Your data matches 6 different statistics following compositions of up to 3 maps.
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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] => [1] => 0
[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
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: St001161
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 81%●distinct values known / distinct values provided: 54%
Values
[1] => [1] => [1,0]
 => 0
[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
[6,7,8,5,4,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 21
[5,6,7,8,4,3,2,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 18
[4,5,6,7,8,3,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[5,6,7,8,3,4,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[6,7,8,3,4,5,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[4,5,6,7,3,8,2,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 12
[5,6,7,8,4,2,3,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 12
[4,5,6,7,8,2,3,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[5,6,7,8,3,2,4,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 12
[6,7,8,5,2,3,4,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 10
[5,6,7,8,2,3,4,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[7,8,3,2,4,5,6,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 10
[4,5,6,7,3,2,8,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[4,5,6,7,2,3,8,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[6,7,2,3,4,5,8,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[3,4,5,2,6,7,8,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[5,2,3,4,6,7,8,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[3,4,2,5,6,7,8,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[4,2,3,5,6,7,8,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[2,3,4,5,6,7,8,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,3,1,2] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[4,5,6,7,8,3,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[5,6,7,8,3,4,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[7,8,3,4,5,6,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[5,6,7,4,3,8,1,2] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[4,5,6,7,3,8,1,2] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[5,6,3,4,7,8,1,2] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[4,5,3,6,7,8,1,2] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[3,4,5,6,7,8,1,2] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,2,1,3] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[4,5,6,7,8,2,1,3] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[6,7,8,5,4,1,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[5,6,7,8,4,1,2,3] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[4,5,6,7,8,1,2,3] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,3,2,1,4] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[5,6,7,8,2,3,1,4] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[5,6,7,8,3,1,2,4] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[6,7,8,5,2,1,3,4] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[5,6,7,8,2,1,3,4] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[6,7,8,5,1,2,3,4] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[5,6,7,8,1,2,3,4] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[6,7,8,4,3,2,1,5] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[6,7,8,3,4,2,1,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[6,7,8,4,2,3,1,5] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[6,7,8,3,2,4,1,5] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[6,7,8,2,3,4,1,5] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[6,7,8,4,3,1,2,5] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[6,7,8,3,4,1,2,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[6,7,8,4,2,1,3,5] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[6,7,8,4,1,2,3,5] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
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: St000947
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 81%●distinct values known / distinct values provided: 54%
Values
[1] => [1] => [1,0]
 => ? = 0
[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
[6,7,8,5,4,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 21
[5,6,7,8,4,3,2,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 18
[4,5,6,7,8,3,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[5,6,7,8,3,4,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[6,7,8,3,4,5,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[4,5,6,7,3,8,2,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 12
[5,6,7,8,4,2,3,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 12
[4,5,6,7,8,2,3,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[5,6,7,8,3,2,4,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 12
[6,7,8,5,2,3,4,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 10
[5,6,7,8,2,3,4,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[7,8,3,2,4,5,6,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 10
[4,5,6,7,3,2,8,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[4,5,6,7,2,3,8,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[6,7,2,3,4,5,8,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[3,4,5,2,6,7,8,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[5,2,3,4,6,7,8,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[3,4,2,5,6,7,8,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[4,2,3,5,6,7,8,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[2,3,4,5,6,7,8,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,3,1,2] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[4,5,6,7,8,3,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[5,6,7,8,3,4,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[7,8,3,4,5,6,1,2] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[5,6,7,4,3,8,1,2] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[4,5,6,7,3,8,1,2] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[5,6,3,4,7,8,1,2] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[4,5,3,6,7,8,1,2] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[3,4,5,6,7,8,1,2] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,2,1,3] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[4,5,6,7,8,2,1,3] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[6,7,8,5,4,1,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[5,6,7,8,4,1,2,3] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[4,5,6,7,8,1,2,3] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,3,2,1,4] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[5,6,7,8,2,3,1,4] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[5,6,7,8,3,1,2,4] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[6,7,8,5,2,1,3,4] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[5,6,7,8,2,1,3,4] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[6,7,8,5,1,2,3,4] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[5,6,7,8,1,2,3,4] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[6,7,8,4,3,2,1,5] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 15
[6,7,8,3,4,2,1,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[6,7,8,4,2,3,1,5] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[6,7,8,3,2,4,1,5] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[6,7,8,2,3,4,1,5] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
[6,7,8,4,3,1,2,5] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[6,7,8,3,4,1,2,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[6,7,8,4,2,1,3,5] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
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: St000081
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 41% ●values known / values provided: 46%●distinct values known / distinct values provided: 41%
Values
[1] => [1] => ([],1)
 => 0
[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
[8,7,6,5,4,3,2,1] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 24
[7,8,6,5,4,3,2,1] => [2,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 23
[7,6,8,5,4,3,2,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 22
[6,7,8,5,4,3,2,1] => [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[7,8,5,6,4,3,2,1] => [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 20
[7,6,5,8,4,3,2,1] => [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[7,5,6,8,4,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[6,5,7,8,4,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[5,6,7,8,4,3,2,1] => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 18
[8,7,6,4,5,3,2,1] => [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 21
[7,8,6,4,5,3,2,1] => [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 20
[8,6,7,4,5,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[7,6,5,4,8,3,2,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 20
[6,5,7,4,8,3,2,1] => [1,3,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 18
[7,6,4,5,8,3,2,1] => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[7,5,4,6,8,3,2,1] => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[7,4,5,6,8,3,2,1] => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 14
[4,5,6,7,8,3,2,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 13
[8,7,6,5,3,4,2,1] => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[7,8,6,5,3,4,2,1] => [2,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 18
[8,6,7,5,3,4,2,1] => [1,2,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 17
[8,7,5,6,3,4,2,1] => [1,1,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[5,6,7,8,3,4,2,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 13
[6,7,8,3,4,5,2,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 13
[8,7,5,4,3,6,2,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 20
[7,6,5,4,3,8,2,1] => [1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 19
[6,4,5,7,3,8,2,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 13
[5,4,6,7,3,8,2,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 13
[4,5,6,7,3,8,2,1] => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 12
[7,6,5,3,4,8,2,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[7,6,4,3,5,8,2,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[7,6,3,4,5,8,2,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 13
[6,3,4,5,7,8,2,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[4,3,5,6,7,8,2,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[8,7,6,5,4,2,3,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 18
[7,8,6,5,4,2,3,1] => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 17
[8,6,7,5,4,2,3,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 16
[8,7,5,6,4,2,3,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[5,6,7,8,4,2,3,1] => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 12
[8,7,6,4,5,2,3,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[8,6,7,4,5,2,3,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 13
[4,5,6,7,8,2,3,1] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 7
[8,7,6,5,3,2,4,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 18
[8,6,5,7,3,2,4,1] => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 15
[5,6,7,8,3,2,4,1] => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 12
[8,7,6,5,2,3,4,1] => [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 13
[6,7,8,5,2,3,4,1] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[8,5,6,7,2,3,4,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[5,6,7,8,2,3,4,1] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 7
[7,8,3,2,4,5,6,1] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
Description
The number of edges of a graph.
Matching statistic: St000462
Mapping
St000462: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 37%
Values
[1] => ? = 0
[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
[1,4,6,5,7,2,3] => ? = 4
[1,4,6,5,7,3,2] => ? = 10
[1,4,6,7,2,3,5] => ? = 1
[1,4,6,7,2,5,3] => ? = 7
[1,4,6,7,3,2,5] => ? = 6
[1,4,6,7,3,5,2] => ? = 7
[1,4,6,7,5,2,3] => ? = 6
[1,4,6,7,5,3,2] => ? = 12
[1,4,7,2,3,5,6] => ? = 1
[1,4,7,2,3,6,5] => ? = 7
[1,4,7,2,5,3,6] => ? = 6
[1,4,7,2,5,6,3] => ? = 7
[1,4,7,2,6,3,5] => ? = 5
[1,4,7,2,6,5,3] => ? = 11
[1,4,7,3,2,5,6] => ? = 5
[1,4,7,3,2,6,5] => ? = 11
[1,4,7,3,5,2,6] => ? = 6
[1,4,7,3,5,6,2] => ? = 7
[1,4,7,3,6,2,5] => ? = 5
[1,4,7,3,6,5,2] => ? = 11
[1,4,7,5,2,3,6] => ? = 4
[1,4,7,5,2,6,3] => ? = 10
[1,4,7,5,3,2,6] => ? = 9
[1,4,7,5,3,6,2] => ? = 10
[1,4,7,5,6,2,3] => ? = 4
[1,4,7,5,6,3,2] => ? = 10
[1,4,7,6,2,3,5] => ? = 4
[1,4,7,6,2,5,3] => ? = 10
[1,4,7,6,3,2,5] => ? = 9
[1,4,7,6,3,5,2] => ? = 10
[1,4,7,6,5,2,3] => ? = 9
[1,4,7,6,5,3,2] => ? = 15
[1,5,2,3,4,6,7] => ? = 1
[1,5,2,3,4,7,6] => ? = 6
[1,5,2,3,6,4,7] => ? = 5
[1,5,2,3,6,7,4] => ? = 5
[1,5,2,3,7,4,6] => ? = 5
[1,5,2,3,7,6,4] => ? = 11
[1,5,2,4,3,6,7] => ? = 5
[1,5,2,4,3,7,6] => ? = 10
[1,5,2,4,6,3,7] => ? = 5
[1,5,2,4,6,7,3] => ? = 5
[1,5,2,4,7,3,6] => ? = 5
[1,5,2,4,7,6,3] => ? = 11
[1,5,2,6,3,4,7] => ? = 4
[1,5,2,6,3,7,4] => ? = 9
[1,5,2,6,4,3,7] => ? = 9
[1,5,2,6,4,7,3] => ? = 9
[1,5,2,6,7,3,4] => ? = 4
Description
The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000005
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 28%
Values
[1] => [1] => [1,0]
 => 0
[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
[1,2,3,4,5,6,7] => [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.