Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000946
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Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000946: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 6
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 5
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 5
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 8
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 6
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 10
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 7
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 6
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 9
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 5
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 10
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 8
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 13
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 9
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 6
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 11
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 15
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 10
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 7
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 12
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 8
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 7
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 11
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 6
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 $H$.
Matching statistic: St000008
Mapping
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ 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
[1,1] => [2] => 10 => [1,1] => 1
[2] => [1,1] => 11 => [2] => 0
[1,1,1] => [3] => 100 => [1,2] => 1
[1,2] => [2,1] => 101 => [1,1,1] => 3
[2,1] => [1,2] => 110 => [2,1] => 2
[3] => [1,1,1] => 111 => [3] => 0
[1,1,1,1] => [4] => 1000 => [1,3] => 1
[1,1,2] => [3,1] => 1001 => [1,2,1] => 4
[1,2,1] => [2,2] => 1010 => [1,1,1,1] => 6
[1,3] => [2,1,1] => 1011 => [1,1,2] => 3
[2,1,1] => [1,3] => 1100 => [2,2] => 2
[2,2] => [1,2,1] => 1101 => [2,1,1] => 5
[3,1] => [1,1,2] => 1110 => [3,1] => 3
[4] => [1,1,1,1] => 1111 => [4] => 0
[1,1,1,1,1] => [5] => 10000 => [1,4] => 1
[1,1,1,2] => [4,1] => 10001 => [1,3,1] => 5
[1,1,2,1] => [3,2] => 10010 => [1,2,1,1] => 8
[1,1,3] => [3,1,1] => 10011 => [1,2,2] => 4
[1,2,1,1] => [2,3] => 10100 => [1,1,1,2] => 6
[1,2,2] => [2,2,1] => 10101 => [1,1,1,1,1] => 10
[1,3,1] => [2,1,2] => 10110 => [1,1,2,1] => 7
[1,4] => [2,1,1,1] => 10111 => [1,1,3] => 3
[2,1,1,1] => [1,4] => 11000 => [2,3] => 2
[2,1,2] => [1,3,1] => 11001 => [2,2,1] => 6
[2,2,1] => [1,2,2] => 11010 => [2,1,1,1] => 9
[2,3] => [1,2,1,1] => 11011 => [2,1,2] => 5
[3,1,1] => [1,1,3] => 11100 => [3,2] => 3
[3,2] => [1,1,2,1] => 11101 => [3,1,1] => 7
[4,1] => [1,1,1,2] => 11110 => [4,1] => 4
[5] => [1,1,1,1,1] => 11111 => [5] => 0
[1,1,1,1,1,1] => [6] => 100000 => [1,5] => 1
[1,1,1,1,2] => [5,1] => 100001 => [1,4,1] => 6
[1,1,1,2,1] => [4,2] => 100010 => [1,3,1,1] => 10
[1,1,1,3] => [4,1,1] => 100011 => [1,3,2] => 5
[1,1,2,1,1] => [3,3] => 100100 => [1,2,1,2] => 8
[1,1,2,2] => [3,2,1] => 100101 => [1,2,1,1,1] => 13
[1,1,3,1] => [3,1,2] => 100110 => [1,2,2,1] => 9
[1,1,4] => [3,1,1,1] => 100111 => [1,2,3] => 4
[1,2,1,1,1] => [2,4] => 101000 => [1,1,1,3] => 6
[1,2,1,2] => [2,3,1] => 101001 => [1,1,1,2,1] => 11
[1,2,2,1] => [2,2,2] => 101010 => [1,1,1,1,1,1] => 15
[1,2,3] => [2,2,1,1] => 101011 => [1,1,1,1,2] => 10
[1,3,1,1] => [2,1,3] => 101100 => [1,1,2,2] => 7
[1,3,2] => [2,1,2,1] => 101101 => [1,1,2,1,1] => 12
[1,4,1] => [2,1,1,2] => 101110 => [1,1,3,1] => 8
[1,5] => [2,1,1,1,1] => 101111 => [1,1,4] => 3
[2,1,1,1,1] => [1,5] => 110000 => [2,4] => 2
[2,1,1,2] => [1,4,1] => 110001 => [2,3,1] => 7
[2,1,2,1] => [1,3,2] => 110010 => [2,2,1,1] => 11
[2,1,3] => [1,3,1,1] => 110011 => [2,2,2] => 6
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]].