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Your data matches 50 different statistics following compositions of up to 3 maps.
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Matching statistic: St001683
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => 1
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 2
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => 3
[1,3,5,2,4] => [1,3,5,2,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => 2
[1,4,3,2,5] => [1,4,2,5,3] => 2
[1,4,3,5,2] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 2
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St000204
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 2
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000012
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[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] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[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] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000024
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [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] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 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: St000394
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[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] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[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] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001007
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [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] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000476
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> ? = 0
[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] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[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] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000984
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000984: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000984: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> ? = 0
[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] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[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] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
Description
The number of boxes below precisely one peak.
Imagine that each peak of the Dyck path, drawn with north and east steps, casts a shadow onto the triangular region between it and the diagonal. This statistic is the number of cells which are in the shade of precisely one peak.
Matching statistic: St001489
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 2
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,2,5,3,4] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,4,3,5] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,4,3,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [1,5,2,3,4] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [1,5,2,4,3] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [1,5,4,2,3] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [1,5,4,3,2] => 3
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [1,4,3,2,5] => 2
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => [1,7,2,3,4,5,6] => ? = 1
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [7,5,6,4,3,1,2] => [1,7,2,3,4,6,5] => ? = 2
[1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [1,7,2,3,6,4,5] => ? = 2
[1,3,2,4,6,7,5] => [1,3,2,4,6,7,5] => [7,5,6,4,2,1,3] => [1,7,2,3,6,5,4] => ? = 3
[1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => [1,7,2,3,5,4,6] => ? = 2
[1,3,2,4,7,6,5] => [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => [1,7,2,3,5,4,6] => ? = 2
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [1,7,2,6,3,4,5] => ? = 2
[1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [1,7,2,6,3,5,4] => ? = 3
[1,3,2,5,6,4,7] => [1,3,2,5,6,4,7] => [7,5,6,3,2,4,1] => [1,7,2,6,5,3,4] => ? = 3
[1,3,2,5,6,7,4] => [1,3,2,5,6,7,4] => [7,5,6,3,2,1,4] => [1,7,2,6,5,4,3] => ? = 4
[1,3,2,5,7,4,6] => [1,3,2,5,7,4,6] => [7,5,6,3,1,4,2] => [1,7,2,6,4,3,5] => ? = 3
[1,3,2,5,7,6,4] => [1,3,2,5,7,4,6] => [7,5,6,3,1,4,2] => [1,7,2,6,4,3,5] => ? = 3
[1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => [7,5,6,2,4,3,1] => [1,7,2,5,3,4,6] => ? = 2
[1,3,2,6,4,7,5] => [1,3,2,6,4,7,5] => [7,5,6,2,4,1,3] => [1,7,2,5,3,6,4] => ? = 3
[1,3,2,6,5,4,7] => [1,3,2,6,4,7,5] => [7,5,6,2,4,1,3] => [1,7,2,5,3,6,4] => ? = 3
[1,3,2,6,5,7,4] => [1,3,2,6,4,5,7] => [7,5,6,2,4,3,1] => [1,7,2,5,3,4,6] => ? = 2
[1,3,2,6,7,4,5] => [1,3,2,6,7,4,5] => [7,5,6,2,1,4,3] => [1,7,2,5,4,3,6] => ? = 3
[1,3,2,6,7,5,4] => [1,3,2,6,7,4,5] => [7,5,6,2,1,4,3] => [1,7,2,5,4,3,6] => ? = 3
[1,3,2,7,4,5,6] => [1,3,2,7,4,5,6] => [7,5,6,1,4,3,2] => [1,7,2,4,3,5,6] => ? = 2
[1,3,2,7,4,6,5] => [1,3,2,7,4,6,5] => [7,5,6,1,4,2,3] => [1,7,2,4,3,6,5] => ? = 3
[1,3,2,7,5,4,6] => [1,3,2,7,4,6,5] => [7,5,6,1,4,2,3] => [1,7,2,4,3,6,5] => ? = 3
[1,3,2,7,5,6,4] => [1,3,2,7,4,5,6] => [7,5,6,1,4,3,2] => [1,7,2,4,3,5,6] => ? = 2
[1,3,2,7,6,4,5] => [1,3,2,7,4,5,6] => [7,5,6,1,4,3,2] => [1,7,2,4,3,5,6] => ? = 2
[1,3,2,7,6,5,4] => [1,3,2,7,4,5,6] => [7,5,6,1,4,3,2] => [1,7,2,4,3,5,6] => ? = 2
[1,3,4,2,5,6,7] => [1,3,4,2,5,6,7] => [7,5,4,6,3,2,1] => [1,7,6,2,3,4,5] => ? = 2
[1,3,4,2,5,7,6] => [1,3,4,2,5,7,6] => [7,5,4,6,3,1,2] => [1,7,6,2,3,5,4] => ? = 3
[1,3,4,2,6,5,7] => [1,3,4,2,6,5,7] => [7,5,4,6,2,3,1] => [1,7,6,2,5,3,4] => ? = 3
[1,3,4,2,6,7,5] => [1,3,4,2,6,7,5] => [7,5,4,6,2,1,3] => [1,7,6,2,5,4,3] => ? = 4
[1,3,4,2,7,5,6] => [1,3,4,2,7,5,6] => [7,5,4,6,1,3,2] => [1,7,6,2,4,3,5] => ? = 3
[1,3,4,2,7,6,5] => [1,3,4,2,7,5,6] => [7,5,4,6,1,3,2] => [1,7,6,2,4,3,5] => ? = 3
[1,3,4,5,2,6,7] => [1,3,4,5,2,6,7] => [7,5,4,3,6,2,1] => [1,7,6,5,2,3,4] => ? = 3
[1,3,4,5,2,7,6] => [1,3,4,5,2,7,6] => [7,5,4,3,6,1,2] => [1,7,6,5,2,4,3] => ? = 4
[1,3,4,5,6,2,7] => [1,3,4,5,6,2,7] => [7,5,4,3,2,6,1] => [1,7,6,5,4,2,3] => ? = 4
[1,3,4,5,6,7,2] => [1,3,4,5,6,7,2] => [7,5,4,3,2,1,6] => [1,7,6,5,4,3,2] => ? = 5
[1,3,4,5,7,2,6] => [1,3,4,5,7,2,6] => [7,5,4,3,1,6,2] => [1,7,6,5,3,2,4] => ? = 4
[1,3,4,5,7,6,2] => [1,3,4,5,7,2,6] => [7,5,4,3,1,6,2] => [1,7,6,5,3,2,4] => ? = 4
[1,3,4,6,2,5,7] => [1,3,4,6,2,5,7] => [7,5,4,2,6,3,1] => [1,7,6,4,2,3,5] => ? = 3
[1,3,4,6,2,7,5] => [1,3,4,6,2,7,5] => [7,5,4,2,6,1,3] => [1,7,6,4,2,5,3] => ? = 4
[1,3,4,6,5,2,7] => [1,3,4,6,2,7,5] => [7,5,4,2,6,1,3] => [1,7,6,4,2,5,3] => ? = 4
[1,3,4,6,5,7,2] => [1,3,4,6,2,5,7] => [7,5,4,2,6,3,1] => [1,7,6,4,2,3,5] => ? = 3
[1,3,4,6,7,2,5] => [1,3,4,6,7,2,5] => [7,5,4,2,1,6,3] => [1,7,6,4,3,2,5] => ? = 4
[1,3,4,6,7,5,2] => [1,3,4,6,7,2,5] => [7,5,4,2,1,6,3] => [1,7,6,4,3,2,5] => ? = 4
[1,3,4,7,2,5,6] => [1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [1,7,6,3,2,4,5] => ? = 3
[1,3,4,7,2,6,5] => [1,3,4,7,2,6,5] => [7,5,4,1,6,2,3] => [1,7,6,3,2,5,4] => ? = 4
[1,3,4,7,5,2,6] => [1,3,4,7,2,6,5] => [7,5,4,1,6,2,3] => [1,7,6,3,2,5,4] => ? = 4
[1,3,4,7,5,6,2] => [1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [1,7,6,3,2,4,5] => ? = 3
[1,3,4,7,6,2,5] => [1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [1,7,6,3,2,4,5] => ? = 3
[1,3,4,7,6,5,2] => [1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [1,7,6,3,2,4,5] => ? = 3
[1,3,5,2,4,6,7] => [1,3,5,2,4,6,7] => [7,5,3,6,4,2,1] => [1,7,5,2,3,4,6] => ? = 2
[1,3,5,2,4,7,6] => [1,3,5,2,4,7,6] => [7,5,3,6,4,1,2] => [1,7,5,2,3,6,4] => ? = 3
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St000470
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1] => [1] => 1 = 0 + 1
[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] => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 3 = 2 + 1
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 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] => [1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,2,5,3,4] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [1,2,4,3,5] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [1,5,2,3,4] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [1,5,2,4,3] => 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [1,5,4,2,3] => 3 = 2 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [1,5,4,3,2] => 4 = 3 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 3 = 2 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [1,4,3,2,5] => 3 = 2 + 1
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => [1,7,2,3,4,5,6] => ? = 1 + 1
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [7,5,6,4,3,1,2] => [1,7,2,3,4,6,5] => ? = 2 + 1
[1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [1,7,2,3,6,4,5] => ? = 2 + 1
[1,3,2,4,6,7,5] => [1,3,2,4,6,7,5] => [7,5,6,4,2,1,3] => [1,7,2,3,6,5,4] => ? = 3 + 1
[1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => [1,7,2,3,5,4,6] => ? = 2 + 1
[1,3,2,4,7,6,5] => [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => [1,7,2,3,5,4,6] => ? = 2 + 1
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [1,7,2,6,3,4,5] => ? = 2 + 1
[1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [1,7,2,6,3,5,4] => ? = 3 + 1
[1,3,2,5,6,4,7] => [1,3,2,5,6,4,7] => [7,5,6,3,2,4,1] => [1,7,2,6,5,3,4] => ? = 3 + 1
[1,3,2,5,6,7,4] => [1,3,2,5,6,7,4] => [7,5,6,3,2,1,4] => [1,7,2,6,5,4,3] => ? = 4 + 1
[1,3,2,5,7,4,6] => [1,3,2,5,7,4,6] => [7,5,6,3,1,4,2] => [1,7,2,6,4,3,5] => ? = 3 + 1
[1,3,2,5,7,6,4] => [1,3,2,5,7,4,6] => [7,5,6,3,1,4,2] => [1,7,2,6,4,3,5] => ? = 3 + 1
[1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => [7,5,6,2,4,3,1] => [1,7,2,5,3,4,6] => ? = 2 + 1
[1,3,2,6,4,7,5] => [1,3,2,6,4,7,5] => [7,5,6,2,4,1,3] => [1,7,2,5,3,6,4] => ? = 3 + 1
[1,3,2,6,5,4,7] => [1,3,2,6,4,7,5] => [7,5,6,2,4,1,3] => [1,7,2,5,3,6,4] => ? = 3 + 1
[1,3,2,6,5,7,4] => [1,3,2,6,4,5,7] => [7,5,6,2,4,3,1] => [1,7,2,5,3,4,6] => ? = 2 + 1
[1,3,2,6,7,4,5] => [1,3,2,6,7,4,5] => [7,5,6,2,1,4,3] => [1,7,2,5,4,3,6] => ? = 3 + 1
[1,3,2,6,7,5,4] => [1,3,2,6,7,4,5] => [7,5,6,2,1,4,3] => [1,7,2,5,4,3,6] => ? = 3 + 1
[1,3,2,7,4,5,6] => [1,3,2,7,4,5,6] => [7,5,6,1,4,3,2] => [1,7,2,4,3,5,6] => ? = 2 + 1
[1,3,2,7,4,6,5] => [1,3,2,7,4,6,5] => [7,5,6,1,4,2,3] => [1,7,2,4,3,6,5] => ? = 3 + 1
[1,3,2,7,5,4,6] => [1,3,2,7,4,6,5] => [7,5,6,1,4,2,3] => [1,7,2,4,3,6,5] => ? = 3 + 1
[1,3,2,7,5,6,4] => [1,3,2,7,4,5,6] => [7,5,6,1,4,3,2] => [1,7,2,4,3,5,6] => ? = 2 + 1
[1,3,2,7,6,4,5] => [1,3,2,7,4,5,6] => [7,5,6,1,4,3,2] => [1,7,2,4,3,5,6] => ? = 2 + 1
[1,3,2,7,6,5,4] => [1,3,2,7,4,5,6] => [7,5,6,1,4,3,2] => [1,7,2,4,3,5,6] => ? = 2 + 1
[1,3,4,2,5,6,7] => [1,3,4,2,5,6,7] => [7,5,4,6,3,2,1] => [1,7,6,2,3,4,5] => ? = 2 + 1
[1,3,4,2,5,7,6] => [1,3,4,2,5,7,6] => [7,5,4,6,3,1,2] => [1,7,6,2,3,5,4] => ? = 3 + 1
[1,3,4,2,6,5,7] => [1,3,4,2,6,5,7] => [7,5,4,6,2,3,1] => [1,7,6,2,5,3,4] => ? = 3 + 1
[1,3,4,2,6,7,5] => [1,3,4,2,6,7,5] => [7,5,4,6,2,1,3] => [1,7,6,2,5,4,3] => ? = 4 + 1
[1,3,4,2,7,5,6] => [1,3,4,2,7,5,6] => [7,5,4,6,1,3,2] => [1,7,6,2,4,3,5] => ? = 3 + 1
[1,3,4,2,7,6,5] => [1,3,4,2,7,5,6] => [7,5,4,6,1,3,2] => [1,7,6,2,4,3,5] => ? = 3 + 1
[1,3,4,5,2,6,7] => [1,3,4,5,2,6,7] => [7,5,4,3,6,2,1] => [1,7,6,5,2,3,4] => ? = 3 + 1
[1,3,4,5,2,7,6] => [1,3,4,5,2,7,6] => [7,5,4,3,6,1,2] => [1,7,6,5,2,4,3] => ? = 4 + 1
[1,3,4,5,6,2,7] => [1,3,4,5,6,2,7] => [7,5,4,3,2,6,1] => [1,7,6,5,4,2,3] => ? = 4 + 1
[1,3,4,5,6,7,2] => [1,3,4,5,6,7,2] => [7,5,4,3,2,1,6] => [1,7,6,5,4,3,2] => ? = 5 + 1
[1,3,4,5,7,2,6] => [1,3,4,5,7,2,6] => [7,5,4,3,1,6,2] => [1,7,6,5,3,2,4] => ? = 4 + 1
[1,3,4,5,7,6,2] => [1,3,4,5,7,2,6] => [7,5,4,3,1,6,2] => [1,7,6,5,3,2,4] => ? = 4 + 1
[1,3,4,6,2,5,7] => [1,3,4,6,2,5,7] => [7,5,4,2,6,3,1] => [1,7,6,4,2,3,5] => ? = 3 + 1
[1,3,4,6,2,7,5] => [1,3,4,6,2,7,5] => [7,5,4,2,6,1,3] => [1,7,6,4,2,5,3] => ? = 4 + 1
[1,3,4,6,5,2,7] => [1,3,4,6,2,7,5] => [7,5,4,2,6,1,3] => [1,7,6,4,2,5,3] => ? = 4 + 1
[1,3,4,6,5,7,2] => [1,3,4,6,2,5,7] => [7,5,4,2,6,3,1] => [1,7,6,4,2,3,5] => ? = 3 + 1
[1,3,4,6,7,2,5] => [1,3,4,6,7,2,5] => [7,5,4,2,1,6,3] => [1,7,6,4,3,2,5] => ? = 4 + 1
[1,3,4,6,7,5,2] => [1,3,4,6,7,2,5] => [7,5,4,2,1,6,3] => [1,7,6,4,3,2,5] => ? = 4 + 1
[1,3,4,7,2,5,6] => [1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [1,7,6,3,2,4,5] => ? = 3 + 1
[1,3,4,7,2,6,5] => [1,3,4,7,2,6,5] => [7,5,4,1,6,2,3] => [1,7,6,3,2,5,4] => ? = 4 + 1
[1,3,4,7,5,2,6] => [1,3,4,7,2,6,5] => [7,5,4,1,6,2,3] => [1,7,6,3,2,5,4] => ? = 4 + 1
[1,3,4,7,5,6,2] => [1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [1,7,6,3,2,4,5] => ? = 3 + 1
[1,3,4,7,6,2,5] => [1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [1,7,6,3,2,4,5] => ? = 3 + 1
[1,3,4,7,6,5,2] => [1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [1,7,6,3,2,4,5] => ? = 3 + 1
[1,3,5,2,4,6,7] => [1,3,5,2,4,6,7] => [7,5,3,6,4,2,1] => [1,7,5,2,3,4,6] => ? = 2 + 1
[1,3,5,2,4,7,6] => [1,3,5,2,4,7,6] => [7,5,3,6,4,1,2] => [1,7,5,2,3,6,4] => ? = 3 + 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.
The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000829The Ulam distance of a permutation to the identity permutation. St000619The number of cyclic descents of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000141The maximum drop size of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000246The number of non-inversions of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001511The minimal number of transpositions needed to sort a permutation in either direction. St000325The width of the tree associated to 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. St001330The hat guessing number of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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 $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001866The nesting alignments of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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