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Your data matches 261 different statistics following compositions of up to 3 maps.
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Matching statistic: St000718
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(load all 28 compositions to match this statistic)
Values
[1] => ([],1)
=> ([(0,1)],2)
=> 2
[1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 3
[2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 4
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
Matching statistic: St001279
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [2,1] => [2]
=> 2
[1,2] => [1,0,1,0]
=> [3,1,2] => [3]
=> 3
[2,1] => [1,1,0,0]
=> [2,3,1] => [3]
=> 3
[1,2,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => [4]
=> 4
[2,3,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => [4]
=> 4
[3,1,2] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> 4
[3,2,1] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> 4
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5]
=> 5
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5]
=> 5
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5]
=> 5
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,2]
=> 5
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> 5
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5]
=> 5
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 5
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 5
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 5
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 5
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 5
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> 5
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6]
=> 6
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6]
=> 6
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6]
=> 6
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,2]
=> 6
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> 6
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6]
=> 6
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6]
=> 6
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,2]
=> 6
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> 6
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> 6
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [6]
=> 6
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> 6
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [3,3]
=> 6
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> 6
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> 6
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6]
=> 6
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,3]
=> 6
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> 6
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,2]
=> 6
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 6
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 6
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 6
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 6
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 6
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [6]
=> 6
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 6
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 6
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 6
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 6
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 6
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6]
=> 6
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000189
(load all 839 compositions to match this statistic)
(load all 839 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
St000189: Posets ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => ([],1)
=> 1 = 2 - 1
[1,2] => ([(0,1)],2)
=> 2 = 3 - 1
[2,1] => ([],2)
=> 2 = 3 - 1
[1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[2,3,1] => ([(1,2)],3)
=> 3 = 4 - 1
[3,1,2] => ([(1,2)],3)
=> 3 = 4 - 1
[3,2,1] => ([],3)
=> 3 = 4 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 4 = 5 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> 4 = 5 - 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> 4 = 5 - 1
[3,4,2,1] => ([(2,3)],4)
=> 4 = 5 - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
=> 4 = 5 - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
=> 4 = 5 - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
=> 4 = 5 - 1
[4,2,3,1] => ([(2,3)],4)
=> 4 = 5 - 1
[4,3,1,2] => ([(2,3)],4)
=> 4 = 5 - 1
[4,3,2,1] => ([],4)
=> 4 = 5 - 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 5 = 6 - 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 5 = 6 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 5 = 6 - 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 5 = 6 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 5 = 6 - 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 5 = 6 - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 5 = 6 - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 5 = 6 - 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 5 = 6 - 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 5 = 6 - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 5 = 6 - 1
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 5 = 6 - 1
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 5 = 6 - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5 = 6 - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 5 = 6 - 1
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 5 = 6 - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 5 = 6 - 1
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 5 = 6 - 1
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> 5 = 6 - 1
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 5 = 6 - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 5 = 6 - 1
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 5 = 6 - 1
[4,5,3,2,1] => ([(3,4)],5)
=> 5 = 6 - 1
[5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 5 = 6 - 1
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> 5 = 6 - 1
[5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> 5 = 6 - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> 5 = 6 - 1
[5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> 5 = 6 - 1
[] => ([],0)
=> ? = 0 - 1
Description
The number of elements in the poset.
Matching statistic: St001622
(load all 429 compositions to match this statistic)
(load all 429 compositions to match this statistic)
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
St001622: Lattices ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 4 - 1
[3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 4 - 1
[3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
[1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> 4 = 5 - 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 4 = 5 - 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 4 = 5 - 1
[2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 4 = 5 - 1
[3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 4 = 5 - 1
[3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 4 = 5 - 1
[4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 4 = 5 - 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 4 = 5 - 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 4 = 5 - 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 4 = 5 - 1
[4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 4 = 5 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> 4 = 5 - 1
[1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> 5 = 6 - 1
[1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> 5 = 6 - 1
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> 5 = 6 - 1
[1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> 5 = 6 - 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> 5 = 6 - 1
[1,4,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> 5 = 6 - 1
[2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> 5 = 6 - 1
[2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> 5 = 6 - 1
[2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> 5 = 6 - 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5 = 6 - 1
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> 5 = 6 - 1
[2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 5 = 6 - 1
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> 5 = 6 - 1
[3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> 5 = 6 - 1
[3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 5 = 6 - 1
[3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> 5 = 6 - 1
[3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> 5 = 6 - 1
[3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> 5 = 6 - 1
[3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> 5 = 6 - 1
[4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> 5 = 6 - 1
[4,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> 5 = 6 - 1
[4,5,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> 5 = 6 - 1
[4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> 5 = 6 - 1
[4,5,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> 5 = 6 - 1
[4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> 5 = 6 - 1
[5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> 5 = 6 - 1
[5,1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> 5 = 6 - 1
[5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> 5 = 6 - 1
[5,1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> 5 = 6 - 1
[5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> 5 = 6 - 1
[5,1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> 5 = 6 - 1
[] => ?
=> ? = 0 - 1
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St001332
(load all 556 compositions to match this statistic)
(load all 556 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St001332: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
St001332: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => 0 = 2 - 2
[1,2] => [1,2] => 1 = 3 - 2
[2,1] => [1,2] => 1 = 3 - 2
[1,2,3] => [1,2,3] => 2 = 4 - 2
[2,3,1] => [1,2,3] => 2 = 4 - 2
[3,1,2] => [1,2,3] => 2 = 4 - 2
[3,2,1] => [1,2,3] => 2 = 4 - 2
[1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,3,4,2] => [1,3,4,2] => 3 = 5 - 2
[2,3,1,4] => [1,4,2,3] => 3 = 5 - 2
[2,3,4,1] => [1,2,3,4] => 3 = 5 - 2
[3,4,1,2] => [1,2,3,4] => 3 = 5 - 2
[3,4,2,1] => [1,2,3,4] => 3 = 5 - 2
[4,1,2,3] => [1,2,3,4] => 3 = 5 - 2
[4,1,3,2] => [1,3,2,4] => 3 = 5 - 2
[4,2,1,3] => [1,3,2,4] => 3 = 5 - 2
[4,2,3,1] => [1,2,3,4] => 3 = 5 - 2
[4,3,1,2] => [1,2,3,4] => 3 = 5 - 2
[4,3,2,1] => [1,2,3,4] => 3 = 5 - 2
[1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,2,4,5,3] => [1,2,4,5,3] => 4 = 6 - 2
[1,3,4,2,5] => [1,3,4,2,5] => 4 = 6 - 2
[1,3,4,5,2] => [1,3,4,5,2] => 4 = 6 - 2
[1,4,5,2,3] => [1,4,5,2,3] => 4 = 6 - 2
[1,4,5,3,2] => [1,4,5,2,3] => 4 = 6 - 2
[2,3,1,4,5] => [1,4,5,2,3] => 4 = 6 - 2
[2,3,4,1,5] => [1,5,2,3,4] => 4 = 6 - 2
[2,3,4,5,1] => [1,2,3,4,5] => 4 = 6 - 2
[2,4,1,5,3] => [1,5,2,4,3] => 4 = 6 - 2
[2,4,3,5,1] => [1,2,4,3,5] => 4 = 6 - 2
[2,4,5,1,3] => [1,3,2,4,5] => 4 = 6 - 2
[2,4,5,3,1] => [1,2,4,5,3] => 4 = 6 - 2
[3,4,1,2,5] => [1,2,5,3,4] => 4 = 6 - 2
[3,4,1,5,2] => [1,5,2,3,4] => 4 = 6 - 2
[3,4,2,1,5] => [1,5,2,3,4] => 4 = 6 - 2
[3,4,2,5,1] => [1,2,5,3,4] => 4 = 6 - 2
[3,4,5,1,2] => [1,2,3,4,5] => 4 = 6 - 2
[3,4,5,2,1] => [1,2,3,4,5] => 4 = 6 - 2
[4,5,1,2,3] => [1,2,3,4,5] => 4 = 6 - 2
[4,5,1,3,2] => [1,3,2,4,5] => 4 = 6 - 2
[4,5,2,1,3] => [1,3,2,4,5] => 4 = 6 - 2
[4,5,2,3,1] => [1,2,3,4,5] => 4 = 6 - 2
[4,5,3,1,2] => [1,2,3,4,5] => 4 = 6 - 2
[4,5,3,2,1] => [1,2,3,4,5] => 4 = 6 - 2
[5,1,2,3,4] => [1,2,3,4,5] => 4 = 6 - 2
[5,1,2,4,3] => [1,2,4,3,5] => 4 = 6 - 2
[5,1,3,2,4] => [1,3,2,4,5] => 4 = 6 - 2
[5,1,3,4,2] => [1,3,4,2,5] => 4 = 6 - 2
[5,1,4,2,3] => [1,4,2,3,5] => 4 = 6 - 2
[5,1,4,3,2] => [1,4,2,3,5] => 4 = 6 - 2
[] => ? => ? = 0 - 2
Description
The number of steps on the non-negative side of the walk associated with the permutation.
Consider the walk taking an up step for each ascent, and a down step for each descent of the permutation. Then this statistic is the number of steps that begin and end at non-negative height.
Matching statistic: St000167
(load all 60 compositions to match this statistic)
(load all 60 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => [.,.]
=> [[],[]]
=> 2
[1,2] => [.,[.,.]]
=> [[],[[],[]]]
=> 3
[2,1] => [[.,.],.]
=> [[[],[]],[]]
=> 3
[1,2,3] => [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 4
[2,3,1] => [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 4
[3,1,2] => [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 4
[3,2,1] => [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 5
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> 5
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 5
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> 5
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 5
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> 5
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 5
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> 5
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 5
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> 5
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 5
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> 5
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> 6
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> 6
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 6
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> 6
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 6
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> 6
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> 6
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> 6
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> 6
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> 6
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> 6
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> 6
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> 6
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> 6
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> 6
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> 6
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> 6
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> 6
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [[[[],[[],[[],[]]]],[]],[]]
=> 6
[4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> 6
[4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> 6
[4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> 6
[4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> 6
[4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> 6
[4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [[[[[],[[],[]]],[]],[]],[]]
=> 6
[5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> 6
[5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> 6
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> 6
[5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> 6
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> 6
[5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> 6
[] => .
=> ?
=> ? = 0
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St000998
(load all 94 compositions to match this statistic)
(load all 94 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> 2
[1,2] => [2] => [1,1,0,0]
=> 3
[2,1] => [2] => [1,1,0,0]
=> 3
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 4
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 4
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 4
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 4
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 5
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 6
[2,3,1,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,3,4,1,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,3,4,5,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,4,1,5,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[2,4,3,5,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[2,4,5,1,3] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,4,5,3,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[3,4,1,2,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,4,1,5,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[3,4,2,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[3,4,2,5,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[3,4,5,1,2] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,4,5,2,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[4,5,1,2,3] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,1,3,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[4,5,2,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[4,5,2,3,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[4,5,3,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[4,5,3,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 6
[5,1,2,3,4] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,2,4,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[5,1,3,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[5,1,3,4,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[5,1,4,2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[5,1,4,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[] => ? => ?
=> ? = 0
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001012
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001012: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001012: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> 2
[1,2] => [1,2] => [1,0,1,0]
=> 3
[2,1] => [2,1] => [1,1,0,0]
=> 3
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> 4
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 4
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 4
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 5
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 5
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 5
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 5
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 5
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 5
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,3,4,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,4,1,5,3] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,4,5,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,4,5,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[3,4,1,2,5] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,1,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,2,1,5] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,2,5,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,5,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[4,5,1,2,3] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,1,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,2,1,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[] => [] => []
=> ? = 0
Description
Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001023
(load all 62 compositions to match this statistic)
(load all 62 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001023: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001023: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> 2
[1,2] => [2] => [1,1,0,0]
=> 3
[2,1] => [2] => [1,1,0,0]
=> 3
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 4
[2,3,1] => [3] => [1,1,1,0,0,0]
=> 4
[3,1,2] => [3] => [1,1,1,0,0,0]
=> 4
[3,2,1] => [2,1] => [1,1,0,0,1,0]
=> 4
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 5
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 5
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 6
[2,3,1,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,3,4,1,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,3,4,5,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,4,1,5,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[2,4,3,5,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[2,4,5,1,3] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,4,5,3,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[3,4,1,2,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,4,1,5,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[3,4,2,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[3,4,2,5,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[3,4,5,1,2] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[3,4,5,2,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[4,5,1,2,3] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[4,5,1,3,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[4,5,2,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[4,5,2,3,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[4,5,3,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[4,5,3,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 6
[5,1,2,3,4] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,2,4,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[5,1,3,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[5,1,3,4,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[5,1,4,2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[5,1,4,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[] => ? => ?
=> ? = 0
Description
Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001179
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001179: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001179: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> 2
[1,2] => [1,2] => [1,0,1,0]
=> 3
[2,1] => [2,1] => [1,1,0,0]
=> 3
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> 4
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 4
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 4
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 5
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 5
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 5
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 5
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 5
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 5
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,3,4,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,4,1,5,3] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,4,5,1,3] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[2,4,5,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 6
[3,4,1,2,5] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,1,5,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,2,1,5] => [3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,2,5,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,5,1,2] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,4,5,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 6
[4,5,1,2,3] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,1,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,2,1,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 6
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[] => [] => []
=> ? = 0
Description
Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra.
The following 251 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000054The first entry of the permutation. St000144The pyramid weight of the Dyck path. St000228The size of a partition. St000288The number of ones in a binary word. St000293The number of inversions of a binary word. St000336The leg major index of a standard tableau. St000395The sum of the heights of the peaks of a Dyck path. St000501The size of the first part in the decomposition of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001746The coalition number of a graph. St000019The cardinality of the support of a permutation. St000081The number of edges of a graph. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000316The number of non-left-to-right-maxima of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001479The number of bridges of a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001958The degree of the polynomial interpolating the values of a permutation. St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000312The number of leaves in a graph. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000050The depth or height of a binary tree. St000171The degree of the graph. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000240The number of indices that are not small excedances. St000290The major index of a binary word. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000503The maximal difference between two elements in a common block. St000505The biggest entry in the block containing the 1. St000553The number of blocks of a graph. St000632The jump number of the poset. St000636The hull number of a graph. St000703The number of deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000839The largest opener of a set partition. St000863The length of the first row of the shifted shape of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001120The length of a longest path in a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001430The number of positive entries in a signed permutation. St001497The position of the largest weak excedence of a permutation. St001523The degree of symmetry of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001725The harmonious chromatic number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St000010The length of the partition. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000210Minimum over maximum difference of elements in cycles. St000384The maximal part of the shifted composition of an integer partition. St000651The maximal size of a rise in a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St000844The size of the largest block in the direct sum decomposition of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000393The number of strictly increasing runs in a binary word. St000653The last descent of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000956The maximal displacement of a permutation. St001267The length of the Lyndon factorization of the binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001437The flex of a binary word. St000519The largest length of a factor maximising the subword complexity. St001925The minimal number of zeros in a row of an alternating sign matrix. St000806The semiperimeter of the associated bargraph. St000294The number of distinct factors of a binary word. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001759The Rajchgot index of a permutation. St000060The greater neighbor of the maximum. St000216The absolute length of a permutation. St000296The length of the symmetric border of a binary word. St000385The number of vertices with out-degree 1 in a binary tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000543The size of the conjugacy class of a binary word. St000627The exponent of a binary word. St000652The maximal difference between successive positions of a permutation. St000730The maximal arc length of a set partition. St000922The minimal number such that all substrings of this length are unique. St000924The number of topologically connected components of a perfect matching. St000957The number of Bruhat lower covers of a permutation. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001077The prefix exchange distance of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001480The number of simple summands of the module J^2/J^3. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000691The number of changes of a binary word. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000837The number of ascents of distance 2 of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000327The number of cover relations in a poset. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000719The number of alignments in a perfect matching. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001645The pebbling number of a connected graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001782The order of rowmotion on the set of order ideals of a poset. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000080The rank of the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the 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. St000784The maximum of the length and the largest part of the integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001917The order of toric promotion on the set of labellings of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000018The number of inversions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001875The number of simple modules with projective dimension at most 1. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St001468The smallest fixpoint of a permutation. St000235The number of indices that are not cyclical small weak excedances. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001401The number of distinct entries in a semistandard tableau. St001429The number of negative entries in a signed permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000029The depth of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000450The number of edges minus the number of vertices plus 2 of a graph. St000809The reduced reflection length of the permutation. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000095The number of triangles of a graph. St001706The number of closed sets in a graph. St000135The number of lucky cars of the parking function. St000044The number of vertices of the unicellular map given by a perfect matching. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000744The length of the path to the largest entry in a standard Young tableau. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001927Sparre Andersen's number of positives of a signed 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 $K[x]/(x^n)$. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001434The number of negative sum pairs of a signed permutation. St001621The number of atoms of a lattice. St000245The number of ascents of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001316The domatic number of a graph. St001494The Alon-Tarsi number of a graph. St000778The metric dimension of a graph.
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