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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000229
(load all 73 compositions to match this statistic)
(load all 73 compositions to match this statistic)
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 1
{{1,2}}
=> 2
{{1},{2}}
=> 2
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 3
{{1,3},{2}}
=> 4
{{1},{2,3}}
=> 3
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 4
{{1,2,4},{3}}
=> 5
{{1,2},{3,4}}
=> 4
{{1,2},{3},{4}}
=> 4
{{1,3,4},{2}}
=> 5
{{1,3},{2,4}}
=> 6
{{1,3},{2},{4}}
=> 5
{{1,4},{2,3}}
=> 6
{{1},{2,3,4}}
=> 4
{{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> 6
{{1},{2,4},{3}}
=> 5
{{1},{2},{3,4}}
=> 4
{{1},{2},{3},{4}}
=> 4
{{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> 5
{{1,2,3,5},{4}}
=> 6
{{1,2,3},{4,5}}
=> 5
{{1,2,3},{4},{5}}
=> 5
{{1,2,4,5},{3}}
=> 6
{{1,2,4},{3,5}}
=> 7
{{1,2,4},{3},{5}}
=> 6
{{1,2,5},{3,4}}
=> 7
{{1,2},{3,4,5}}
=> 5
{{1,2},{3,4},{5}}
=> 5
{{1,2,5},{3},{4}}
=> 7
{{1,2},{3,5},{4}}
=> 6
{{1,2},{3},{4,5}}
=> 5
{{1,2},{3},{4},{5}}
=> 5
{{1,3,4,5},{2}}
=> 6
{{1,3,4},{2,5}}
=> 8
{{1,3,4},{2},{5}}
=> 6
{{1,3,5},{2,4}}
=> 8
{{1,3},{2,4,5}}
=> 7
{{1,3},{2,4},{5}}
=> 7
{{1,3,5},{2},{4}}
=> 7
{{1,3},{2,5},{4}}
=> 8
{{1,3},{2},{4,5}}
=> 6
{{1,3},{2},{4},{5}}
=> 6
{{1,4,5},{2,3}}
=> 7
{{1,4},{2,3,5}}
=> 8
Description
Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition.
This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is
$$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$
This statistic is called ''dimension index'' in [2]
Matching statistic: St001894
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 58%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 58%
Values
{{1}}
=> [1] => [1] => [-1] => 1
{{1,2}}
=> [2,1] => [2,1] => [1,-2] => 2
{{1},{2}}
=> [1,2] => [1,2] => [2,-1] => 2
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [1,2,-3] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,3,-2] => 3
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,1,-3] => 4
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [3,2,-1] => 3
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [2,3,-1] => 3
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 4
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 4
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => 5
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 6
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 5
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 6
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 6
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => 5
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => 6
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => 5
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => 6
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => 7
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => 7
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => 5
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => 5
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => 7
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => 6
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => 5
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => 5
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => ? = 6
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => ? = 8
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => ? = 6
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,-5] => ? = 8
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => ? = 7
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 7
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,-5] => ? = 7
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => ? = 8
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => ? = 6
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => ? = 6
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [3,2,1,4,-5] => ? = 7
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => [5,2,1,3,-4] => ? = 8
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,-4] => ? = 7
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,3,4,2,1] => [4,2,3,1,-5] => ? = 8
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => ? = 5
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => ? = 5
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,1,-5] => ? = 8
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => ? = 6
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => ? = 5
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => ? = 5
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,5,1] => [2,3,1,4,-5] => ? = 7
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => [5,3,1,2,-4] => ? = 9
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => ? = 8
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => [2,3,1,5,-4] => ? = 7
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,-5] => ? = 9
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => ? = 6
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => ? = 7
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => ? = 6
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => [2,4,3,1,-5] => ? = 8
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => ? = 7
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => ? = 5
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => ? = 5
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,1,-5] => ? = 8
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => ? = 7
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => ? = 6
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => ? = 5
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 5
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,-6] => ? = 6
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,6,-5] => ? = 6
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,6,5,1] => [1,2,3,5,4,-6] => ? = 7
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,3,4,1,6,5] => [1,2,3,6,5,-4] => ? = 6
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,5,6,-4] => ? = 6
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,6,1] => [1,2,4,3,5,-6] => ? = 7
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,6,1,4] => [1,2,6,3,4,-5] => ? = 8
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1,6] => [1,2,4,3,6,-5] => ? = 7
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,6,5,4,1] => [1,2,5,4,3,-6] => ? = 8
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [2,3,1,5,6,4] => [1,2,6,4,5,-3] => ? = 6
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [2,3,1,5,4,6] => [1,2,5,4,6,-3] => ? = 6
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,6,4,5,1] => [1,2,4,5,3,-6] => ? = 8
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [2,3,1,6,5,4] => [1,2,6,5,4,-3] => ? = 7
Description
The depth of a signed permutation.
The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.
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