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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000004
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 4
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 4
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 6
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 2
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000008
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [2,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [3,1] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [3,1] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [2,2] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [2,2] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [2,2] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [2,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [3,1] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [4,1] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [4,1] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [3,2] => 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [3,2] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [3,2] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [3,2] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [4,1] => 4
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [4,1] => 4
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [3,2] => 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [3,2] => 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [3,2] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [2,3] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [2,3] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [3,2] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [2,2,1] => 6
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [2,3] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [2,3] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [3,2] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [2,3] => 2
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]].
Matching statistic: St000018
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [2,3,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [2,4,1,3] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [2,3,1,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [2,3,1,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [2,1,4,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [2,1,4,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [2,3,4,1] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,5,1] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [2,3,4,1,5] => 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [2,3,4,1,5] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => 4
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [2,5,1,3,4] => 4
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [2,4,1,3,5] => 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [2,4,1,3,5] => 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [2,1,4,5,3] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [2,3,1,4,5] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [2,3,1,4,5] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [2,1,4,5,3] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [3,4,2,5,1] => 6
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [2,3,1,4,5] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [2,1,5,3,4] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [2,1,4,3,5] => 2
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000305
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,4,5,3] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 4
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => 4
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,4,2,3,5] => 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 6
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 2
Description
The inverse major index of a permutation.
This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000330
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [[1]]
=> 0
{{1,2}}
=> [2,1] => [1,2] => [[1,2]]
=> 0
{{1},{2}}
=> [1,2] => [1,2] => [[1,2]]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [[1,2,3]]
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [[1,2],[3]]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [[1,2,3]]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 2
Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000748
St000748: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 3
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 3
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 2
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> 3
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 4
{{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 4
{{1,2,4},{3,5}}
=> 3
{{1,2,4},{3},{5}}
=> 3
{{1,2,5},{3,4}}
=> 3
{{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> 3
{{1,2},{3,5},{4}}
=> 4
{{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 4
{{1,3,4},{2,5}}
=> 3
{{1,3,4},{2},{5}}
=> 3
{{1,3,5},{2,4}}
=> 3
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 3
{{1,3},{2,5},{4}}
=> 6
{{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> 2
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 2
Description
The major index of the permutation obtained by flattening the set partition.
A set partition can be represented by a sequence of blocks where the first entries of the blocks and the blocks themselves are increasing. This statistic is then the major index of the permutation obtained by flattening the set partition in this canonical form.
Matching statistic: St000391
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => => ? = 0
{{1,2}}
=> [2,1] => [1,2] => 0 => 0
{{1},{2}}
=> [1,2] => [1,2] => 0 => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 00 => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 00 => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 01 => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 00 => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 00 => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 000 => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 001 => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 000 => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 000 => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 001 => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 010 => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 010 => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 010 => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 000 => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 000 => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 010 => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 001 => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 000 => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 0001 => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0000 => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0000 => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 0001 => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 0010 => 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 0010 => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 0010 => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0000 => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0000 => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 0010 => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 0001 => 4
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0000 => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0000 => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 0001 => 4
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 0010 => 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 0010 => 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 0010 => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 0100 => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 0100 => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 0010 => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 0101 => 6
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 0100 => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 0100 => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 0010 => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 0100 => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => 0100 => 2
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000833
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => ? = 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [3,1,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => [3,1,2,4] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => [4,1,2,3,5] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => [1,4,2,3,5] => 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,4,2,3,5] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => 4
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => [3,1,2,4,5] => 4
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => [1,3,2,4,5] => 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,3,2,4,5] => 3
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => [3,5,1,2,4] => 3
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => [1,3,4,2,5] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => [1,3,4,2,5] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => [3,5,1,2,4] => 3
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => [3,1,5,2,4] => 6
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => [1,3,4,2,5] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => [4,5,1,2,3] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,2,4,3,5] => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,2,4,3,5] => 2
Description
The comajor index of a permutation.
This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 50%●distinct values known / distinct values provided: 38%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 50%●distinct values known / distinct values provided: 38%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> 0
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([],3)
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([],3)
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3}
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3}
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([],4)
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {3,3,3}
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,6,6,6}
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8}
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000777
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 62%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 62%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [2] => ([],2)
=> ? ∊ {0,0} + 1
{{1},{2}}
=> [1,2] => [2] => ([],2)
=> ? ∊ {0,0} + 1
{{1,2,3}}
=> [2,3,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
{{1,2},{3}}
=> [2,1,3] => [3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
{{1,3},{2}}
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
{{1},{2,3}}
=> [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0} + 1
{{1},{2},{3}}
=> [1,2,3] => [3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,3} + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,6,6,6} + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3,6},{2},{4,5}}
=> [3,2,6,5,4,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3,6},{2},{4},{5}}
=> [3,2,6,4,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,3},{2},{4,6},{5}}
=> [3,2,1,6,5,4] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000264The girth of a graph, which is not a tree. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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