- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000008

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => [1] => 0
[-1] => 1 => [1] => 0
[1,2] => 00 => [2] => 0
[1,-2] => 01 => [1,1] => 1
[-1,2] => 10 => [1,1] => 1
[-1,-2] => 11 => [2] => 0
[2,1] => 00 => [2] => 0
[2,-1] => 01 => [1,1] => 1
[-2,1] => 10 => [1,1] => 1
[-2,-1] => 11 => [2] => 0
[1,2,3] => 000 => [3] => 0
[1,2,-3] => 001 => [2,1] => 2
[1,-2,3] => 010 => [1,1,1] => 3
[1,-2,-3] => 011 => [1,2] => 1
[-1,2,3] => 100 => [1,2] => 1
[-1,2,-3] => 101 => [1,1,1] => 3
[-1,-2,3] => 110 => [2,1] => 2
[-1,-2,-3] => 111 => [3] => 0
[1,3,2] => 000 => [3] => 0
[1,3,-2] => 001 => [2,1] => 2
[1,-3,2] => 010 => [1,1,1] => 3
[1,-3,-2] => 011 => [1,2] => 1
[-1,3,2] => 100 => [1,2] => 1
[-1,3,-2] => 101 => [1,1,1] => 3
[-1,-3,2] => 110 => [2,1] => 2
[-1,-3,-2] => 111 => [3] => 0
[2,1,3] => 000 => [3] => 0
[2,1,-3] => 001 => [2,1] => 2
[2,-1,3] => 010 => [1,1,1] => 3
[2,-1,-3] => 011 => [1,2] => 1
[-2,1,3] => 100 => [1,2] => 1
[-2,1,-3] => 101 => [1,1,1] => 3
[-2,-1,3] => 110 => [2,1] => 2
[-2,-1,-3] => 111 => [3] => 0
[2,3,1] => 000 => [3] => 0
[2,3,-1] => 001 => [2,1] => 2
[2,-3,1] => 010 => [1,1,1] => 3
[2,-3,-1] => 011 => [1,2] => 1
[-2,3,1] => 100 => [1,2] => 1
[-2,3,-1] => 101 => [1,1,1] => 3
[-2,-3,1] => 110 => [2,1] => 2
[-2,-3,-1] => 111 => [3] => 0
[3,1,2] => 000 => [3] => 0
[3,1,-2] => 001 => [2,1] => 2
[3,-1,2] => 010 => [1,1,1] => 3
[3,-1,-2] => 011 => [1,2] => 1
[-3,1,2] => 100 => [1,2] => 1
[-3,1,-2] => 101 => [1,1,1] => 3
[-3,-1,2] => 110 => [2,1] => 2
[-3,-1,-2] => 111 => [3] => 0

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

Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => [1] => [1,0]
 => 0
[-1] => 1 => [1] => [1,0]
 => 0
[1,2] => 00 => [2] => [1,1,0,0]
 => 0
[1,-2] => 01 => [1,1] => [1,0,1,0]
 => 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
 => 1
[-1,-2] => 11 => [2] => [1,1,0,0]
 => 0
[2,1] => 00 => [2] => [1,1,0,0]
 => 0
[2,-1] => 01 => [1,1] => [1,0,1,0]
 => 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
 => 1
[-2,-1] => 11 => [2] => [1,1,0,0]
 => 0
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[5,6,7,8,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,3,1,2,4] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,1,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,1,4,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,8,5,6,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[3,2,8,-6,-5,1,4,7] => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 16
[4,5,2,3,-8,-6,1,7] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[4,3,7,-6,2,5,-8,1] => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 20
[5,6,3,4,-7,2,-8,1] => 00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 22
[6,7,4,5,-8,-2,1,3] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[5,6,7,1,8,2,3,4] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[-4,-8,-7,1,-6,2,3,5] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[-4,-8,-7,2,-5,1,3,6] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[-8,-4,-6,2,3,-7,1,5] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[3,4,7,8,1,2,5,6] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,2,6,-8,-5,1,4] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[-7,-8,-3,2,5,-6,1,4] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[7,2,6,-8,-5,1,3,4] => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[-8,-6,-5,4,2,-7,1,3] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[4,7,6,-5,2,3,-8,1] => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 20
[3,2,5,4,7,6,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[4,5,2,3,7,6,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[5,4,2,3,7,6,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[6,4,2,7,3,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[7,4,2,6,3,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[8,4,2,6,3,-1,5,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[8,5,6,2,3,-1,4,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[7,5,6,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[6,5,7,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[5,6,7,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[4,6,2,7,3,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[3,2,6,7,4,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[3,2,7,6,4,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[4,7,2,6,3,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[5,7,6,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[6,7,5,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[7,6,5,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[8,6,5,2,3,-1,4,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[8,7,5,2,-1,3,4,6] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[7,8,5,2,-1,3,4,6] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[6,8,5,2,3,-1,4,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[5,8,6,2,3,-1,4,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[4,8,2,6,3,-1,5,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[3,2,8,6,4,-1,5,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[3,2,8,7,-1,4,5,6] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[4,8,2,7,-1,3,5,6] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[5,8,7,2,-1,3,4,6] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[6,8,7,-1,2,3,4,5] => 00010000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 7

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

Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1] => 0 => [1] => [1,0]
 => ? = 0
[-1] => 1 => [1] => [1,0]
 => ? = 0
[1,2] => 00 => [2] => [1,1,0,0]
 => 0
[1,-2] => 01 => [1,1] => [1,0,1,0]
 => 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
 => 1
[-1,-2] => 11 => [2] => [1,1,0,0]
 => 0
[2,1] => 00 => [2] => [1,1,0,0]
 => 0
[2,-1] => 01 => [1,1] => [1,0,1,0]
 => 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
 => 1
[-2,-1] => 11 => [2] => [1,1,0,0]
 => 0
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[5,6,7,8,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,3,1,2,4] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,1,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,1,4,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,8,5,6,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[3,2,8,-6,-5,1,4,7] => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 16
[4,5,2,3,-8,-6,1,7] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[4,3,7,-6,2,5,-8,1] => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 20
[5,6,3,4,-7,2,-8,1] => 00001010 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 22
[6,7,4,5,-8,-2,1,3] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[5,6,7,1,8,2,3,4] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[-4,-8,-7,1,-6,2,3,5] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[-4,-8,-7,2,-5,1,3,6] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 12
[-8,-4,-6,2,3,-7,1,5] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[3,4,7,8,1,2,5,6] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[3,7,2,6,-8,-5,1,4] => 00001100 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 10
[-7,-8,-3,2,5,-6,1,4] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[7,2,6,-8,-5,1,3,4] => 00011000 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 8
[-8,-6,-5,4,2,-7,1,3] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 14
[4,7,6,-5,2,3,-8,1] => 00010010 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 20
[3,2,5,4,7,6,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[4,5,2,3,7,6,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[5,4,2,3,7,6,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[6,4,2,7,3,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[7,4,2,6,3,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[8,4,2,6,3,-1,5,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[8,5,6,2,3,-1,4,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[7,5,6,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[6,5,7,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[5,6,7,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[4,6,2,7,3,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[3,2,6,7,4,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[3,2,7,6,4,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[4,7,2,6,3,5,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[5,7,6,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[6,7,5,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[7,6,5,2,3,4,-1,8] => 00000010 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 13
[8,6,5,2,3,-1,4,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[8,7,5,2,-1,3,4,6] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[7,8,5,2,-1,3,4,6] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[6,8,5,2,3,-1,4,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[5,8,6,2,3,-1,4,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[4,8,2,6,3,-1,5,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[3,2,8,6,4,-1,5,7] => 00000100 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 11
[3,2,8,7,-1,4,5,6] => 00001000 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 9
[4,8,2,7,-1,3,5,6] => 00001000 => [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

Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 93%

Values

[1] => 0 => [1] => ([],1)
 => 0
[-1] => 1 => [1] => ([],1)
 => 0
[1,2] => 00 => [2] => ([],2)
 => 0
[1,-2] => 01 => [1,1] => ([(0,1)],2)
 => 1
[-1,2] => 10 => [1,1] => ([(0,1)],2)
 => 1
[-1,-2] => 11 => [2] => ([],2)
 => 0
[2,1] => 00 => [2] => ([],2)
 => 0
[2,-1] => 01 => [1,1] => ([(0,1)],2)
 => 1
[-2,1] => 10 => [1,1] => ([(0,1)],2)
 => 1
[-2,-1] => 11 => [2] => ([],2)
 => 0
[1,2,3] => 000 => [3] => ([],3)
 => 0
[1,2,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,-2,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,-2,-3] => 011 => [1,2] => ([(1,2)],3)
 => 1
[-1,2,3] => 100 => [1,2] => ([(1,2)],3)
 => 1
[-1,2,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[-1,-2,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[-1,-2,-3] => 111 => [3] => ([],3)
 => 0
[1,3,2] => 000 => [3] => ([],3)
 => 0
[1,3,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,-3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,-3,-2] => 011 => [1,2] => ([(1,2)],3)
 => 1
[-1,3,2] => 100 => [1,2] => ([(1,2)],3)
 => 1
[-1,3,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[-1,-3,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[-1,-3,-2] => 111 => [3] => ([],3)
 => 0
[2,1,3] => 000 => [3] => ([],3)
 => 0
[2,1,-3] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,-1,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,-1,-3] => 011 => [1,2] => ([(1,2)],3)
 => 1
[-2,1,3] => 100 => [1,2] => ([(1,2)],3)
 => 1
[-2,1,-3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[-2,-1,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[-2,-1,-3] => 111 => [3] => ([],3)
 => 0
[2,3,1] => 000 => [3] => ([],3)
 => 0
[2,3,-1] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,-3,1] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,-3,-1] => 011 => [1,2] => ([(1,2)],3)
 => 1
[-2,3,1] => 100 => [1,2] => ([(1,2)],3)
 => 1
[-2,3,-1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[-2,-3,1] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[-2,-3,-1] => 111 => [3] => ([],3)
 => 0
[3,1,2] => 000 => [3] => ([],3)
 => 0
[3,1,-2] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,-1,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,-1,-2] => 011 => [1,2] => ([(1,2)],3)
 => 1
[-3,1,2] => 100 => [1,2] => ([(1,2)],3)
 => 1
[-3,1,-2] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[-3,-1,2] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[-3,-1,-2] => 111 => [3] => ([],3)
 => 0
[5,6,7,8,3,4,1,2] => 00000000 => [8] => ([],8)
 => ? = 0
[5,6,7,8,3,1,2,4] => 00000000 => [8] => ([],8)
 => ? = 0
[5,6,7,8,4,1,2,3] => 00000000 => [8] => ([],8)
 => ? = 0
[5,6,7,8,1,4,2,3] => 00000000 => [8] => ([],8)
 => ? = 0
[7,8,5,6,3,4,1,2] => 00000000 => [8] => ([],8)
 => ? = 0
[-8,-6,-4,-2,1,3,5,7] => 11110000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 4
[6,8,-4,-3,1,2,5,7] => 00110000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[-3,2,-8,-6,-4,1,5,7] => 10111000 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[3,2,8,-6,-5,1,4,7] => 00011000 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[-5,-3,2,4,-8,-6,1,7] => 11001100 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 12
[5,-3,2,4,8,-7,1,6] => 01000100 => [1,1,3,1,2] => ([(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)
 => ? = 14
[-7,-5,-3,2,4,6,-8,1] => 11100010 => [3,3,1,1] => ([(0,6),(0,7),(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)
 => ? = 16
[5,7,-4,2,3,6,-8,1] => 00100010 => [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
[-4,5,2,3,8,-7,1,6] => 10000100 => [1,4,1,2] => ([(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)
 => ? = 12
[4,5,2,3,-8,-6,1,7] => 00001100 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[2,-8,-4,-6,-5,1,3,7] => 01111000 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[-8,-6,1,-5,-4,2,3,7] => 11011000 => [2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[-3,6,8,-5,1,2,4,7] => 10010000 => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[2,8,-6,-5,1,3,4,7] => 00110000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[-5,4,-8,-6,-2,1,3,7] => 10111000 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[4,-5,-8,-6,-3,1,2,7] => 01111000 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[3,-8,-5,-6,-4,1,2,7] => 01111000 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[-6,4,3,8,-5,1,2,7] => 10001000 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[-4,3,-5,2,-8,-6,1,7] => 10101100 => [1,1,1,1,2,2] => ([(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)
 => ? = 16
[4,3,-5,2,8,-7,1,6] => 00100100 => [2,1,2,1,2] => ([(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)
 => ? = 16
[-4,3,-7,-5,2,6,-8,1] => 10110010 => [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
[4,3,7,-6,2,5,-8,1] => 00010010 => [3,1,2,1,1] => ([(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)
 => ? = 20
[3,-4,-8,-6,2,-7,1,5] => 01110100 => [1,3,1,1,2] => ([(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)
 => ? = 16
[8,4,-7,-6,3,1,2,5] => 00110000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[-5,4,-7,3,8,1,2,6] => 10100000 => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[5,8,-4,3,-7,1,2,6] => 00101000 => [2,1,1,1,3] => ([(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)
 => ? = 14
[5,-3,4,8,-7,1,2,6] => 01001000 => [1,1,2,1,3] => ([(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)
 => ? = 12
[-5,2,4,8,-7,1,3,6] => 10001000 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[-7,-5,4,6,-8,-2,1,3] => 11001100 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 12
[4,-7,-5,6,-8,-3,1,2] => 01101100 => [1,2,1,2,2] => ([(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)
 => ? = 14
[-7,3,-5,6,-8,-4,1,2] => 10101100 => [1,1,1,1,2,2] => ([(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)
 => ? = 16
[-4,3,6,-7,-8,-5,1,2] => 10011100 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[-4,3,5,8,7,-6,1,2] => 10000100 => [1,4,1,2] => ([(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)
 => ? = 12
[4,-8,6,-7,3,5,1,2] => 01010000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[-6,-4,3,5,-7,2,-8,1] => 11001010 => [2,2,1,1,1,1] => ([(0,4),(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)
 => ? = 24
[5,6,3,4,-7,2,-8,1] => 00001010 => [4,1,1,1,1] => ([(0,4),(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)
 => ? = 22
[3,-8,5,7,4,-6,1,2] => 01000100 => [1,1,3,1,2] => ([(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)
 => ? = 14
[-7,-6,3,-8,4,-5,1,2] => 11010100 => [2,1,1,1,1,2] => ([(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)
 => ? = 20
[-7,-6,5,3,-8,-4,1,2] => 11001100 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 12
[-7,4,-6,5,-8,-3,1,2] => 10101100 => [1,1,1,1,2,2] => ([(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)
 => ? = 16
[6,7,4,5,-8,-2,1,3] => 00001100 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 10
[8,-6,2,4,-7,1,3,5] => 01001000 => [1,1,2,1,3] => ([(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)
 => ? = 12
[4,-8,-6,-3,-7,1,2,5] => 01111000 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[-4,3,-8,-6,-7,1,2,5] => 10111000 => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[-7,6,8,1,3,-5,2,4] => 10000100 => [1,4,1,2] => ([(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)
 => ? = 12

Description

The number of edges of a graph.

Matching statistic: St000005

Mapping

Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 85%●distinct values known / distinct values provided: 57%

Values

[1] => 0 => [1] => [1,0]
 => 0
[-1] => 1 => [1] => [1,0]
 => 0
[1,2] => 00 => [2] => [1,1,0,0]
 => 0
[1,-2] => 01 => [1,1] => [1,0,1,0]
 => 1
[-1,2] => 10 => [1,1] => [1,0,1,0]
 => 1
[-1,-2] => 11 => [2] => [1,1,0,0]
 => 0
[2,1] => 00 => [2] => [1,1,0,0]
 => 0
[2,-1] => 01 => [1,1] => [1,0,1,0]
 => 1
[-2,1] => 10 => [1,1] => [1,0,1,0]
 => 1
[-2,-1] => 11 => [2] => [1,1,0,0]
 => 0
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
 => 0
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
 => 0
[1,2,3,4,7,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,2,5,3,7,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,2,7,3,4,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,2,7,3,6,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,4,7,2,3,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,5,7,2,3,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,6,7,2,3,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,6,7,2,5,3,4] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,6,7,4,2,3,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,1,7,2,4,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,1,7,5,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,4,7,1,2,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,4,7,1,6,2,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,4,7,5,1,2,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,4,7,5,6,1,2] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,5,7,1,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,6,1,2,7,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,7,4,1,2,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,7,4,5,1,2,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,7,4,5,6,1,2] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,7,4,6,1,2,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,7,5,1,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,7,5,6,1,2,4] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[3,7,6,1,2,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[4,7,5,6,1,2,3] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[5,3,4,7,1,2,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[5,3,7,1,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[5,4,7,1,2,3,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,2,5,3,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,3,2,4,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,4,2,3,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,4,2,6,3,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,4,5,2,3,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,4,6,2,3,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,5,2,3,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,5,6,2,3,4] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,6,2,3,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,1,6,4,5,2,3] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,2,4,1,3,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,2,5,1,3,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,2,6,1,3,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,3,1,2,4,5,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,3,5,1,2,4,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,3,5,6,1,2,4] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,3,6,1,2,4,5] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[7,5,1,4,2,3,6] => 0000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,3,4,1,2] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,3,1,2,4] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,1,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,1,4,2,3] => 00000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0

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: St001434
(load all 2 compositions to match this statistic)

Mapping

Mp00161: Signed permutations —reverse⟶ Signed permutations
Mp00271: Signed permutations —flag negatives to negatives⟶ Signed permutations
Mp00162: Signed permutations —inverse⟶ Signed permutations
St001434: Signed permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%

Values

[1] => [1] => [1] => [1] => 0
[-1] => [-1] => [-1] => [-1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[1,-2] => [-2,1] => [2,-1] => [-2,1] => 1
[-1,2] => [2,-1] => [-2,-1] => [-2,-1] => 1
[-1,-2] => [-2,-1] => [-2,1] => [2,-1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[2,-1] => [-1,2] => [1,-2] => [1,-2] => 1
[-2,1] => [1,-2] => [-1,-2] => [-1,-2] => 1
[-2,-1] => [-1,-2] => [-1,2] => [-1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,-3] => [-3,2,1] => [3,2,-1] => [-3,2,1] => 2
[1,-2,3] => [3,-2,1] => [3,-2,-1] => [-3,-2,1] => 3
[1,-2,-3] => [-3,-2,1] => [3,-2,1] => [3,-2,1] => 1
[-1,2,3] => [3,2,-1] => [-3,-2,1] => [3,-2,-1] => 1
[-1,2,-3] => [-3,2,-1] => [-3,-2,-1] => [-3,-2,-1] => 3
[-1,-2,3] => [3,-2,-1] => [-3,2,-1] => [-3,2,-1] => 2
[-1,-2,-3] => [-3,-2,-1] => [-3,2,1] => [3,2,-1] => 0
[1,3,2] => [2,3,1] => [2,3,1] => [3,1,2] => 0
[1,3,-2] => [-2,3,1] => [2,3,-1] => [-3,1,2] => 2
[1,-3,2] => [2,-3,1] => [2,-3,-1] => [-3,1,-2] => 3
[1,-3,-2] => [-2,-3,1] => [2,-3,1] => [3,1,-2] => 1
[-1,3,2] => [2,3,-1] => [-2,-3,1] => [3,-1,-2] => 1
[-1,3,-2] => [-2,3,-1] => [-2,-3,-1] => [-3,-1,-2] => 3
[-1,-3,2] => [2,-3,-1] => [-2,3,-1] => [-3,-1,2] => 2
[-1,-3,-2] => [-2,-3,-1] => [-2,3,1] => [3,-1,2] => 0
[2,1,3] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[2,1,-3] => [-3,1,2] => [3,1,-2] => [2,-3,1] => 2
[2,-1,3] => [3,-1,2] => [3,-1,-2] => [-2,-3,1] => 3
[2,-1,-3] => [-3,-1,2] => [3,-1,2] => [-2,3,1] => 1
[-2,1,3] => [3,1,-2] => [-3,-1,2] => [-2,3,-1] => 1
[-2,1,-3] => [-3,1,-2] => [-3,-1,-2] => [-2,-3,-1] => 3
[-2,-1,3] => [3,-1,-2] => [-3,1,-2] => [2,-3,-1] => 2
[-2,-1,-3] => [-3,-1,-2] => [-3,1,2] => [2,3,-1] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,3,-1] => [-1,3,2] => [1,3,-2] => [1,-3,2] => 2
[2,-3,1] => [1,-3,2] => [1,-3,-2] => [1,-3,-2] => 3
[2,-3,-1] => [-1,-3,2] => [1,-3,2] => [1,3,-2] => 1
[-2,3,1] => [1,3,-2] => [-1,-3,2] => [-1,3,-2] => 1
[-2,3,-1] => [-1,3,-2] => [-1,-3,-2] => [-1,-3,-2] => 3
[-2,-3,1] => [1,-3,-2] => [-1,3,-2] => [-1,-3,2] => 2
[-2,-3,-1] => [-1,-3,-2] => [-1,3,2] => [-1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,1,-2] => [-2,1,3] => [2,1,-3] => [2,1,-3] => 2
[3,-1,2] => [2,-1,3] => [2,-1,-3] => [-2,1,-3] => 3
[3,-1,-2] => [-2,-1,3] => [2,-1,3] => [-2,1,3] => 1
[-3,1,2] => [2,1,-3] => [-2,-1,3] => [-2,-1,3] => 1
[-3,1,-2] => [-2,1,-3] => [-2,-1,-3] => [-2,-1,-3] => 3
[-3,-1,2] => [2,-1,-3] => [-2,1,-3] => [2,-1,-3] => 2
[-3,-1,-2] => [-2,-1,-3] => [-2,1,3] => [2,-1,3] => 0
[1,2,3,4,-5] => [-5,4,3,2,1] => [5,4,3,2,-1] => [-5,4,3,2,1] => ? = 4
[1,2,3,-4,5] => [5,-4,3,2,1] => [5,4,3,-2,-1] => [-5,-4,3,2,1] => ? = 7
[1,2,3,-4,-5] => [-5,-4,3,2,1] => [5,4,3,-2,1] => [5,-4,3,2,1] => ? = 3
[1,2,-3,4,5] => [5,4,-3,2,1] => [5,4,-3,-2,1] => [5,-4,-3,2,1] => ? = 5
[1,2,-3,4,-5] => [-5,4,-3,2,1] => [5,4,-3,-2,-1] => [-5,-4,-3,2,1] => ? = 9
[1,2,-3,-4,5] => [5,-4,-3,2,1] => [5,4,-3,2,-1] => [-5,4,-3,2,1] => ? = 6
[1,2,-3,-4,-5] => [-5,-4,-3,2,1] => [5,4,-3,2,1] => [5,4,-3,2,1] => ? = 2
[1,-2,3,4,5] => [5,4,3,-2,1] => [5,-4,-3,2,1] => [5,4,-3,-2,1] => ? = 3
[1,-2,3,4,-5] => [-5,4,3,-2,1] => [5,-4,-3,2,-1] => [-5,4,-3,-2,1] => ? = 7
[1,-2,3,-4,5] => [5,-4,3,-2,1] => [5,-4,-3,-2,-1] => [-5,-4,-3,-2,1] => ? = 10
[1,-2,3,-4,-5] => [-5,-4,3,-2,1] => [5,-4,-3,-2,1] => [5,-4,-3,-2,1] => ? = 6
[1,-2,-3,4,5] => [5,4,-3,-2,1] => [5,-4,3,-2,1] => [5,-4,3,-2,1] => ? = 4
[1,-2,-3,4,-5] => [-5,4,-3,-2,1] => [5,-4,3,-2,-1] => [-5,-4,3,-2,1] => ? = 8
[1,-2,-3,-4,5] => [5,-4,-3,-2,1] => [5,-4,3,2,-1] => [-5,4,3,-2,1] => ? = 5
[1,-2,-3,-4,-5] => [-5,-4,-3,-2,1] => [5,-4,3,2,1] => [5,4,3,-2,1] => ? = 1
[-1,2,3,4,5] => [5,4,3,2,-1] => [-5,-4,3,2,1] => [5,4,3,-2,-1] => ? = 1
[-1,2,3,4,-5] => [-5,4,3,2,-1] => [-5,-4,3,2,-1] => [-5,4,3,-2,-1] => ? = 5
[-1,2,3,-4,5] => [5,-4,3,2,-1] => [-5,-4,3,-2,-1] => [-5,-4,3,-2,-1] => ? = 8
[-1,2,3,-4,-5] => [-5,-4,3,2,-1] => [-5,-4,3,-2,1] => [5,-4,3,-2,-1] => ? = 4
[-1,2,-3,4,5] => [5,4,-3,2,-1] => [-5,-4,-3,-2,1] => [5,-4,-3,-2,-1] => ? = 6
[-1,2,-3,4,-5] => [-5,4,-3,2,-1] => [-5,-4,-3,-2,-1] => [-5,-4,-3,-2,-1] => ? = 10
[-1,2,-3,-4,5] => [5,-4,-3,2,-1] => [-5,-4,-3,2,-1] => [-5,4,-3,-2,-1] => ? = 7
[-1,2,-3,-4,-5] => [-5,-4,-3,2,-1] => [-5,-4,-3,2,1] => [5,4,-3,-2,-1] => ? = 3
[-1,-2,3,4,5] => [5,4,3,-2,-1] => [-5,4,-3,2,1] => [5,4,-3,2,-1] => ? = 2
[-1,-2,3,4,-5] => [-5,4,3,-2,-1] => [-5,4,-3,2,-1] => [-5,4,-3,2,-1] => ? = 6
[-1,-2,3,-4,5] => [5,-4,3,-2,-1] => [-5,4,-3,-2,-1] => [-5,-4,-3,2,-1] => ? = 9
[-1,-2,3,-4,-5] => [-5,-4,3,-2,-1] => [-5,4,-3,-2,1] => [5,-4,-3,2,-1] => ? = 5
[-1,-2,-3,4,5] => [5,4,-3,-2,-1] => [-5,4,3,-2,1] => [5,-4,3,2,-1] => ? = 3
[-1,-2,-3,4,-5] => [-5,4,-3,-2,-1] => [-5,4,3,-2,-1] => [-5,-4,3,2,-1] => ? = 7
[-1,-2,-3,-4,5] => [5,-4,-3,-2,-1] => [-5,4,3,2,-1] => [-5,4,3,2,-1] => ? = 4
[1,2,3,5,-4] => [-4,5,3,2,1] => [4,5,3,2,-1] => [-5,4,3,1,2] => ? = 4
[1,2,3,-5,4] => [4,-5,3,2,1] => [4,5,3,-2,-1] => [-5,-4,3,1,2] => ? = 7
[1,2,3,-5,-4] => [-4,-5,3,2,1] => [4,5,3,-2,1] => [5,-4,3,1,2] => ? = 3
[1,2,-3,5,4] => [4,5,-3,2,1] => [4,5,-3,-2,1] => [5,-4,-3,1,2] => ? = 5
[1,2,-3,5,-4] => [-4,5,-3,2,1] => [4,5,-3,-2,-1] => [-5,-4,-3,1,2] => ? = 9
[1,2,-3,-5,4] => [4,-5,-3,2,1] => [4,5,-3,2,-1] => [-5,4,-3,1,2] => ? = 6
[1,2,-3,-5,-4] => [-4,-5,-3,2,1] => [4,5,-3,2,1] => [5,4,-3,1,2] => ? = 2
[1,-2,3,5,4] => [4,5,3,-2,1] => [4,-5,-3,2,1] => [5,4,-3,1,-2] => ? = 3
[1,-2,3,5,-4] => [-4,5,3,-2,1] => [4,-5,-3,2,-1] => [-5,4,-3,1,-2] => ? = 7
[1,-2,3,-5,4] => [4,-5,3,-2,1] => [4,-5,-3,-2,-1] => [-5,-4,-3,1,-2] => ? = 10
[1,-2,3,-5,-4] => [-4,-5,3,-2,1] => [4,-5,-3,-2,1] => [5,-4,-3,1,-2] => ? = 6
[1,-2,-3,5,4] => [4,5,-3,-2,1] => [4,-5,3,-2,1] => [5,-4,3,1,-2] => ? = 4
[1,-2,-3,5,-4] => [-4,5,-3,-2,1] => [4,-5,3,-2,-1] => [-5,-4,3,1,-2] => ? = 8
[1,-2,-3,-5,4] => [4,-5,-3,-2,1] => [4,-5,3,2,-1] => [-5,4,3,1,-2] => ? = 5
[-1,2,3,5,4] => [4,5,3,2,-1] => [-4,-5,3,2,1] => [5,4,3,-1,-2] => ? = 1
[-1,2,3,5,-4] => [-4,5,3,2,-1] => [-4,-5,3,2,-1] => [-5,4,3,-1,-2] => ? = 5
[-1,2,3,-5,4] => [4,-5,3,2,-1] => [-4,-5,3,-2,-1] => [-5,-4,3,-1,-2] => ? = 8
[-1,2,3,-5,-4] => [-4,-5,3,2,-1] => [-4,-5,3,-2,1] => [5,-4,3,-1,-2] => ? = 4
[-1,2,-3,5,4] => [4,5,-3,2,-1] => [-4,-5,-3,-2,1] => [5,-4,-3,-1,-2] => ? = 6
[-1,2,-3,5,-4] => [-4,5,-3,2,-1] => [-4,-5,-3,-2,-1] => [-5,-4,-3,-1,-2] => ? = 10

Description

The number of negative sum pairs of a signed permutation. The number of negative sum pairs of a signed permutation $\sigma$ is: $$\operatorname{nsp}(\sigma)=\big|\{1\leq i < j\leq n \mid \sigma(i)+\sigma(j) < 0\}\big|,$$ see [1, Eq.(8.1)].