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Your data matches 108 different statistics following compositions of up to 3 maps.
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Matching statistic: St000328
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St000328: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> 1 = 0 + 1
[[],[]]
=> 2 = 1 + 1
[[[]]]
=> 1 = 0 + 1
[[],[],[]]
=> 3 = 2 + 1
[[],[[]]]
=> 2 = 1 + 1
[[[]],[]]
=> 2 = 1 + 1
[[[],[]]]
=> 2 = 1 + 1
[[[[]]]]
=> 1 = 0 + 1
[[],[],[],[]]
=> 4 = 3 + 1
[[],[],[[]]]
=> 3 = 2 + 1
[[],[[]],[]]
=> 3 = 2 + 1
[[],[[],[]]]
=> 2 = 1 + 1
[[],[[[]]]]
=> 2 = 1 + 1
[[[]],[],[]]
=> 3 = 2 + 1
[[[]],[[]]]
=> 2 = 1 + 1
[[[],[]],[]]
=> 2 = 1 + 1
[[[[]]],[]]
=> 2 = 1 + 1
[[[],[],[]]]
=> 3 = 2 + 1
[[[],[[]]]]
=> 2 = 1 + 1
[[[[]],[]]]
=> 2 = 1 + 1
[[[[],[]]]]
=> 2 = 1 + 1
[[[[[]]]]]
=> 1 = 0 + 1
[[[[[[]]]]]]
=> 1 = 0 + 1
Description
The maximum number of child nodes in a tree.
Matching statistic: St001058
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St001058: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> 1 = 0 + 1
[[],[]]
=> 2 = 1 + 1
[[[]]]
=> 1 = 0 + 1
[[],[],[]]
=> 3 = 2 + 1
[[],[[]]]
=> 2 = 1 + 1
[[[]],[]]
=> 2 = 1 + 1
[[[],[]]]
=> 2 = 1 + 1
[[[[]]]]
=> 1 = 0 + 1
[[],[],[],[]]
=> 4 = 3 + 1
[[],[],[[]]]
=> 3 = 2 + 1
[[],[[]],[]]
=> 3 = 2 + 1
[[],[[],[]]]
=> 2 = 1 + 1
[[],[[[]]]]
=> 2 = 1 + 1
[[[]],[],[]]
=> 3 = 2 + 1
[[[]],[[]]]
=> 2 = 1 + 1
[[[],[]],[]]
=> 2 = 1 + 1
[[[[]]],[]]
=> 2 = 1 + 1
[[[],[],[]]]
=> 3 = 2 + 1
[[[],[[]]]]
=> 2 = 1 + 1
[[[[]],[]]]
=> 2 = 1 + 1
[[[[],[]]]]
=> 2 = 1 + 1
[[[[[]]]]]
=> 1 = 0 + 1
[[[[[[]]]]]]
=> 1 = 0 + 1
Description
The breadth of the ordered tree.
This is the maximal number of nodes having the same depth.
Matching statistic: St000846
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000846: 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)
=> 2 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 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)
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 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)
=> 2 = 1 + 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)
=> 3 = 2 + 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)
=> 2 = 1 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St001294
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1,0]
=> 0
[[],[]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1
[[[]]]
=> [[.,.],.]
=> [1,1,0,0]
=> 0
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra.
See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].
The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.
Matching statistic: St000308
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => 1 = 0 + 1
[[],[]]
=> [[.,.],.]
=> [1,2] => 2 = 1 + 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => 1 = 0 + 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 3 = 2 + 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2 = 1 + 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2 = 1 + 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2 = 1 + 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 1 = 0 + 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 4 = 3 + 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 2 + 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3 = 2 + 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 1 + 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 3 = 2 + 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2 = 1 + 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 1 + 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 1 + 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3 = 2 + 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 1 + 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2 = 1 + 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 1 + 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1 = 0 + 1
[[[[[[]]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1 = 0 + 1
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000845
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(load all 3 compositions to match this statistic)
Values
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> 3 = 2 + 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4 = 3 + 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3 = 2 + 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3 = 2 + 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3 = 2 + 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3 = 2 + 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St001530
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001530: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001530: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
Description
The depth of a Dyck path. That is the depth of the corresponding Nakayama algebra with a linear quiver.
Matching statistic: St000662
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [1] => 0
[[],[]]
=> [.,[.,.]]
=> [2,1] => [2,1] => 1
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,2] => 0
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 2
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 3
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => 2
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => 2
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => 2
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => 1
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001192
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1,0]
=> [1,0]
=> 0
[[],[]]
=> [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[[]]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[[],[],[]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[[[[[]]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001431
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[[[[[]]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
The following 98 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001949The rigidity index of a graph. St000013The height of a Dyck path. St000058The order of a permutation. St000335The difference of lower and upper interactions. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000392The length of the longest run of ones in a binary word. St000444The length of the maximal rise of a Dyck path. St000469The distinguishing number of a graph. St000686The finitistic dominant dimension of a Dyck path. St000808The number of up steps of the associated bargraph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001366The maximal multiplicity of a degree of a vertex of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000442The maximal area to the right of an up step of a Dyck path. St000485The length of the longest cycle of a permutation. St001062The maximal size of a block of a set partition. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000806The semiperimeter of the associated bargraph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000871The number of very big ascents of a permutation. St000035The number of left outer peaks of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000023The number of inner peaks of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001712The number of natural descents of a standard Young tableau. St001840The number of descents of a set partition. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001061The number of indices that are both descents and recoils of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001489The maximum of the number of descents and the number of inverse descents. St001737The number of descents of type 2 in a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St000824The sum of the number of descents and the number of recoils of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001645The pebbling number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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 $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element.
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