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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: St000005

Mapping

Mp00248: Permutations —DEX composition⟶ 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] => [1,0]
 => 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

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: 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] => [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

Description

The number of edges of a graph.

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] => [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

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: St000462

Mapping

St000462: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

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

Description

The major index minus the number of excedences of a permutation. This occurs in the context of Eulerian polynomials [1].

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] => [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

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

$\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$ .