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Your data matches 424 different statistics following compositions of up to 3 maps.
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Matching statistic: St000001
(load all 33 compositions to match this statistic)
(load all 33 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 7 compositions to match this statistic)
(load all 7 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 9 compositions to match this statistic)
(load all 9 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: St000100
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[1,1,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[1,1,1,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 16
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 6
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 5
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> ? = 1
[1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
Description
The number of linear extensions of a poset.
Matching statistic: St000036
(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
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 67%
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> [3,5,7,1,2,4,6,8] => ? = 16
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => ? = 5
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => ? = 6
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => ? = 3
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => ? = 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => 5
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ? = 2
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => 3
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => ? = 2
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => ? = 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 1
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.
Matching statistic: St000405
(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
Mp00252: Permutations —restriction⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,4,1,3,5] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [2,5,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [3,4,1,2,5] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [3,5,1,2,4] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,1,2,3] => 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] => [2,4,6,1,3,5,7] => ? = 16 - 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] => [2,4,7,1,3,5,6] => ? = 5 - 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] => [2,5,6,1,3,4,7] => ? = 6 - 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] => [2,5,7,1,3,4,6] => 2 = 3 - 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] => [2,6,7,1,3,4,5] => 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] => [3,4,6,1,2,5,7] => ? = 5 - 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] => [3,4,7,1,2,5,6] => ? = 2 - 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] => [3,5,6,1,2,4,7] => 2 = 3 - 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] => [3,5,7,1,2,4,6] => ? = 2 - 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] => [3,6,7,1,2,4,5] => ? = 1 - 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] => [4,5,6,1,2,3,7] => 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] => [4,5,7,1,2,3,6] => ? = 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] => [4,6,7,1,2,3,5] => ? = 1 - 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] => [5,6,7,1,2,3,4] => 0 = 1 - 1
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St001686
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> []
=> []
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> [[1]]
=> [[1]]
=> ? = 1
[1,1,0,0]
=> []
=> []
=> ?
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> ? = 1
[1,1,1,0,0,0]
=> []
=> []
=> ?
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? = 16
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? = 5
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? = 6
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 5
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> ? = 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ?
=> ? = 1
Description
The order of promotion on a Gelfand-Tsetlin pattern.
Matching statistic: St000031
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? = 16
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ? = 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 6
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? = 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? = 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ? = 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ? = 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000153
(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
Mp00239: Permutations —Corteel⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,2,4,3,1,5] => 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6,2,5,3,4,1] => 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,4,3,2,1,5] => 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [6,5,3,2,4,1] => 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => 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] => [8,2,6,4,3,5,1,7] => ? = 16
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => [8,2,7,4,3,5,6,1] => ? = 5
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [2,5,6,8,1,3,4,7] => [8,2,6,5,4,3,1,7] => ? = 6
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => [8,2,7,5,4,3,6,1] => ? = 3
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => [8,2,7,6,4,5,3,1] => ? = 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] => [8,6,3,4,2,5,1,7] => ? = 5
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [3,4,7,8,1,2,5,6] => [8,7,3,4,2,5,6,1] => ? = 2
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [3,5,6,8,1,2,4,7] => [8,6,3,5,4,2,1,7] => ? = 3
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => [8,7,3,5,4,2,6,1] => ? = 2
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [3,6,7,8,1,2,4,5] => [8,7,3,6,4,5,2,1] => ? = 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] => [8,6,5,4,3,2,1,7] => 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] => [8,7,5,4,3,2,6,1] => ? = 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] => [8,7,6,4,3,5,2,1] => ? = 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] => [8,7,6,5,4,3,2,1] => 1
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
The following 414 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001095The number of non-isomorphic posets with precisely one further covering relation. St000842The breadth of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001625The Möbius invariant of a lattice. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001344The neighbouring number of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001857The number of edges in the reduced word graph of a signed 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. St000570The Edelman-Greene number of a permutation. St000668The least common multiple of the parts of the partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000933The number of multipartitions of sizes given by an integer partition. St001061The number of indices that are both descents and recoils of a permutation. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. 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. St001413Half the length of the longest even length palindromic prefix of a binary word. 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. 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. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. 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. St000091The descent 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. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. 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. St000308The height of the tree associated to a permutation. St000338The number of pixed points of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000370The genus of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000478Another weight of a partition according to Alladi. 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. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000622The number of occurrences of the patterns 2143 or 4231 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. St000666The number of right tethers 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. St000779The tier of 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. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000934The 2-degree of an integer partition. 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. St001062The maximal size of a block of a set partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. 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. 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. 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. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001470The cyclic holeyness of a permutation. 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. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. 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. 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. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in 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. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001871The number of triconnected components of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000264The girth of a graph, which is not a tree. St000181The number of connected components of the Hasse diagram for the poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001890The maximum magnitude of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000879The number of long braid edges in the graph of braid moves of a permutation. 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 $eA$. 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 $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001060The distinguishing index of a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. 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 $eA$. St001330The hat guessing number of a graph. St001964The interval resolution global dimension of a poset. St000260The radius of a connected graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000124The cardinality of the preimage of the Simion-Schmidt map. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001720The minimal length of a chain of small intervals in a lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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 $K[x]/(x^n)$. 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. St000253The crossing number of a set partition. St000254The nesting number of a set partition. 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. 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. St000456The monochromatic index of a connected graph. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. 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. St000694The number of affine bounded permutations that project to a given 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St000883The number of longest increasing subsequences of a permutation. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. 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. St001050The number of terminal closers of a set partition. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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. St001260The permanent of an alternating sign matrix. St001273The projective dimension of the first term in an injective coresolution of the regular module. 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001410The minimal entry of a semistandard tableau. 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. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. 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. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001928The number of non-overlapping descents in a permutation. St001937The size of the center of a parking function. St001946The number of descents in a parking function. St000022The number of fixed points of a permutation. St000039The number of crossings of a permutation. St000058The order of a permutation. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000089The absolute variation of a composition. St000090The 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. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St000251The number of nonsingleton blocks of a set partition. St000259The diameter of a connected graph. 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. St000295The length of the border of a binary word. 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. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. 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. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of 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. St000563The number of overlapping pairs of blocks of 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. 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. 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. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. 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. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. 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. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000679The pruning number of an ordered tree. St000709The number of occurrences of 14-2-3 or 14-3-2. St000731The number of double exceedences of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. 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. 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. St000893The number of distinct diagonal sums of an alternating sign matrix. St000894The trace of an alternating sign matrix. 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. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. 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. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. 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. 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 $K[x]/(x^n)$. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. 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. St001423The number of distinct cubes in a binary word. St001429The number of negative entries in a signed permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001513The number of nested exceedences of a permutation. St001517The length of a longest pair of twins in a permutation. St001520The number of strict 3-descents. St001524The degree of symmetry of a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001536The number of cyclic misalignments of a permutation. St001552The number of inversions between excedances and fixed points 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. 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. St001597The Frobenius rank of a skew partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. 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. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001783The number of odd automorphisms of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE 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. St000891The number of distinct diagonal sums of a permutation matrix. 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. St001875The number of simple modules with projective dimension at most 1. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001645The pebbling number of a connected graph.
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