cars of the parking function. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000521The number of distinct subtrees of an ordered tree. St000744The length of the path to the largest entry in a standard Young tableau. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000988The orbit size of a permutation under Foata's bijection. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over 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. St001417The length of a longest palindromic subword of a binary word. 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. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000735The last entry on the main diagonal of a standard tableau.
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