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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000021
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,3,2] => [1,3,2] => 1
[2,1,1] => [2,3,1] => [3,1,2] => 1
[1,1,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,1] => [2,3,1] => [3,1,2] => 1
[1,2,2] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => 1
[2,2,1] => [3,1,2] => [2,3,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [2,3,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[2,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[3,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,2,1] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,1,2] => [2,3,1,4] => [3,1,2,4] => 1
[2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 2
[2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,3,1] => [1,4,2,3] => [1,3,4,2] => 1
[1,3,1,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,1,3] => [2,3,1,4] => [3,1,2,4] => 1
[2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[3,1,1,2] => [2,3,4,1] => [4,1,2,3] => 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000157
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(load all 4 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ 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] => [1] => [[1]]
=> 0
[1,1] => [1,2] => [[1,2]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,1,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,2] => [1,2,3] => [[1,2,3]]
=> 0
[1,2,1] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,1] => [2,3,1] => [[1,3],[2]]
=> 1
[1,1,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,1] => [2,3,1] => [[1,3],[2]]
=> 1
[1,2,2] => [1,2,3] => [[1,2,3]]
=> 0
[2,1,2] => [2,1,3] => [[1,3],[2]]
=> 1
[2,2,1] => [3,1,2] => [[1,2],[3]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [3,1,2] => [[1,2],[3]]
=> 1
[3,1,2] => [2,3,1] => [[1,3],[2]]
=> 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,1,1,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,2,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,2,1,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[2,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,1,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,1,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[3,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,4,1,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[4,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,2,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,1,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,2,2,1] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[2,1,1,2] => [2,3,1,4] => [[1,3,4],[2]]
=> 1
[2,1,2,1] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[2,2,1,1] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,3,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,2,3,1] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,3,1,2] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[1,3,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[2,1,1,3] => [2,3,1,4] => [[1,3,4],[2]]
=> 1
[2,1,3,1] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[2,3,1,1] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[3,1,1,2] => [2,3,4,1] => [[1,3,4],[2]]
=> 1
[3,1,2,1] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
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: St000325
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(load all 8 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,1,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,1,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,2,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,1,2] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,2,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[2,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[3,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,2,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[2,1,1,2] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,3,1,2] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[2,1,1,3] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[3,1,1,2] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 3 = 2 + 1
Description
The width 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 width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
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(load all 8 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,1,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,1,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,2,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,1,2] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,2,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[2,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[3,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,2,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[2,1,1,2] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,3,1,2] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[2,1,1,3] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[3,1,1,2] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 3 = 2 + 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000024
Mp00055: Parking functions —to labelling permutation⟶ 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] => [1] => [.,.]
=> [1,0]
=> 0
[1,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,2,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[2,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,2,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,2,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,2,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,1,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,1,3] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,1,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
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: St000053
Mp00055: Parking functions —to labelling permutation⟶ 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] => [1] => [.,.]
=> [1,0]
=> 0
[1,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,1,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[3,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,2,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,2,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,4,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,2,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,1,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,2,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,2,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,1,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,2,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,1,3] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000155
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,1] => [2,3,1] => [3,1,2] => [3,1,2] => 1
[1,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,1,1] => [2,3,1] => [3,1,2] => [3,1,2] => 1
[1,2,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,2,1] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 2
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,1,1] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,1,1] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,1,1] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[3,1,1,1] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,2,1] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
[2,1,1,2] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[2,1,2,1] => [2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 2
[2,2,1,1] => [3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
[1,3,1,2] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,3,2,1] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[2,1,1,3] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[2,1,3,1] => [2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 2
[2,3,1,1] => [3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 1
[3,1,1,2] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => [3,4,1,2] => 2
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000168
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000168: 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
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [[]]
=> 0
[1,1] => [1,2] => [.,[.,.]]
=> [[],[]]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> [[],[]]
=> 0
[2,1] => [2,1] => [[.,.],.]
=> [[[]]]
=> 1
[1,1,1] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 0
[1,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 0
[1,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 1
[2,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [[[]],[]]
=> 1
[1,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 0
[1,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 1
[3,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [[[]],[]]
=> 1
[1,2,2] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 0
[2,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 1
[2,2,1] => [3,1,2] => [[.,[.,.]],.]
=> [[[],[]]]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 1
[2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> [[[],[]]]
=> 1
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> [[[]],[]]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 2
[1,1,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 0
[1,1,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 0
[1,1,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 1
[1,2,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 1
[2,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 1
[1,1,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 0
[1,1,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 1
[1,3,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 1
[3,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 1
[1,1,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 0
[1,1,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 1
[1,4,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 1
[4,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 1
[1,1,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 0
[1,2,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 1
[1,2,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 1
[2,1,1,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 1
[2,1,2,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2
[2,2,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 1
[1,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 0
[1,1,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 1
[1,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 1
[1,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 1
[1,3,1,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 1
[1,3,2,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 2
[2,1,1,3] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 1
[2,1,3,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2
[2,3,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 1
[3,1,1,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 1
[3,1,2,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 2
Description
The number of internal nodes of an ordered tree.
A node is internal if it is neither the root nor a leaf.
Matching statistic: St000245
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [2,1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 1
[1,1,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,1,2] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,1] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,1] => [2,3,1] => [3,1,2] => [2,1,3] => 1
[1,1,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,1] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[3,1,1] => [2,3,1] => [3,1,2] => [2,1,3] => 1
[1,2,2] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[2,2,1] => [3,1,2] => [2,3,1] => [1,3,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 2
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,2,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,2,1,1] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 1
[2,1,1,1] => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,3,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,3,1,1] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 1
[3,1,1,1] => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,1,2] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,2,2,1] => [1,4,2,3] => [1,3,4,2] => [2,4,3,1] => 1
[2,1,1,2] => [2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 1
[2,1,2,1] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[2,2,1,1] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,3,2] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,2,1,3] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,2,3,1] => [1,4,2,3] => [1,3,4,2] => [2,4,3,1] => 1
[1,3,1,2] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 1
[1,3,2,1] => [1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 2
[2,1,1,3] => [2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 1
[2,1,3,1] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[2,3,1,1] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[3,1,1,2] => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => [2,3,1,4] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000329
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000329: 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
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 1
[1,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 1
[1,2,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[2,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 1
[2,2,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 1
[2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,2,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,1,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1,3] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,1] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,1,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,1,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. 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. St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000167The number of leaves of an ordered tree. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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. St000354The number of recoils of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000288The number of ones in a binary word. St001427The number of descents of a signed permutation. St001935The number of ascents in a parking function. St001896The number of right descents of a signed permutations. St000761The number of ascents in an integer composition. St001330The hat guessing number of a graph. St001946The number of descents in a parking function.
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