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Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St000354
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 1
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 1
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => 1
[1,4,5,3,2] => [1,2,4,3,5] => 1
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation π is a value i such that i+1 appears to the left of i in π1,π2,…,πn.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern ([2,1],(0,1),(1,1),(2,1)), i.e., the middle row is shaded.
Matching statistic: St000157
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [1,2] => [[1,2]]
=> 0
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,2,3] => [[1,2,3]]
=> 0
[2,1,3] => [1,2,3] => [[1,2,3]]
=> 0
[2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[3,2,1] => [1,3,2] => [[1,2],[3]]
=> 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,3,2,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,3,4,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,4,2,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[2,1,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,1,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,3,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,4,1,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[3,1,4,2] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[3,2,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[3,2,4,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[3,4,1,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[3,4,2,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,1,3,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[4,2,1,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,2,3,1] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[4,3,1,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[4,3,2,1] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
Description
The number of descents of a standard tableau.
Entry i of a standard Young tableau is a descent if i+1 appears in a row below the row of i.
Matching statistic: St000024
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000052
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000053
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000288
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,2,3] => [1,2,3] => 00 => 0
[2,1,3] => [1,2,3] => [1,2,3] => 00 => 0
[2,3,1] => [1,2,3] => [1,2,3] => 00 => 0
[3,1,2] => [1,3,2] => [1,3,2] => 01 => 1
[3,2,1] => [1,3,2] => [1,3,2] => 01 => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 010 => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 011 => 2
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => 010 => 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 011 => 2
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 010 => 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 010 => 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 010 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000167
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1 = 0 + 1
[2,1] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [3,1,2] => [[1,2],[3]]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [3,1,2] => [[1,2],[3]]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 2 = 1 + 1
Description
The number of ascents of a standard tableau.
Entry i of a standard Young tableau is an '''ascent''' if i+1 appears to the right or above i in the tableau (with respect to the English notation for tableaux).
Matching statistic: St001007
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001068
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000703The number of deficiencies of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000245The number of ascents of a permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001489The maximum of the number of descents and the number of inverse descents. St000996The number of exclusive left-to-right maxima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001427The number of descents of a signed permutation. St001896The number of right descents of a signed permutations. St001960The number of descents of a permutation minus one if its first entry is not one. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000260The radius 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. 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. St001645The pebbling number 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. St001624The breadth of a lattice. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function.
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