Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001094
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Mapping
St001094: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => 0
{{1,2}}
 => 1
{{1},{2}}
 => 0
{{1,2,3}}
 => 3
{{1,2},{3}}
 => 2
{{1,3},{2}}
 => 1
{{1},{2,3}}
 => 2
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 6
{{1,2,3},{4}}
 => 5
{{1,2,4},{3}}
 => 4
{{1,2},{3,4}}
 => 5
{{1,2},{3},{4}}
 => 3
{{1,3,4},{2}}
 => 4
{{1,3},{2,4}}
 => 3
{{1,3},{2},{4}}
 => 2
{{1,4},{2,3}}
 => 4
{{1},{2,3,4}}
 => 5
{{1},{2,3},{4}}
 => 3
{{1,4},{2},{3}}
 => 1
{{1},{2,4},{3}}
 => 2
{{1},{2},{3,4}}
 => 3
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 10
{{1,2,3,4},{5}}
 => 9
{{1,2,3,5},{4}}
 => 8
{{1,2,3},{4,5}}
 => 9
{{1,2,3},{4},{5}}
 => 7
{{1,2,4,5},{3}}
 => 8
{{1,2,4},{3,5}}
 => 7
{{1,2,4},{3},{5}}
 => 6
{{1,2,5},{3,4}}
 => 8
{{1,2},{3,4,5}}
 => 9
{{1,2},{3,4},{5}}
 => 7
{{1,2,5},{3},{4}}
 => 5
{{1,2},{3,5},{4}}
 => 6
{{1,2},{3},{4,5}}
 => 7
{{1,2},{3},{4},{5}}
 => 4
{{1,3,4,5},{2}}
 => 8
{{1,3,4},{2,5}}
 => 7
{{1,3,4},{2},{5}}
 => 6
{{1,3,5},{2,4}}
 => 6
{{1,3},{2,4,5}}
 => 7
{{1,3},{2,4},{5}}
 => 5
{{1,3,5},{2},{4}}
 => 5
{{1,3},{2,5},{4}}
 => 4
{{1,3},{2},{4,5}}
 => 6
{{1,3},{2},{4},{5}}
 => 3
{{1,4,5},{2,3}}
 => 8
{{1,4},{2,3,5}}
 => 7
Description
The depth index of a set partition. For a set partition $\Pi$ of $\{1,\dots,n\}$ with arcs $\mathcal A$, this is $$\sum_{i=1}^{|\mathcal A|} (n-i) - \sum_{j=1}^n depth(j) + \sum_{\alpha\in\mathcal A} depth(\alpha),$$ where the depth of an element $i$ is the number of arcs $(k,\ell)$ with $k < i < \ell$, and the depth of an arc $(i,j)$ is the number of arcs $(k,\ell)$ with $k < i$ and $j < \ell$.
Matching statistic: St000008
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 62% ●values known / values provided: 88%●distinct values known / distinct values provided: 62%
Values
{{1}}
 => {{1}}
 => [1] => [1] => 0
{{1,2}}
 => {{1,2}}
 => [2] => [1,1] => 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [2] => 0
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [1,1,1] => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [2,1] => 2
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,2] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [2,1] => 2
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [3] => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1] => 6
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [2,1,1] => 5
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [1,2,1] => 4
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [2,1,1] => 5
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [3,1] => 3
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => [1,2,1] => 4
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => [1,1,2] => 3
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 2
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [1,2,1] => 4
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [2,1,1] => 5
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [3,1] => 3
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,3] => 1
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [3,1] => 3
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 10
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [2,1,1,1] => 9
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,2,1,1] => 8
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [2,1,1,1] => 9
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [3,1,1] => 7
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [1,2,1,1] => 8
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [1,1,2,1] => 7
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [2,2,1] => 6
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,2,1,1] => 8
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [2,1,1,1] => 9
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [3,1,1] => 7
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 5
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [2,2,1] => 6
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [3,1,1] => 7
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [4,1] => 4
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => [1,2,1,1] => 8
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,2,1] => 7
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [2,2,1] => 6
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => [1,1,1,2] => 6
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [1,1,2,1] => 7
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [2,1,2] => 5
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [2,1,2] => 5
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,2,2] => 4
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [2,2,1] => 6
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [3,2] => 3
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [1,2,1,1] => 8
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [1,1,2,1] => 7
{{1,2},{3,4},{5,6},{7,8},{9,10}}
 => {{1,2,4,6,8,10},{3},{5},{7},{9}}
 => [6,1,1,1,1] => [5,1,1,1,1,1] => ? = 35
{{1,4},{2,3},{5,6},{7,8},{9,10}}
 => {{1,3,6,8,10},{2,4},{5},{7},{9}}
 => [5,2,1,1,1] => [4,2,1,1,1,1] => ? = 34
{{1,8},{2,3},{4,5},{6,7},{9,10}}
 => {{1,3,5,7,10},{2},{4},{6,8},{9}}
 => [5,1,1,2,1] => [2,4,1,1,1,1] => ? = 32
{{1,2},{3,6},{4,5},{7,8},{9,10}}
 => {{1,2,5,8,10},{3,6},{4},{7},{9}}
 => [5,2,1,1,1] => [4,2,1,1,1,1] => ? = 34
{{1,6},{2,5},{3,4},{7,8},{9,10}}
 => {{1,4,8,10},{2,5},{3,6},{7},{9}}
 => [4,2,2,1,1] => [3,2,2,1,1,1] => ? = 32
{{1,8},{2,5},{3,4},{6,7},{9,10}}
 => {{1,4,7,10},{2,5},{3},{6,8},{9}}
 => [4,2,1,2,1] => [2,3,2,1,1,1] => ? = 31
{{1,10},{2,5},{3,4},{6,7},{8,9}}
 => {{1,4,7,9},{2,5},{3},{6},{8,10}}
 => [4,2,1,1,2] => [1,4,2,1,1,1] => ? = 30
{{1,8},{2,7},{3,4},{5,6},{9,10}}
 => {{1,4,6,10},{2},{3,7},{5,8},{9}}
 => [4,1,2,2,1] => [2,2,3,1,1,1] => ? = 30
{{1,10},{2,7},{3,4},{5,6},{8,9}}
 => {{1,4,6,9},{2},{3,7},{5},{8,10}}
 => [4,1,2,1,2] => [1,3,3,1,1,1] => ? = 29
{{1,2},{3,10},{4,5},{6,7},{8,9}}
 => {{1,2,5,7,9},{3},{4},{6,10},{8}}
 => [5,1,1,2,1] => [2,4,1,1,1,1] => ? = 32
{{1,10},{2,9},{3,4},{5,6},{7,8}}
 => {{1,4,6,8},{2},{3},{5,9},{7,10}}
 => [4,1,1,2,2] => [1,2,4,1,1,1] => ? = 28
{{1,2},{3,4},{5,8},{6,7},{9,10}}
 => {{1,2,4,7,10},{3,8},{5},{6},{9}}
 => [5,2,1,1,1] => [4,2,1,1,1,1] => ? = 34
{{1,4},{2,3},{5,8},{6,7},{9,10}}
 => {{1,3,7,10},{2,4,8},{5},{6},{9}}
 => [4,3,1,1,1] => [4,1,2,1,1,1] => ? = 33
{{1,8},{2,3},{4,7},{5,6},{9,10}}
 => {{1,3,6,10},{2,7},{4},{5,8},{9}}
 => [4,2,1,2,1] => [2,3,2,1,1,1] => ? = 31
{{1,10},{2,3},{4,7},{5,6},{8,9}}
 => {{1,3,6,9},{2,7},{4},{5},{8,10}}
 => [4,2,1,1,2] => [1,4,2,1,1,1] => ? = 30
{{1,2},{3,8},{4,7},{5,6},{9,10}}
 => {{1,2,6,10},{3,7},{4,8},{5},{9}}
 => [4,2,2,1,1] => [3,2,2,1,1,1] => ? = 32
{{1,10},{2,7},{3,6},{4,5},{8,9}}
 => {{1,5,9},{2,6},{3,7},{4},{8,10}}
 => [3,2,2,1,2] => [1,3,2,2,1,1] => ? = 28
{{1,2},{3,10},{4,7},{5,6},{8,9}}
 => {{1,2,6,9},{3,7},{4},{5,10},{8}}
 => [4,2,1,2,1] => [2,3,2,1,1,1] => ? = 31
{{1,10},{2,9},{3,6},{4,5},{7,8}}
 => {{1,5,8},{2,6},{3},{4,9},{7,10}}
 => [3,2,1,2,2] => [1,2,3,2,1,1] => ? = 27
{{1,4},{2,3},{5,10},{6,7},{8,9}}
 => {{1,3,7,9},{2,4},{5,10},{6},{8}}
 => [4,2,2,1,1] => [3,2,2,1,1,1] => ? = 32
{{1,10},{2,3},{4,9},{5,6},{7,8}}
 => {{1,3,6,8},{2},{4,9},{5},{7,10}}
 => [4,1,2,1,2] => [1,3,3,1,1,1] => ? = 29
{{1,2},{3,10},{4,9},{5,6},{7,8}}
 => {{1,2,6,8},{3},{4,9},{5,10},{7}}
 => [4,1,2,2,1] => [2,2,3,1,1,1] => ? = 30
{{1,10},{2,9},{3,8},{4,5},{6,7}}
 => {{1,5,7},{2},{3,8},{4,9},{6,10}}
 => [3,1,2,2,2] => [1,2,2,3,1,1] => ? = 26
{{1,2},{3,4},{5,6},{7,10},{8,9}}
 => {{1,2,4,6,9},{3,10},{5},{7},{8}}
 => [5,2,1,1,1] => [4,2,1,1,1,1] => ? = 34
{{1,4},{2,3},{5,6},{7,10},{8,9}}
 => {{1,3,6,9},{2,4,10},{5},{7},{8}}
 => [4,3,1,1,1] => [4,1,2,1,1,1] => ? = 33
{{1,6},{2,3},{4,5},{7,10},{8,9}}
 => {{1,3,5,9},{2,10},{4,6},{7},{8}}
 => ? => ? => ? = 32
{{1,10},{2,3},{4,5},{6,9},{7,8}}
 => {{1,3,5,8},{2,9},{4},{6},{7,10}}
 => [4,2,1,1,2] => [1,4,2,1,1,1] => ? = 30
{{1,2},{3,6},{4,5},{7,10},{8,9}}
 => {{1,2,5,9},{3,6,10},{4},{7},{8}}
 => [4,3,1,1,1] => [4,1,2,1,1,1] => ? = 33
{{1,10},{2,5},{3,4},{6,9},{7,8}}
 => {{1,4,8},{2,5,9},{3},{6},{7,10}}
 => [3,3,1,1,2] => [1,4,1,2,1,1] => ? = 29
{{1,2},{3,10},{4,5},{6,9},{7,8}}
 => {{1,2,5,8},{3,9},{4},{6,10},{7}}
 => [4,2,1,2,1] => [2,3,2,1,1,1] => ? = 31
{{1,10},{2,9},{3,4},{5,8},{6,7}}
 => {{1,4,7},{2,8},{3},{5,9},{6,10}}
 => [3,2,1,2,2] => [1,2,3,2,1,1] => ? = 27
{{1,2},{3,4},{5,10},{6,9},{7,8}}
 => {{1,2,4,8},{3,9},{5,10},{6},{7}}
 => [4,2,2,1,1] => [3,2,2,1,1,1] => ? = 32
{{1,10},{2,3},{4,9},{5,8},{6,7}}
 => {{1,3,7},{2,8},{4,9},{5},{6,10}}
 => [3,2,2,1,2] => [1,3,2,2,1,1] => ? = 28
{{1,10},{2,9},{3,8},{4,7},{5,6}}
 => {{1,6},{2,7},{3,8},{4,9},{5,10}}
 => [2,2,2,2,2] => [1,2,2,2,2,1] => ? = 25
{{1,4,5,6,7,8},{2},{3}}
 => {{1,5,6,7,8},{2},{3,4}}
 => ? => ? => ? = 23
{{1,2,3,4,5,6,7,8},{9}}
 => {{1,2,3,4,5,6,7,8},{9}}
 => [8,1] => [2,1,1,1,1,1,1,1] => ? = 35
{{1},{2,3,4,5,6,7,8,9}}
 => {{1,3,4,5,6,7,8,9},{2}}
 => [8,1] => [2,1,1,1,1,1,1,1] => ? = 35
{{1,2,3,4,5,6,7,8,9},{10}}
 => {{1,2,3,4,5,6,7,8,9},{10}}
 => [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 44
{{1,2,3,4,5,6,8},{7},{9}}
 => {{1,2,3,4,5,6},{7,8},{9}}
 => [6,2,1] => [2,2,1,1,1,1,1] => ? = 32
{{1},{2,3,4,5,6,7,9},{8}}
 => {{1,3,4,5,6,7},{2,9},{8}}
 => [6,2,1] => [2,2,1,1,1,1,1] => ? = 32
{{1},{2,3,4,5,6,7,8,9,10}}
 => {{1,3,4,5,6,7,8,9,10},{2}}
 => [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 44
{{1,2,3,4,5,6,7,8,9,10},{11}}
 => {{1,2,3,4,5,6,7,8,9,10},{11}}
 => [10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 54
{{1,2,3,4,5,8},{6,7},{9}}
 => {{1,2,3,4,5,7},{6,8},{9}}
 => [6,2,1] => [2,2,1,1,1,1,1] => ? = 32
{{1,2,3,4,5,7,8},{6},{9}}
 => {{1,2,3,4,5,8},{6,7},{9}}
 => [6,2,1] => [2,2,1,1,1,1,1] => ? = 32
{{1},{2,3,4,5,6,9},{7,8}}
 => {{1,3,4,5,6,8},{2,9},{7}}
 => [6,2,1] => [2,2,1,1,1,1,1] => ? = 32
{{1},{2,3,4,5,6,8,9},{7}}
 => {{1,3,4,5,6,9},{2,8},{7}}
 => [6,2,1] => [2,2,1,1,1,1,1] => ? = 32
{{1},{2,3,4,5,6,7,8,9,10,11}}
 => {{1,3,4,5,6,7,8,9,10,11},{2}}
 => [10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 54
{{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
 => {{1,2,4,6,8,10,12},{3},{5},{7},{9},{11}}
 => [7,1,1,1,1,1] => ? => ? = 51
{{1,2},{3,4},{5,6},{7,8},{9,12},{10,11}}
 => {{1,2,4,6,8,11},{3,12},{5},{7},{9},{10}}
 => [6,2,1,1,1,1] => ? => ? = 50
{{1,2},{3,4},{5,6},{7,10},{8,9},{11,12}}
 => {{1,2,4,6,9,12},{3,10},{5},{7},{8},{11}}
 => [6,2,1,1,1,1] => ? => ? = 50
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]].