Processing math: 38%

Your data matches 18 different statistics following compositions of up to 3 maps.
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Mp00252: Permutations restrictionPermutations
Mp00061: Permutations to increasing treeBinary trees
St000051: Binary trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> 0
[2,1] => [1] => [.,.]
=> 0
[1,2,3] => [1,2] => [.,[.,.]]
=> 0
[1,3,2] => [1,2] => [.,[.,.]]
=> 0
[2,1,3] => [2,1] => [[.,.],.]
=> 1
[2,3,1] => [2,1] => [[.,.],.]
=> 1
[3,1,2] => [1,2] => [.,[.,.]]
=> 0
[3,2,1] => [2,1] => [[.,.],.]
=> 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> 0
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> 2
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> 2
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 2
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> 2
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> 2
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> 2
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 2
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 2
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
Description
The size of the left subtree of a binary tree.
Mp00252: Permutations restrictionPermutations
Mp00066: Permutations inversePermutations
St000054: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [1,2] => 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1,3] => [2,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [2,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [3,1,2] => 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [3,1,2] => 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [3,1,2] => 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 0 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [3,1,2] => 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 0 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 1 = 0 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => 1 = 0 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => 1 = 0 + 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation π of n, together with its rotations, obtained by conjugating with the long cycle (1,,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. Then, \pi(1)-1 is the number of rotations of \pi where the arc (1, \pi(1)) is a deficiency. In particular, if O(\pi) is the orbit of rotations of \pi, then the number of deficiencies of \pi equals \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
Mp00252: Permutations restrictionPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St000740: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [2,1] => 1 = 0 + 1
[1,3,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1,3] => [2,1] => [1,2] => 2 = 1 + 1
[2,3,1] => [2,1] => [1,2] => 2 = 1 + 1
[3,1,2] => [1,2] => [2,1] => 1 = 0 + 1
[3,2,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[1,3,4,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[1,4,2,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [2,4,3,1] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,3] => [2,4,3,1] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [2,4,3,1] => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4] => [3,2,4,1] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,4] => [3,2,4,1] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2] => [4,2,3,1] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,2] => [4,2,3,1] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,2,4] => [3,2,4,1] => 1 = 0 + 1
[1,3,5,4,2] => [1,3,4,2] => [4,2,3,1] => 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,2,1] => 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,2,1] => 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,4,3,5,2] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,3] => [3,4,2,1] => 1 = 0 + 1
[1,4,5,3,2] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St000133
Mp00064: Permutations reversePermutations
Mp00252: Permutations restrictionPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
St000133: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1] => [1] => 0
[2,1] => [1,2] => [1] => [1] => 0
[1,2,3] => [3,2,1] => [2,1] => [2,1] => 0
[1,3,2] => [2,3,1] => [2,1] => [2,1] => 0
[2,1,3] => [3,1,2] => [1,2] => [1,2] => 1
[2,3,1] => [1,3,2] => [1,2] => [1,2] => 1
[3,1,2] => [2,1,3] => [2,1] => [2,1] => 0
[3,2,1] => [1,2,3] => [1,2] => [1,2] => 1
[1,2,3,4] => [4,3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [3,2,1] => [3,2,1] => 0
[1,3,2,4] => [4,2,3,1] => [2,3,1] => [2,3,1] => 0
[1,3,4,2] => [2,4,3,1] => [2,3,1] => [2,3,1] => 0
[1,4,2,3] => [3,2,4,1] => [3,2,1] => [3,2,1] => 0
[1,4,3,2] => [2,3,4,1] => [2,3,1] => [2,3,1] => 0
[2,1,3,4] => [4,3,1,2] => [3,1,2] => [3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => [3,1,2] => [3,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [1,3,2] => [1,3,2] => 2
[2,3,4,1] => [1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,4,1,3] => [3,1,4,2] => [3,1,2] => [3,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,3,2] => [1,3,2] => 2
[3,1,2,4] => [4,2,1,3] => [2,1,3] => [2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [1,2,3] => [1,3,2] => 2
[3,2,4,1] => [1,4,2,3] => [1,2,3] => [1,3,2] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[3,4,2,1] => [1,2,4,3] => [1,2,3] => [1,3,2] => 2
[4,1,2,3] => [3,2,1,4] => [3,2,1] => [3,2,1] => 0
[4,1,3,2] => [2,3,1,4] => [2,3,1] => [2,3,1] => 0
[4,2,1,3] => [3,1,2,4] => [3,1,2] => [3,1,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2] => [1,3,2] => 2
[4,3,1,2] => [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3] => [1,3,2] => 2
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3,5] => [5,3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 0
[1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => [3,4,2,1] => 0
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [3,4,2,1] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
[1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => [2,4,3,1] => 0
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => [2,4,3,1] => 0
[1,4,2,3,5] => [5,3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[1,4,2,5,3] => [3,5,2,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => [2,4,3,1] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => [2,4,3,1] => 0
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,4,1] => [3,2,4,1] => 0
[1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => [2,4,3,1] => 0
Description
The "bounce" of a permutation.
Matching statistic: St000025
Mp00252: Permutations restrictionPermutations
Mp00066: Permutations inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1,0]
=> 1 = 0 + 1
[2,1] => [1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1] => [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [2,1] => [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of D.
Matching statistic: St000026
Mp00252: Permutations restrictionPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[2,1] => [1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St001291
Mp00252: Permutations restrictionPermutations
Mp00069: Permutations complementPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1,0]
=> 1 = 0 + 1
[2,1] => [1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,1] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [2,1] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [2,1] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let A be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of D(A) \otimes D(A), where D(A) is the natural dual of A.
Matching statistic: St001497
Mp00252: Permutations restrictionPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00069: Permutations complementPermutations
St001497: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1,3] => [2,1] => [2,1] => [1,2] => 2 = 1 + 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[3,2,1] => [2,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,4,2,3] => [1,2,3] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,3,1] => [2,1,3] => 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [2,3,1] => [2,1,3] => 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,3,1] => [2,1,3] => 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,3,1] => [2,1,3] => 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St000439
Mp00252: Permutations restrictionPermutations
Mp00066: Permutations inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1,0]
=> 2 = 0 + 2
[2,1] => [1] => [1] => [1,0]
=> 2 = 0 + 2
[1,2,3] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[1,3,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[2,1,3] => [2,1] => [2,1] => [1,1,0,0]
=> 3 = 1 + 2
[2,3,1] => [2,1] => [2,1] => [1,1,0,0]
=> 3 = 1 + 2
[3,1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[3,2,1] => [2,1] => [2,1] => [1,1,0,0]
=> 3 = 1 + 2
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,3,2,4] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,3,4,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,4,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,3,4] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[2,1,4,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[2,3,1,4] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[2,3,4,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[2,4,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[2,4,3,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[3,1,2,4] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[3,1,4,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[3,2,1,4] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[3,2,4,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[3,4,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[3,4,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 0 + 2
[4,1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[4,2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 1 + 2
[4,2,3,1] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[4,3,1,2] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 1 + 2
[4,3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000727
Mp00252: Permutations restrictionPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St000727: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => ? = 0 + 1
[2,1] => [1] => [1] => [1] => ? = 0 + 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1,3] => [2,1] => [2,1] => [1,2] => 2 = 1 + 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[3,2,1] => [2,1] => [2,1] => [1,2] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[1,4,2,3] => [1,2,3] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[2,3,4,1] => [2,3,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[3,1,2,4] => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [2,3,1] => [1,2,3] => 3 = 2 + 1
[4,3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [2,1,3] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,5,2,3,4] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[1,5,2,4,3] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
Description
The largest label of a leaf in the binary search tree associated with the permutation. Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000066The column of the unique '1' in the first row of the alternating sign matrix. St000260The radius of a connected graph. St001434The number of negative sum pairs of a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001557The number of inversions of the second entry of a permutation.