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Your data matches 233 different statistics following compositions of up to 3 maps.
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Matching statistic: St000413
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(load all 2 compositions to match this statistic)
St000413: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> 1 = 0 + 1
[[],[]]
=> 1 = 0 + 1
[[[]]]
=> 1 = 0 + 1
[[],[],[]]
=> 1 = 0 + 1
[[],[[]]]
=> 2 = 1 + 1
[[[]],[]]
=> 2 = 1 + 1
[[[],[]]]
=> 1 = 0 + 1
[[[[]]]]
=> 1 = 0 + 1
[[],[],[],[]]
=> 1 = 0 + 1
[[],[],[[]]]
=> 3 = 2 + 1
[[],[[]],[]]
=> 3 = 2 + 1
[[],[[],[]]]
=> 2 = 1 + 1
[[],[[[]]]]
=> 2 = 1 + 1
[[[]],[],[]]
=> 3 = 2 + 1
[[[]],[[]]]
=> 1 = 0 + 1
[[[],[]],[]]
=> 2 = 1 + 1
[[[[]]],[]]
=> 2 = 1 + 1
[[[],[],[]]]
=> 1 = 0 + 1
[[[],[[]]]]
=> 2 = 1 + 1
[[[[]],[]]]
=> 2 = 1 + 1
[[[[],[]]]]
=> 1 = 0 + 1
[[[[[]]]]]
=> 1 = 0 + 1
Description
The number of ordered trees with the same underlying unordered tree.
Matching statistic: St000282
Mp00047: Ordered trees —to poset⟶ Posets
St000282: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000282: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1 = 0 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1 = 0 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
Description
The size of the preimage of the map 'to poset' from Ordered trees to Posets.
Matching statistic: St001083
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(load all 2 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [2,1] => 0
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => 0
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St001683
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [2,1] => 0
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => 0
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St001687
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [2,1] => 0
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => 0
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St001689
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(load all 2 compositions to match this statistic)
Values
[[]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 0
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
Description
The number of celebrities in a graph.
A celebrity is a vertex such that more than half of its neighbours have smaller degree.
Matching statistic: St000089
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
St000089: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
St000089: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [[1],[]]
=> [1] => 0
[[],[]]
=> [1,0,1,0]
=> [[1,1],[]]
=> [1,1] => 0
[[[]]]
=> [1,1,0,0]
=> [[2],[]]
=> [2] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> [3] => 0
[[[[]]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => 0
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [2,3] => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3] => 0
Description
The absolute variation of a composition.
Matching statistic: St000317
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [2,1] => [2,1] => 0
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => [2,3,1] => 0
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => [1,3,2] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,3,4,2] => 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [3,4,2,1] => 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,4,3] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,2,3] => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,5,2,3,1] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,2,4] => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,3,2,1,4] => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,5,1,3,2] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,2,5,3,1] => 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 0
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Matching statistic: St000360
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000360: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000360: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [2,1] => [2,1] => 0
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => [2,3,1] => 0
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,3,1] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [3,4,1,2] => 0
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [2,4,3,1] => 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,3,4,1] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,5,2,3,1] => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,4,2,5,1] => 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,3,1,4] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [3,4,5,1,2] => 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,3,4,1,2] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [2,4,5,3,1] => 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [4,5,1,2,3] => 0
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [2,3,5,4,1] => 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5,2,4,3,1] => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,5,4,1,2] => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,4,1,5,3] => 0
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
Description
The number of occurrences of the pattern 32-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $32\!\!-\!\!1$.
Matching statistic: St000663
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [2,1] => [2,1] => 0
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[[[]]]
=> [1,1,0,0]
=> [2,3,1] => [3,2,1] => 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 0
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 0
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 0
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,4,2,1,3] => 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 0
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,5,2,4] => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,5,3] => 0
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,4,5,2,1] => 0
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0
Description
The number of right floats of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
The following 223 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000779The tier of a permutation. St000837The number of ascents of distance 2 of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001520The number of strict 3-descents. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000120The number of left tunnels of a Dyck path. St000647The number of big descents of a permutation. St000886The number of permutations with the same antidiagonal sums. St001052The length of the exterior of a permutation. St000526The number of posets with combinatorially isomorphic order polytopes. St000538The number of even inversions of a permutation. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000710The number of big deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001557The number of inversions of the second entry of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001806The upper middle entry of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000148The number of odd parts of a partition. St000475The number of parts equal to 1 in a partition. St000549The number of odd partial sums of an integer partition. St000885The number of critical steps in the Catalan decomposition of a binary word. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000741The Colin de Verdière graph invariant. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000455The second largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000661The number of rises of length 3 of a Dyck path. St001638The book thickness of a graph. St000137The Grundy value of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000233The number of nestings of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000365The number of double ascents of a permutation. St000650The number of 3-rises of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001470The cyclic holeyness of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001845The number of join irreducibles minus the rank of a lattice. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001964The interval resolution global dimension of a poset. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001128The exponens consonantiae of a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001527The cyclic permutation representation number of an integer partition. St001541The Gini index of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001587Half of the largest even part of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001488The number of corners of a skew partition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000181The number of connected components of the Hasse diagram for the poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001890The maximum magnitude of the Möbius function of a poset. St001330The hat guessing number of a graph. St001644The dimension of a graph. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001828The Euler characteristic of a graph. St000822The Hadwiger number of the graph. St001060The distinguishing index of a graph. St001626The number of maximal proper sublattices of a lattice. St001734The lettericity of a graph.
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