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Your data matches 156 different statistics following compositions of up to 3 maps.
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Matching statistic: St000011
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 2 - 1
[1,2] => [1,0,1,0]
=> 2 = 3 - 1
[2,1] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000439
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 2
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 3
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
Description
The position of the first down step of a Dyck path.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> 1 = 2 - 1
[1,2] => ([],2)
=> [1,1]
=> 2 = 3 - 1
[2,1] => ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[1,2,3] => ([],3)
=> [1,1,1]
=> 3 = 4 - 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> 2 = 3 - 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> 2 = 3 - 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 4 = 5 - 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 3 = 4 - 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 3 = 4 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> 3 = 4 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 3 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1 = 2 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 5 = 6 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 4 = 5 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 4 = 5 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 4 = 5 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3 = 4 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 3 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2 = 3 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 3 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2 = 3 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 3 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2 = 3 - 1
Description
The length of the partition.
Matching statistic: St000025
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000383
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1] => 1 = 2 - 1
[1,2] => ([],2)
=> [2] => 2 = 3 - 1
[2,1] => ([(0,1)],2)
=> [1,1] => 1 = 2 - 1
[1,2,3] => ([],3)
=> [3] => 3 = 4 - 1
[1,3,2] => ([(1,2)],3)
=> [1,2] => 2 = 3 - 1
[2,1,3] => ([(1,2)],3)
=> [1,2] => 2 = 3 - 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1 = 2 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1 = 2 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 1 = 2 - 1
[1,2,3,4] => ([],4)
=> [4] => 4 = 5 - 1
[1,2,4,3] => ([(2,3)],4)
=> [1,3] => 3 = 4 - 1
[1,3,2,4] => ([(2,3)],4)
=> [1,3] => 3 = 4 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2 = 3 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2 = 3 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2 = 3 - 1
[2,1,3,4] => ([(2,3)],4)
=> [1,3] => 3 = 4 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => 2 = 3 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2 = 3 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1 = 2 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1 = 2 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 2 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2 = 3 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1 = 2 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2 = 3 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 2 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1 = 2 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1 = 2 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1 = 2 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 2 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 2 - 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1 = 2 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1 = 2 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1 = 2 - 1
[1,2,3,4,5] => ([],5)
=> [5] => 5 = 6 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> [1,4] => 4 = 5 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> [1,4] => 4 = 5 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [1,1,3] => 3 = 4 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [1,1,3] => 3 = 4 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3 = 4 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> [1,4] => 4 = 5 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,3] => 3 = 4 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => 3 = 4 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2 = 3 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2 = 3 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2 = 3 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [1,1,3] => 3 = 4 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2 = 3 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 3 = 4 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2 = 3 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 2 = 3 - 1
Description
The last part of an integer composition.
Matching statistic: St001462
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 1 = 2 - 1
[1,2] => [1,2] => [[1,2]]
=> 2 = 3 - 1
[2,1] => [2,1] => [[1],[2]]
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 3 = 4 - 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2 = 3 - 1
[2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 2 - 1
[3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 4 - 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 4 - 1
[1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 4 - 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 3 - 1
[2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 3 - 1
[2,3,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[2,4,1,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 2 - 1
[2,4,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 2 - 1
[3,1,2,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 3 - 1
[3,2,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[3,4,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 2 - 1
[3,4,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 2 - 1
[4,1,2,3] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[4,1,3,2] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1 = 2 - 1
[4,2,1,3] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 2 - 1
[4,2,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 2 - 1
[4,3,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3 = 4 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3 = 4 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 2 = 3 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 3 = 4 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 3 = 4 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2 = 3 - 1
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St001028
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001028: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> 2
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 3
[2,1] => 0 => [2] => [1,1,0,0]
=> 2
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 4
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 3
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 4
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 2
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
Description
Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000097
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => ([],1)
=> 1 = 2 - 1
[1,2] => 1 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[2,1] => 0 => [2] => ([],2)
=> 1 = 2 - 1
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,3,1] => 00 => [3] => ([],3)
=> 1 = 2 - 1
[3,1,2] => 00 => [3] => ([],3)
=> 1 = 2 - 1
[3,2,1] => 00 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,4,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[2,4,1,3] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[2,4,3,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,4,2] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,4,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[3,4,1,2] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[3,4,2,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,1,2,3] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,1,3,2] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,2,1,3] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,2,3,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,3,1,2] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,3,2,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => ([],1)
=> 1 = 2 - 1
[1,2] => 1 => [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[2,1] => 0 => [2] => ([],2)
=> 1 = 2 - 1
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,3,1] => 00 => [3] => ([],3)
=> 1 = 2 - 1
[3,1,2] => 00 => [3] => ([],3)
=> 1 = 2 - 1
[3,2,1] => 00 => [3] => ([],3)
=> 1 = 2 - 1
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 2 = 3 - 1
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,4,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[2,4,1,3] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[2,4,3,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,4,2] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,4,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[3,4,1,2] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[3,4,2,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,1,2,3] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,1,3,2] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,2,1,3] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,2,3,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,3,1,2] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[4,3,2,1] => 000 => [4] => ([],4)
=> 1 = 2 - 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
=> 2 = 3 - 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000147
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1]
=> [1]
=> 1 = 2 - 1
[1,2] => ([],2)
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,1]
=> 1 = 2 - 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
Description
The largest part of an integer partition.
The following 146 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000733The row containing the largest entry of a standard tableau. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama 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:
St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000234The number of global ascents of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000925The number of topologically connected components of a set partition. St000502The number of successions of a set partitions. St000932The number of occurrences of the pattern UDU in a Dyck path. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000306The bounce count of a Dyck path. St001363The Euler characteristic of a graph according to Knill. St000717The number of ordinal summands of a poset. St000069The number of maximal elements of a poset. St000245The number of ascents of a permutation. St000287The number of connected components of a graph. St001828The Euler characteristic of a graph. St000286The number of connected components of the complement of a graph. St000153The number of adjacent cycles of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001479The number of bridges of a graph. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000068The number of minimal elements in a poset. St000237The number of small exceedances. St000553The number of blocks of a graph. St000906The length of the shortest maximal chain in a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000542The number of left-to-right-minima 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. St001322The size of a minimal independent dominating set in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001339The irredundance number of a graph. St001118The acyclic chromatic index of a graph. St001530The depth of a Dyck path. St001060The distinguishing index of a graph. 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 $eA$. St000648The number of 2-excedences of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000214The number of adjacencies of a permutation. St000054The first entry of the permutation. St000989The number of final rises of a permutation. St000843The decomposition number of a perfect matching. St000990The first ascent of a permutation. St000654The first descent of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000996The number of exclusive left-to-right maxima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001330The hat guessing number of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000260The radius of a connected graph. St000740The last entry of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000308The height of the tree associated to a permutation. St000738The first entry in the last row of a standard tableau. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000056The decomposition (or block) number of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000084The number of subtrees. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000883The number of longest increasing subsequences of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000822The Hadwiger number of the graph. 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. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St001812The biclique partition number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000264The girth of a graph, which is not a tree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001657The number of twos in an integer partition. St000455The second largest eigenvalue of a graph if it is integral. 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 $eA$. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000924The number of topologically connected components of a perfect matching. St000454The largest eigenvalue of a graph if it is integral. St000181The number of connected components of the Hasse diagram for the poset. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000732The number of double deficiencies of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001570The minimal number of edges to add to make a graph Hamiltonian. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001889The size of the connectivity set of a signed permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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