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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000008
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2] => [2] => 0
{{1},{2}}
=> [1,1] => [1,1] => 1
{{1,2,3}}
=> [3] => [3] => 0
{{1,2},{3}}
=> [2,1] => [1,2] => 1
{{1,3},{2}}
=> [2,1] => [1,2] => 1
{{1},{2,3}}
=> [1,2] => [2,1] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => 3
{{1,2,3,4}}
=> [4] => [4] => 0
{{1,2,3},{4}}
=> [3,1] => [1,3] => 1
{{1,2,4},{3}}
=> [3,1] => [1,3] => 1
{{1,2},{3,4}}
=> [2,2] => [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => 3
{{1,3,4},{2}}
=> [3,1] => [1,3] => 1
{{1,3},{2,4}}
=> [2,2] => [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => 3
{{1,4},{2,3}}
=> [2,2] => [2,2] => 2
{{1},{2,3,4}}
=> [1,3] => [3,1] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => 4
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => 6
{{1,2,3,4,5}}
=> [5] => [5] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => 1
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => 1
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => 3
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => 1
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => 3
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => 2
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => 6
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => 1
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => 3
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => 2
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => 6
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => 2
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => 3
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000004
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 6
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 6
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000005
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: 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}}
=> [2] => [2] => [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
{{1,2,3}}
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
{{1,3,4},{2}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
{{1,4},{2,3}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2,3,4}}
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
{{1,2,3,4,5}}
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
Description
The bounce statistic of a Dyck path.
The '''bounce path''' $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$.
The points where $D'$ touches the $x$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$.
In particular, the bounce statistics of $D$ and $D'$ coincide.
Matching statistic: St000012
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ 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]
=> [1,0]
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
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: St000081
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2] => [2] => ([],2)
=> 0
{{1},{2}}
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
{{1,2,3}}
=> [3] => [3] => ([],3)
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
{{1,2,3,4}}
=> [4] => [4] => ([],4)
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
{{1,2},{3,4}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
{{1,3,4},{2}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> 1
{{1,3},{2,4}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
{{1,4},{2,3}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,3,4}}
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
{{1,2,3,4,5}}
=> [5] => [5] => ([],5)
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
Description
The number of edges of a graph.
Matching statistic: St000133
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 6
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 6
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
Description
The "bounce" of a permutation.
Matching statistic: St000161
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [.,.]
=> 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [[.,.],.]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [.,[.,.]]
=> 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 3
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 3
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 4
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 6
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 6
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 3
Description
The sum of the sizes of the right subtrees of a binary tree.
This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the $312$-avoiding permutation corresponding to the binary tree.
It is also the sum of all heights $j$ of the coordinates $(i,j)$ of the Dyck path corresponding to the binary tree.
Matching statistic: St000304
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 6
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 6
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
Description
The load of a permutation.
The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.
Matching statistic: St000446
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000446: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000446: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 0
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 6
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 6
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 6
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
Description
The disorder of a permutation.
Consider a permutation $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The '''disorder''' of $\pi$ is defined to be the number of times a position was not removed in this process.
For example, the disorder of $[3,5,2,1,4]$ is $8$ since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.
Matching statistic: St001161
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: 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}}
=> [2] => [2] => [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
{{1,2,3}}
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
{{1,3,4},{2}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
{{1,4},{2,3}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2,3,4}}
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
{{1,2,3,4,5}}
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
Description
The major index north count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000490The intertwining number of a set partition. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000947The major index east count of a Dyck path. St000456The monochromatic index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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