Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000946
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000946: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 00 => [2] => [1,1,0,0]
 => 1
[1,-2] => 01 => [1,1] => [1,0,1,0]
 => 0
[-1,2] => 10 => [1,1] => [1,0,1,0]
 => 0
[-1,-2] => 11 => [2] => [1,1,0,0]
 => 1
[2,1] => 00 => [2] => [1,1,0,0]
 => 1
[2,-1] => 01 => [1,1] => [1,0,1,0]
 => 0
[-2,1] => 10 => [1,1] => [1,0,1,0]
 => 0
[-2,-1] => 11 => [2] => [1,1,0,0]
 => 1
[1,2,3] => 000 => [3] => [1,1,1,0,0,0]
 => 1
[1,2,-3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 3
[1,-2,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,-2,-3] => 011 => [1,2] => [1,0,1,1,0,0]
 => 2
[-1,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 2
[-1,2,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[-1,-2,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 3
[-1,-2,-3] => 111 => [3] => [1,1,1,0,0,0]
 => 1
[1,3,2] => 000 => [3] => [1,1,1,0,0,0]
 => 1
[1,3,-2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 3
[1,-3,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,-3,-2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 2
[-1,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 2
[-1,3,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[-1,-3,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 3
[-1,-3,-2] => 111 => [3] => [1,1,1,0,0,0]
 => 1
[2,1,3] => 000 => [3] => [1,1,1,0,0,0]
 => 1
[2,1,-3] => 001 => [2,1] => [1,1,0,0,1,0]
 => 3
[2,-1,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[2,-1,-3] => 011 => [1,2] => [1,0,1,1,0,0]
 => 2
[-2,1,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 2
[-2,1,-3] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[-2,-1,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 3
[-2,-1,-3] => 111 => [3] => [1,1,1,0,0,0]
 => 1
[2,3,1] => 000 => [3] => [1,1,1,0,0,0]
 => 1
[2,3,-1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 3
[2,-3,1] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[2,-3,-1] => 011 => [1,2] => [1,0,1,1,0,0]
 => 2
[-2,3,1] => 100 => [1,2] => [1,0,1,1,0,0]
 => 2
[-2,3,-1] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[-2,-3,1] => 110 => [2,1] => [1,1,0,0,1,0]
 => 3
[-2,-3,-1] => 111 => [3] => [1,1,1,0,0,0]
 => 1
[3,1,2] => 000 => [3] => [1,1,1,0,0,0]
 => 1
[3,1,-2] => 001 => [2,1] => [1,1,0,0,1,0]
 => 3
[3,-1,2] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[3,-1,-2] => 011 => [1,2] => [1,0,1,1,0,0]
 => 2
[-3,1,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 2
[-3,1,-2] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 0
[-3,-1,2] => 110 => [2,1] => [1,1,0,0,1,0]
 => 3
[-3,-1,-2] => 111 => [3] => [1,1,1,0,0,0]
 => 1
[3,2,1] => 000 => [3] => [1,1,1,0,0,0]
 => 1
[3,2,-1] => 001 => [2,1] => [1,1,0,0,1,0]
 => 3
Description
The sum of the skew hook positions in a Dyck path. A skew hook is an occurrence of a down step followed by two up steps or of an up step followed by a down step. Write $U_i$ for the $i$-th up step and $D_j$ for the $j$-th down step in the Dyck path. Then the skew hook set is the set $$H = \{j: U_{i−1} U_i D_j \text{ is a skew hook}\} \cup \{i: D_{i−1} D_i U_j\text{ is a skew hook}\}.$$ This statistic is the sum of all elements in $H$.