Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000462
Mapping
St000462: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 4
[2,1,3,4] => 0
[2,1,4,3] => 2
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 3
[3,1,2,4] => 0
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 0
[3,4,2,1] => 3
[4,1,2,3] => 0
[4,1,3,2] => 3
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 1
[4,3,2,1] => 4
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 6
[1,3,2,4,5] => 1
[1,3,2,5,4] => 4
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 5
[1,4,2,3,5] => 1
[1,4,2,5,3] => 4
[1,4,3,2,5] => 4
[1,4,3,5,2] => 4
[1,4,5,2,3] => 1
[1,4,5,3,2] => 5
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000008
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => 0
[2,1] => [2] => 0
[1,2,3] => [3] => 0
[1,3,2] => [1,2] => 1
[2,1,3] => [3] => 0
[2,3,1] => [3] => 0
[3,1,2] => [3] => 0
[3,2,1] => [2,1] => 2
[1,2,3,4] => [4] => 0
[1,2,4,3] => [2,2] => 2
[1,3,2,4] => [1,3] => 1
[1,3,4,2] => [1,3] => 1
[1,4,2,3] => [1,3] => 1
[1,4,3,2] => [1,2,1] => 4
[2,1,3,4] => [4] => 0
[2,1,4,3] => [2,2] => 2
[2,3,1,4] => [4] => 0
[2,3,4,1] => [4] => 0
[2,4,1,3] => [4] => 0
[2,4,3,1] => [3,1] => 3
[3,1,2,4] => [4] => 0
[3,1,4,2] => [2,2] => 2
[3,2,1,4] => [2,2] => 2
[3,2,4,1] => [2,2] => 2
[3,4,1,2] => [4] => 0
[3,4,2,1] => [3,1] => 3
[4,1,2,3] => [4] => 0
[4,1,3,2] => [3,1] => 3
[4,2,1,3] => [2,2] => 2
[4,2,3,1] => [3,1] => 3
[4,3,1,2] => [1,3] => 1
[4,3,2,1] => [1,2,1] => 4
[1,2,3,4,5] => [5] => 0
[1,2,3,5,4] => [3,2] => 3
[1,2,4,3,5] => [2,3] => 2
[1,2,4,5,3] => [2,3] => 2
[1,2,5,3,4] => [2,3] => 2
[1,2,5,4,3] => [2,2,1] => 6
[1,3,2,4,5] => [1,4] => 1
[1,3,2,5,4] => [1,2,2] => 4
[1,3,4,2,5] => [1,4] => 1
[1,3,4,5,2] => [1,4] => 1
[1,3,5,2,4] => [1,4] => 1
[1,3,5,4,2] => [1,3,1] => 5
[1,4,2,3,5] => [1,4] => 1
[1,4,2,5,3] => [1,2,2] => 4
[1,4,3,2,5] => [1,2,2] => 4
[1,4,3,5,2] => [1,2,2] => 4
[1,4,5,2,3] => [1,4] => 1
[1,4,5,3,2] => [1,3,1] => 5
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: St000081
Mapping
Mp00248: Permutations —DEX composition⟶ 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
[1,2] => [2] => ([],2)
 => 0
[2,1] => [2] => ([],2)
 => 0
[1,2,3] => [3] => ([],3)
 => 0
[1,3,2] => [1,2] => ([(1,2)],3)
 => 1
[2,1,3] => [3] => ([],3)
 => 0
[2,3,1] => [3] => ([],3)
 => 0
[3,1,2] => [3] => ([],3)
 => 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [4] => ([],4)
 => 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,3,2,4] => [1,3] => ([(2,3)],4)
 => 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
 => 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
 => 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,1,3,4] => [4] => ([],4)
 => 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,3,1,4] => [4] => ([],4)
 => 0
[2,3,4,1] => [4] => ([],4)
 => 0
[2,4,1,3] => [4] => ([],4)
 => 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => [4] => ([],4)
 => 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[3,4,1,2] => [4] => ([],4)
 => 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => [4] => ([],4)
 => 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,3,1,2] => [1,3] => ([(2,3)],4)
 => 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => [5] => ([],5)
 => 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
 => 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
 => 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
 => 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
 => 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
 => 1
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
Description
The number of edges of a graph.
Matching statistic: St000947
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
Description
The major index east 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 east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Matching statistic: St001161
Mapping
Mp00248: Permutations —DEX composition⟶ 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
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
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 $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
Matching statistic: St000005
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 76%
Values
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 6
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,2,3,4,5,7,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 5
[1,2,3,4,6,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,4,6,7,5] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,4,7,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,2,3,4,7,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 10
[1,2,3,5,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,3,5,4,7,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,3,5,6,4,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,3,5,6,7,4] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,3,5,7,4,6] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,3,5,7,6,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,2,3,6,4,5,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,3,6,4,7,5] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,3,6,5,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,3,6,5,7,4] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,3,6,7,4,5] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,3,6,7,5,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,2,3,7,4,5,6] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,2,3,7,4,6,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,2,3,7,5,4,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 8
[1,2,3,7,5,6,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 9
[1,2,3,7,6,4,5] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 7
[1,2,3,7,6,5,4] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 13
[1,2,4,3,5,6,7] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,4,3,5,7,6] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 7
[1,2,4,3,6,5,7] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6
[1,2,4,3,6,7,5] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6
[1,2,4,3,7,5,6] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6
[1,2,4,3,7,6,5] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 12
[1,2,4,5,3,6,7] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,4,5,3,7,6] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 7
[1,2,4,5,6,3,7] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,4,5,6,7,3] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,4,5,7,3,6] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,4,5,7,6,3] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 8
[1,2,4,6,3,5,7] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,4,6,3,7,5] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 7
[1,2,4,6,5,3,7] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 7
[1,2,4,6,5,7,3] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 7
[1,2,4,6,7,3,5] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,4,6,7,5,3] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 8
[1,2,4,7,3,5,6] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,4,7,3,6,5] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 8
[1,2,4,7,5,3,6] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 7
[1,2,4,7,5,6,3] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 8
[1,2,4,7,6,3,5] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6
[1,2,4,7,6,5,3] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 12
[1,2,5,3,4,6,7] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,5,3,4,7,6] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 7
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.