Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000006
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000006: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
Description
The dinv of a Dyck path. Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see [[St000012]]). The dinv statistic of $D$ is $$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$ Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose ''arm length'' is one larger or equal to its ''leg length''. There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2]. Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by $$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$