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Your data matches 856 different statistics following compositions of up to 3 maps.
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Matching statistic: St000001
(load all 59 compositions to match this statistic)
(load all 59 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 16
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 6
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 5
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[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] => 1
Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Matching statistic: St000003
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> 1
[1,0,1,0]
=> [1]
=> 1
[1,1,0,0]
=> []
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1,0,0,0]
=> []
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 16
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 5
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 6
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St001768
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001768: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001768: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 16
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => 5
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 6
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,1,2] => 5
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [4,2,1,3] => 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
Description
The number of reduced words of a signed permutation.
This is the number of ways to write a permutation as a minimal length product of simple reflections.
Matching statistic: St001595
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001595: Skew partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> 16
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> 5
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> 6
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> 5
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 1
Description
The number of standard Young tableaux of the skew partition.
Matching statistic: St001435
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [[1],[]]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {5,5,6,16} - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {5,5,6,16} - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {5,5,6,16} - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {5,5,6,16} - 1
Description
The number of missing boxes in the first row.
Matching statistic: St000233
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => {{1,2}}
=> 0 = 1 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0 = 1 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => {{1,2,4},{3,5,6}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => {{1,2,3,4,5,6}}
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => {{1,2,3,4,5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => {{1,3,4,6},{2,5}}
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => {{1,2,4,5,7,8},{3,6}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => {{1,2,5},{3,6},{4,7,8}}
=> ? ∊ {2,5,5,6,16} - 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => {{1,2,5},{3,4,6,7,8}}
=> ? ∊ {2,5,5,6,16} - 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => {{1,3,5,7},{2,4,6,8}}
=> ? ∊ {2,5,5,6,16} - 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => {{1,2,3,5,6},{4,7,8}}
=> ? ∊ {2,5,5,6,16} - 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => {{1,3,4,5,7,8},{2,6}}
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => {{1,2,4,5,6,8},{3,7}}
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => {{1,4,5,8},{2,6},{3,7}}
=> ? ∊ {2,5,5,6,16} - 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> 0 = 1 - 1
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000496
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => {{1,2}}
=> 0 = 1 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0 = 1 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => {{1,2,4},{3,5,6}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => {{1,2,3,4,5,6}}
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => {{1,2,3,4,5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => {{1,3,4,6},{2,5}}
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => {{1,2,4,5,7,8},{3,6}}
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => {{1,2,5},{3,6},{4,7,8}}
=> ? ∊ {2,5,5,6,16} - 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => {{1,2,5},{3,4,6,7,8}}
=> ? ∊ {2,5,5,6,16} - 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => {{1,3,5,7},{2,4,6,8}}
=> ? ∊ {2,5,5,6,16} - 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => {{1,2,3,5,6},{4,7,8}}
=> ? ∊ {2,5,5,6,16} - 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => {{1,3,4,5,7,8},{2,6}}
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => {{1,2,4,5,6,8},{3,7}}
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => {{1,4,5,8},{2,6},{3,7}}
=> ? ∊ {2,5,5,6,16} - 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> 0 = 1 - 1
Description
The rcs statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St000884
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 77%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [3,2,1,4] => 0 = 1 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [3,2,1,4,5,6] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,5,4,2,3,6] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [3,2,5,4,1,6,7,8] => ? ∊ {3,5,5,6,16} - 1
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,2,1,7,6,4,5,8] => ? ∊ {3,5,5,6,16} - 1
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [5,4,3,2,1,6,7,8] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,5,4,7,6,2,3,8] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,5,4,2,3,6,7,8] => ? ∊ {3,5,5,6,16} - 1
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,7,6,5,4,2,3,8] => ? ∊ {3,5,5,6,16} - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,7,6,3,4,5,8] => ? ∊ {3,5,5,6,16} - 1
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St000589
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 33% ●values known / values provided: 73%●distinct values known / distinct values provided: 33%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 33% ●values known / values provided: 73%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => {{1,2}}
=> 0 = 1 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0 = 1 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => {{1,2,4},{3,5,6}}
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => {{1,2,3,4,5,6}}
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => {{1,2,3,4,5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => {{1,3,4,6},{2,5}}
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => {{1,2,4,5,7,8},{3,6}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => {{1,2,5},{3,6},{4,7,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => {{1,2,5},{3,4,6,7,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => {{1,3,5,7},{2,4,6,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => {{1,2,3,5,6},{4,7,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => {{1,3,4,5,7,8},{2,6}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => {{1,2,4,5,6,8},{3,7}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => {{1,4,5,8},{2,6},{3,7}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000606
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000606: Set partitions ⟶ ℤResult quality: 33% ●values known / values provided: 73%●distinct values known / distinct values provided: 33%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000606: Set partitions ⟶ ℤResult quality: 33% ●values known / values provided: 73%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => {{1,2}}
=> 0 = 1 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => {{1,2,3,4}}
=> 0 = 1 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => {{1,2,4},{3,5,6}}
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => {{1,2,3,4,5,6}}
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => {{1,2,3,4,5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => {{1,3,4,6},{2,5}}
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => {{1,2,4,5,7,8},{3,6}}
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => {{1,2,5},{3,6},{4,7,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => {{1,2,5},{3,4,6,7,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [3,4,6,8,1,2,5,7] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => {{1,3,5,7},{2,4,6,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => {{1,2,3,5,6},{4,7,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => {{1,3,4,5,7,8},{2,6}}
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => {{1,2,3,4,5,6,7,8}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => {{1,2,4,5,6,8},{3,7}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => {{1,4,5,8},{2,6},{3,7}}
=> ? ∊ {3,3,5,5,6,16} - 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> ? ∊ {3,3,5,5,6,16} - 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block.
The following 846 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St001438The number of missing boxes of a skew partition. St000100The number of linear extensions of a poset. St000665The number of rafts of a permutation. St001050The number of terminal closers of a set partition. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000237The number of small exceedances. St000534The number of 2-rises of a permutation. St001862The number of crossings of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001964The interval resolution global dimension of a poset. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000834The number of right outer peaks of a permutation. St001621The number of atoms of a lattice. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St001083The number of boxed occurrences of 132 in a permutation. St001115The number of even descents of a permutation. St001846The number of elements which do not have a complement in the lattice. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000028The number of stack-sorts needed to sort a permutation. St000352The Elizalde-Pak rank of a permutation. St000667The greatest common divisor of the parts of the partition. St000669The number of permutations obtained by switching ascents or descents of size 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001527The cyclic permutation representation number of an integer partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000022The number of fixed points of 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. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000563The number of overlapping pairs of blocks of a set partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000842The breadth of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001330The hat guessing number of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000010The length of the partition. St000014The number of parking functions supported by a Dyck path. St000147The largest part of an integer partition. St000160The multiplicity of the smallest part of a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000295The length of the border of a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000548The number of different non-empty partial sums of an integer partition. St000734The last entry in the first row of a standard tableau. St000764The number of strong records in an integer composition. St000935The number of ordered refinements of an integer partition. St000947The major index east count of a Dyck path. St000993The multiplicity of the largest part of an integer partition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001161The major index north count of a 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. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001571The Cartan determinant of the integer partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001933The largest multiplicity of a part in an integer partition. St000007The number of saliances of the permutation. St000359The number of occurrences of the pattern 23-1. St000455The second largest eigenvalue of a graph if it is integral. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000647The number of big descents of a permutation. St000731The number of double exceedences of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001394The genus of a permutation. St001875The number of simple modules with projective dimension at most 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000045The number of linear extensions of a binary tree. St000181The number of connected components of the Hasse diagram for the poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. 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. St001490The number of connected components of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001890The maximum magnitude of the Möbius function of a poset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St000456The monochromatic index of a connected graph. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000441The number of successions of a permutation. 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. St000891The number of distinct diagonal sums of a permutation matrix. St001095The number of non-isomorphic posets with precisely one further covering relation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St000068The number of minimal elements in a poset. St000260The radius of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000460The hook length of the last cell along the main diagonal of an integer partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000762The sum of the positions of the weak records of an integer composition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001118The acyclic chromatic index of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001498The normalised height of a Nakayama algebra with magnitude 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000648The number of 2-excedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001153The number of blocks with even minimum in a set partition. St001625The Möbius invariant of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001867The number of alignments of type EN of a signed permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St000091The descent variation of a composition. St000217The number of occurrences of the pattern 312 in a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000562The number of internal points of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000779The tier of a permutation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000492The rob statistic of a set partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000535The rank-width of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000567The sum of the products of all pairs of parts. St000570The Edelman-Greene number of a permutation. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000681The Grundy value of Chomp on Ferrers diagrams. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000694The number of affine bounded permutations that project to a given permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000805The number of peaks of the associated bargraph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000942The number of critical left to right maxima of the parking functions. St000989The number of final rises of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001151The number of blocks with odd minimum. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001220The width of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001340The cardinality of a minimal non-edge isolating set of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001378The product of the cohook lengths of the integer partition. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001729The number of visible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001884The number of borders of a binary word. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001937The size of the center of a parking function. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001943The sum of the squares of the hook lengths of an integer partition. St001946The number of descents in a parking function. St001949The rigidity index of a graph. St000017The number of inversions of a standard tableau. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000058The order of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000090The variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000236The number of cyclical small weak excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000370The genus of a graph. St000372The number of mid points of increasing 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 $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000469The distinguishing number of a graph. St000485The length of the longest cycle of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000487The length of the shortest cycle 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. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000623The number of occurrences of the pattern 52341 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000636The hull number of a graph. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000679The pruning number of an ordered tree. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000758The length of the longest staircase fitting into an integer composition. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000839The largest opener of a set partition. St000872The number of very big descents 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. St000895The number of ones on the main diagonal of an alternating sign matrix. St000906The length of the shortest maximal chain in a poset. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001093The detour number of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001309The number of four-cliques in 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. St001323The independence gap of a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001352The number of internal nodes in the modular decomposition of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001434The number of negative sum pairs of a signed permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001470The cyclic holeyness of a permutation. St001513The number of nested exceedences of 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. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. 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. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001674The number of vertices of the largest induced star graph in the graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001691The number of kings in a graph. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001797The number of overfull subgraphs of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001847The number of occurrences of the pattern 1432 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. 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. St000230Sum of the minimal elements of the blocks of a set partition. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001487The number of inner corners of a skew partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000088The row sums of the character table of the symmetric group. St000108The number of partitions contained in the given partition. St000120The number of left tunnels of a Dyck path. St000144The pyramid weight of the Dyck path. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000284The Plancherel distribution on integer partitions. St000288The number of ones in a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000306The bounce count of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000347The inversion sum of a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000418The number of Dyck paths that are weakly below a 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. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000517The Kreweras number of an integer partition. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000627The exponent of a binary word. St000628The balance of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. 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. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000706The product of the factorials of the multiplicities of an integer partition. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000759The smallest missing part in an integer partition. St000792The Grundy value for the game of ruler on a binary word. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000913The number of ways to refine the partition into singletons. St000914The sum of the values of the Möbius function of a poset. St000922The minimal number such that all substrings of this length are unique. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000939The number of characters of the symmetric group whose value on the partition is positive. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules 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. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two 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. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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:
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 $eA$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. 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. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. 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. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. 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. St001372The length of a longest cyclic run of ones of a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001462The number of factors of a standard tableaux under concatenation. St001471The magnitude of a Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001485The modular major index of a binary word. St001488The number of corners of a skew partition. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001597The Frobenius rank of a skew partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001614The cyclic permutation representation number of a skew partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001721The degree of a binary word. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001881The number of factors of a lattice as a Cartesian product of lattices. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001930The weak major index of a binary word. St001955The number of natural descents for set-valued two row standard Young tableaux. St001959The product of the heights of the peaks of a Dyck path. St000219The number of occurrences of the pattern 231 in a permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000422The energy of a graph, if it is integral. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St001301The first Betti number of the order complex associated with the poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001616The number of neutral elements in a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000717The number of ordinal summands of a poset. St001618The cardinality of the Frattini sublattice of a lattice. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001060The distinguishing index of a graph. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001260The permanent of an alternating sign matrix. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St000894The trace of an alternating sign matrix. St000071The number of maximal chains in a poset. St000259The diameter of a connected graph. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000527The width of the poset. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000909The number of maximal chains of maximal size in a poset. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000124The cardinality of the preimage of the Simion-Schmidt map. St001644The dimension of a graph. St001645The pebbling number of a connected graph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000096The number of spanning trees of a graph. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000307The number of rowmotion orbits of a poset. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. 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. St000353The number of inner valleys of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000654The first descent of a permutation. St000729The minimal arc length of a set partition. St000739The first entry in the last row of a semistandard tableau. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000883The number of longest increasing subsequences of a permutation. St000958The number of Bruhat factorizations of a permutation. St000968We 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 $c_{n−1}$. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001204Call 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001410The minimal entry of a semistandard tableau. St001778The largest greatest common divisor of an element and its image in a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001828The Euler characteristic of a graph. St001928The number of non-overlapping descents in a permutation. St000084The number of subtrees. St000089The absolute variation of a composition. St000092The number of outer peaks of a permutation. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000105The number of blocks in the set partition. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000134The size of the orbit of an alternating sign matrix under gyration. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000251The number of nonsingleton blocks of a set partition. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000401The size of the symmetry class of a permutation. St000402Half the size of the symmetry class of a permutation. St000417The size of the automorphism group of the ordered tree. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000632The jump number of the poset. St000663The number of right floats of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000836The number of descents of distance 2 of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000943The number of spots the most unlucky car had to go further in a parking function. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001075The minimal size of a block of a set partition. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001423The number of distinct cubes in a binary word. St001429The number of negative entries in a signed permutation. St001517The length of a longest pair of twins in a permutation. St001524The degree of symmetry of a binary word. St001536The number of cyclic misalignments of a permutation. St001555The order of a signed permutation. St001557The number of inversions of the second entry of a permutation. 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. St001623The number of doubly irreducible elements of a lattice. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001730The number of times the path corresponding to a binary word crosses the base line. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001783The number of odd automorphisms of a graph. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001893The flag descent of a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001926Sparre Andersen's position of the maximum of a signed permutation. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one.
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