Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000579
(load all 99 compositions to match this statistic)
Mapping
St000579: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => 0
{{1},{2}}
 => 1
{{1,2,3}}
 => 0
{{1,2},{3}}
 => 2
{{1,3},{2}}
 => 2
{{1},{2,3}}
 => 1
{{1},{2},{3}}
 => 3
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 3
{{1,2,4},{3}}
 => 3
{{1,2},{3,4}}
 => 2
{{1,2},{3},{4}}
 => 5
{{1,3,4},{2}}
 => 2
{{1,3},{2,4}}
 => 3
{{1,3},{2},{4}}
 => 5
{{1,4},{2,3}}
 => 3
{{1},{2,3,4}}
 => 1
{{1},{2,3},{4}}
 => 4
{{1,4},{2},{3}}
 => 5
{{1},{2,4},{3}}
 => 4
{{1},{2},{3,4}}
 => 3
{{1},{2},{3},{4}}
 => 6
{{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => 4
{{1,2,3,5},{4}}
 => 4
{{1,2,3},{4,5}}
 => 3
{{1,2,3},{4},{5}}
 => 7
{{1,2,4,5},{3}}
 => 3
{{1,2,4},{3,5}}
 => 4
{{1,2,4},{3},{5}}
 => 7
{{1,2,5},{3,4}}
 => 4
{{1,2},{3,4,5}}
 => 2
{{1,2},{3,4},{5}}
 => 6
{{1,2,5},{3},{4}}
 => 7
{{1,2},{3,5},{4}}
 => 6
{{1,2},{3},{4,5}}
 => 5
{{1,2},{3},{4},{5}}
 => 9
{{1,3,4,5},{2}}
 => 2
{{1,3,4},{2,5}}
 => 4
{{1,3,4},{2},{5}}
 => 6
{{1,3,5},{2,4}}
 => 4
{{1,3},{2,4,5}}
 => 3
{{1,3},{2,4},{5}}
 => 7
{{1,3,5},{2},{4}}
 => 6
{{1,3},{2,5},{4}}
 => 7
{{1,3},{2},{4,5}}
 => 5
{{1,3},{2},{4},{5}}
 => 9
{{1,4,5},{2,3}}
 => 3
{{1,4},{2,3,5}}
 => 4
{{1,4},{2,3},{5}}
 => 7
Description
The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. This is the number of pairs $i\lt j$ in different blocks such that $j$ is the maximal element of a block.
Matching statistic: St001161
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [2,1] => [1,1,0,0]
 => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 5
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 3
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 5
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 4
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 5
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 4
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 6
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 4
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 7
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 3
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 4
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 7
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 4
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 6
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 7
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 6
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 5
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 9
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
 => 4
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 6
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 4
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => 3
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 7
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 6
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 7
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 5
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 9
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
 => 4
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => 7
{{1},{2,3,4,6,7,8},{5}}
 => [1,3,4,6,5,7,8,2] => [1,5,8,2,3,4,6,7] => ?
 => ? = 6
{{1},{2,3,4,5,6,7},{8}}
 => ? => ? => ?
 => ? = 8
{{1,4,5,6,7,8},{2},{3}}
 => [4,2,3,5,6,7,8,1] => [2,3,8,1,4,5,6,7] => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5
{{1,3,5,6,7,8},{2},{4}}
 => [3,2,5,4,6,7,8,1] => [2,4,8,1,3,5,6,7] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 6
{{1,3,4,5,6,7,8},{2}}
 => [3,2,4,5,6,7,8,1] => [2,8,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
{{1,4,5,6,7,8},{2,3}}
 => [4,3,2,5,6,7,8,1] => [3,2,8,1,4,5,6,7] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
{{1,2,4,5,6,7,8},{3}}
 => [2,4,3,5,6,7,8,1] => [3,8,1,2,4,5,6,7] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 3
{{1,2,5,6,7,8},{3,4}}
 => [2,5,4,3,6,7,8,1] => [4,3,8,1,2,5,6,7] => ?
 => ? = 4
{{1,2,3,5,6,7,8},{4}}
 => [2,3,5,4,6,7,8,1] => [4,8,1,2,3,5,6,7] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 4
{{1,2,3,6,7,8},{4,5}}
 => [2,3,6,5,4,7,8,1] => [5,4,8,1,2,3,6,7] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 5
{{1,2,3,4,6,7,8},{5}}
 => [2,3,4,6,5,7,8,1] => [5,8,1,2,3,4,6,7] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 5
{{1,2,3,4,5,6},{7,8}}
 => [2,3,4,5,6,1,8,7] => [6,1,2,3,4,5,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
{{1,2,3,4,7,8},{5,6}}
 => [2,3,4,7,6,5,8,1] => [6,5,8,1,2,3,4,7] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 6
{{1,2,3,4,5,7,8},{6}}
 => [2,3,4,5,7,6,8,1] => [6,8,1,2,3,4,5,7] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 6
{{1,2,3,4,5,6,7},{8}}
 => [2,3,4,5,6,7,1,8] => [7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
{{1,8},{2,3,4,5,6,7}}
 => [8,3,4,5,6,7,2,1] => [7,2,3,4,5,6,8,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 7
{{1,2,3,4,5,8},{6,7}}
 => [2,3,4,5,8,7,6,1] => [7,6,8,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 7
{{1,2,3,4,5,6,8},{7}}
 => [2,3,4,5,6,8,7,1] => [7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 7
{{1,2,3,4,5,6,7,8}}
 => [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
{{1,3,5,6,7,8},{2,4}}
 => [3,4,5,2,6,7,8,1] => [4,2,8,1,3,5,6,7] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 4
{{1,3,4,6,7,8},{2,5}}
 => [3,5,4,6,2,7,8,1] => [5,2,8,1,3,4,6,7] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 5
{{1,2,4,6,7,8},{3,5}}
 => [2,4,5,6,3,7,8,1] => [5,3,8,1,2,4,6,7] => ?
 => ? = 5
{{1,3,4,5,7,8},{2,6}}
 => [3,6,4,5,7,2,8,1] => [6,2,8,1,3,4,5,7] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 6
{{1,2,4,5,7,8},{3,6}}
 => [2,4,6,5,7,3,8,1] => [6,3,8,1,2,4,5,7] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 6
{{1,2,3,5,7,8},{4,6}}
 => [2,3,5,6,7,4,8,1] => [6,4,8,1,2,3,5,7] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 6
{{1,3,4,5,6,8},{2,7}}
 => [3,7,4,5,6,8,2,1] => [7,2,8,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 7
{{1,2,4,5,6,8},{3,7}}
 => [2,4,7,5,6,8,3,1] => [7,3,8,1,2,4,5,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 7
{{1,2,3,5,6,8},{4,7}}
 => [2,3,5,7,6,8,4,1] => [7,4,8,1,2,3,5,6] => ?
 => ? = 7
{{1,2,3,4,6,8},{5,7}}
 => [2,3,4,6,7,8,5,1] => [7,5,8,1,2,3,4,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 7
{{1,3,4,5,6,7},{2,8}}
 => [3,8,4,5,6,7,1,2] => [7,1,3,4,5,6,8,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 7
{{1,2,4,5,6,7},{3,8}}
 => [2,4,8,5,6,7,1,3] => [7,1,2,4,5,6,8,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 7
{{1,2,3,5,6,7},{4,8}}
 => [2,3,5,8,6,7,1,4] => [7,1,2,3,5,6,8,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 7
{{1,2,3,4,6,7},{5,8}}
 => [2,3,4,6,8,7,1,5] => [7,1,2,3,4,6,8,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 7
{{1,2,3,4,5,7},{6,8}}
 => [2,3,4,5,7,8,1,6] => [7,1,2,3,4,5,8,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 7
{{1,3},{2,4,5,6,7,8}}
 => [3,4,1,5,6,7,8,2] => [3,1,8,2,4,5,6,7] => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
{{1,4},{2,3,5,6,7,8}}
 => [4,3,5,1,6,7,8,2] => [4,1,8,2,3,5,6,7] => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 4
{{1,5},{2,3,4,6,7,8}}
 => [5,3,4,6,1,7,8,2] => [5,1,8,2,3,4,6,7] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 5
{{1,6},{2,3,4,5,7,8}}
 => [6,3,4,5,7,1,8,2] => [6,1,8,2,3,4,5,7] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 6
{{1,7},{2,3,4,5,6,8}}
 => [7,3,4,5,6,8,1,2] => [7,1,8,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 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 $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.