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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St001696
St001696: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0
[[1],[2]]
=> 0
[[1,2,3]]
=> 0
[[1,3],[2]]
=> 2
[[1,2],[3]]
=> 0
[[1],[2],[3]]
=> 0
[[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> 2
[[1,2,4],[3]]
=> 3
[[1,2,3],[4]]
=> 0
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 0
[[1,4],[2],[3]]
=> 3
[[1,3],[2],[4]]
=> 2
[[1,2],[3],[4]]
=> 0
[[1],[2],[3],[4]]
=> 0
[[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> 2
[[1,2,4,5],[3]]
=> 3
[[1,2,3,5],[4]]
=> 4
[[1,2,3,4],[5]]
=> 0
[[1,3,5],[2,4]]
=> 6
[[1,2,5],[3,4]]
=> 4
[[1,3,4],[2,5]]
=> 2
[[1,2,4],[3,5]]
=> 3
[[1,2,3],[4,5]]
=> 0
[[1,4,5],[2],[3]]
=> 3
[[1,3,5],[2],[4]]
=> 6
[[1,2,5],[3],[4]]
=> 4
[[1,3,4],[2],[5]]
=> 2
[[1,2,4],[3],[5]]
=> 3
[[1,2,3],[4],[5]]
=> 0
[[1,4],[2,5],[3]]
=> 3
[[1,3],[2,5],[4]]
=> 6
[[1,2],[3,5],[4]]
=> 4
[[1,3],[2,4],[5]]
=> 2
[[1,2],[3,4],[5]]
=> 0
[[1,5],[2],[3],[4]]
=> 4
[[1,4],[2],[3],[5]]
=> 3
[[1,3],[2],[4],[5]]
=> 2
[[1,2],[3],[4],[5]]
=> 0
[[1],[2],[3],[4],[5]]
=> 0
[[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> 2
[[1,2,4,5,6],[3]]
=> 3
[[1,2,3,5,6],[4]]
=> 4
[[1,2,3,4,6],[5]]
=> 5
[[1,2,3,4,5],[6]]
=> 0
[[1,3,5,6],[2,4]]
=> 6
[[1,2,5,6],[3,4]]
=> 4
Description
The natural major index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.
Matching statistic: St000305
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(load all 4 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => [1,3,2] => 2
[[1,2],[3]]
=> [3,1,2] => [1,2,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,2,4,3] => 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3,4] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,2,4] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,3,4] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,2,4,5,3] => 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,3,5,4] => 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,5,2,4] => 6
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,2,5] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,3,5] => 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => 6
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,2,5,3,4] => 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,3,4,2,5] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,2,4,3,5] => 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,2,5,3] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => 6
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,3,5,4] => 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,2,4,5] => 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,3,5] => 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,3,2,4,5] => 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,2,4,5,6,3] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,3,5,6,4] => 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,4,6,5] => 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5,6] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => 6
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => 4
Description
The inverse major index of a permutation.
This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000462
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000462: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000462: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> {{1,2}}
=> [2,1] => 0
[[1],[2]]
=> {{1},{2}}
=> [1,2] => 0
[[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 0
[[1,3],[2]]
=> {{1,3},{2}}
=> [3,2,1] => 2
[[1,2],[3]]
=> {{1,2},{3}}
=> [2,1,3] => 0
[[1],[2],[3]]
=> {{1},{2},{3}}
=> [1,2,3] => 0
[[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> [3,2,4,1] => 2
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> [2,4,3,1] => 3
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> [2,3,1,4] => 0
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> [3,4,1,2] => 0
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 3
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 2
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 0
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 2
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => 3
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => 4
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => 4
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 6
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => 2
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 3
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 3
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 6
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 4
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 2
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 3
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => 3
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 4
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 6
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 0
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 4
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 3
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 2
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 0
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 2
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 3
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 4
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 5
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => 0
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => 4
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 6
Description
The major index minus the number of excedences of a permutation.
This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000004
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2
[[1,2],[3]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 6
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 6
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,2,5,3] => [1,3,5,2,4] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => 6
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => [1,5,2,6,3,4] => 6
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => 4
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: St000304
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [2,1] => 0
[[1],[2]]
=> [2,1] => [1,2] => [2,1] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
[[1,3],[2]]
=> [2,1,3] => [1,3,2] => [3,1,2] => 2
[[1,2],[3]]
=> [3,1,2] => [1,2,3] => [3,2,1] => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [3,2,1] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => [5,3,2,1,4] => 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,3,5,4] => [5,4,3,1,2] => 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,5,2,4] => [5,3,1,4,2] => 6
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => [5,4,1,3,2] => 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,2,5] => [5,3,2,4,1] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => [5,2,1,4,3] => 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [5,3,1,4,2] => 6
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,2,5,3,4] => [5,4,1,3,2] => 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [5,3,2,4,1] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,2,5,3] => [5,2,4,1,3] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => [5,3,4,1,2] => 6
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [5,4,3,1,2] => 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [5,2,4,3,1] => 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => [6,4,3,2,1,5] => 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,2,4,5,6,3] => [6,5,3,2,1,4] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,4,6,5] => [6,5,4,3,1,2] => 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => [6,4,2,1,5,3] => 6
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => [6,5,2,1,4,3] => 4
Description
The load of a permutation.
The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.
Matching statistic: St000330
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[[1],[2]]
=> [2,1] => [1,2] => [[1,2]]
=> 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[[1,3],[2]]
=> [2,1,3] => [1,3,2] => [[1,2],[3]]
=> 2
[[1,2],[3]]
=> [3,1,2] => [1,2,3] => [[1,2,3]]
=> 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [[1,2,3]]
=> 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => [[1,2,4],[3]]
=> 2
[[1,2,4],[3]]
=> [3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => [[1,2,3],[4]]
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => [[1,2,4,5],[3]]
=> 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 6
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 6
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
=> 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,2,4,5,6,3] => [[1,2,3,5,6],[4]]
=> 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,3,5,6,4] => [[1,2,3,4,6],[5]]
=> 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,5,6,2,4] => [[1,2,4,6],[3,5]]
=> 6
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => [[1,2,3,4],[5,6]]
=> 4
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: St000961
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000961: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000961: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => [2,1] => 0
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [2,3,1] => 2
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2
[[1,2,4],[3]]
=> [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 3
[[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => 2
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => 3
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5,3,2,1,4] => [4,3,2,5,1] => 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,1,4,2] => [3,5,2,4,1] => 6
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [3,2,5,4,1] => 4
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [3,5,2,1,4] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [3,2,5,1,4] => 3
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,3,1,2,4] => [3,4,2,5,1] => 6
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,2,1,3,4] => [3,2,4,5,1] => 4
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [3,4,2,1,5] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [3,2,4,1,5] => 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1,5,2,3] => [2,4,5,1,3] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,1,5,2,4] => [2,4,1,5,3] => 6
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,4,5,3] => 4
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,1,4,2,5] => [2,4,1,3,5] => 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [2,3,4,1,5] => 3
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [2,3,1,4,5] => 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,5,4,3,1,2] => [5,6,4,3,2,1] => 2
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [6,5,4,2,1,3] => [5,4,6,3,2,1] => 3
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [6,5,3,2,1,4] => [5,4,3,6,2,1] => 4
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [6,4,3,2,1,5] => [5,4,3,2,6,1] => 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,5,3,1,4,2] => [4,6,3,5,2,1] => 6
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [6,5,2,1,4,3] => [4,3,6,5,2,1] => 4
Description
The shifted major index of a permutation.
This is given by the sum of all indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
Summing with [[St000354]] yields Rawlings' Mahonian statistic, see [1, p. 50].
Matching statistic: St000422
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 43%●distinct values known / distinct values provided: 40%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 43%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2] => [2] => ([],2)
=> 0
[[1],[2]]
=> [2] => [2] => ([],2)
=> 0
[[1,2,3]]
=> [3] => [3] => ([],3)
=> 0
[[1,3],[2]]
=> [2,1] => [1,2] => ([(1,2)],3)
=> 2
[[1,2],[3]]
=> [3] => [3] => ([],3)
=> 0
[[1],[2],[3]]
=> [3] => [3] => ([],3)
=> 0
[[1,2,3,4]]
=> [4] => [4] => ([],4)
=> 0
[[1,3,4],[2]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,3,3}
[[1,2,4],[3]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2
[[1,2,3],[4]]
=> [4] => [4] => ([],4)
=> 0
[[1,3],[2,4]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,3,3}
[[1,2],[3,4]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [3,1] => [1,3] => ([(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,3,3}
[[1,2],[3],[4]]
=> [4] => [4] => ([],4)
=> 0
[[1],[2],[3],[4]]
=> [4] => [4] => ([],4)
=> 0
[[1,2,3,4,5]]
=> [5] => [5] => ([],5)
=> 0
[[1,3,4,5],[2]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4,5],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3,5],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2
[[1,2,3,4],[5]]
=> [5] => [5] => ([],5)
=> 0
[[1,3,5],[2,4]]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,5],[3,4]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3,4],[2,5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4],[3,5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3],[4,5]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2
[[1,4,5],[2],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3,5],[2],[4]]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,5],[3],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2
[[1,3,4],[2],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4],[3],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3],[4],[5]]
=> [5] => [5] => ([],5)
=> 0
[[1,4],[2,5],[3]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3],[2,5],[4]]
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2],[3,5],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2
[[1,3],[2,4],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2],[3,4],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,5],[2],[3],[4]]
=> [4,1] => [1,4] => ([(3,4)],5)
=> 2
[[1,4],[2],[3],[5]]
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3],[2],[4],[5]]
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2],[3],[4],[5]]
=> [5] => [5] => ([],5)
=> 0
[[1],[2],[3],[4],[5]]
=> [5] => [5] => ([],5)
=> 0
[[1,2,3,4,5,6]]
=> [6] => [6] => ([],6)
=> 0
[[1,3,4,5,6],[2]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5,6],[3]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,5,6],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,4,6],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2
[[1,2,3,4,5],[6]]
=> [6] => [6] => ([],6)
=> 0
[[1,3,5,6],[2,4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5,6],[3,4]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4,6],[2,5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,4,6],[3,5]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,6],[4,5]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4,5],[2,6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5],[3,6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,5],[4,6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,4],[5,6]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2
[[1,4,5,6],[2],[3]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5,6],[3],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,6],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2
[[1,3,4,5],[2],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4,5],[3],[6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,5],[4],[6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,4],[5],[6]]
=> [6] => [6] => ([],6)
=> 0
[[1,3,5],[2,4,6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5],[3,4,6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4],[2,5,6]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,4],[3,5,6]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3],[4,5,6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,6],[2,5],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,6],[3,5],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,4,5],[2,6],[3]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,5],[2,6],[4]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5],[3,6],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3],[4,6],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2
[[1,3,5],[2,4],[6]]
=> [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5],[3,4],[6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4],[2,5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,4],[3,5],[6]]
=> [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3],[4,5],[6]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,5,6],[2],[3],[4]]
=> [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2,6],[3],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2
[[1,3,4],[2],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,3],[4],[5],[6]]
=> [6] => [6] => ([],6)
=> 0
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[[1,2],[3,6],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2
[[1,3],[2,4],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,6],[2],[3],[4],[5]]
=> [5,1] => [1,5] => ([(4,5)],6)
=> 2
[[1,3],[2],[4],[5],[6]]
=> [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2],[3],[4],[5],[6]]
=> [6] => [6] => ([],6)
=> 0
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: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 30%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 30%
Values
[[1,2]]
=> [[1,2]]
=> [2] => ([],2)
=> 0
[[1],[2]]
=> [[1,2]]
=> [2] => ([],2)
=> 0
[[1,2,3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> 0
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {0,2}
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> 0
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {0,2}
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 0
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,3,3}
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,3,3}
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 0
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,3,3}
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 0
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,3,3}
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,3,3}
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,3,3}
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,3,3}
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 0
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 0
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 0
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 0
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 0
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 0
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4,5],[2],[6]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4,5],[3],[6]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 0
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[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)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,6],[2,5],[4]]
=> [[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,6],[3,5],[4]]
=> [[1,2,3,5],[4],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [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)
=> ? ∊ {0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001060
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 20%
Values
[[1,2]]
=> [1,2] => [2] => ([],2)
=> ? ∊ {0,0}
[[1],[2]]
=> [2,1] => [2] => ([],2)
=> ? ∊ {0,0}
[[1,2,3]]
=> [1,2,3] => [3] => ([],3)
=> ? ∊ {0,0,0}
[[1,3],[2]]
=> [2,1,3] => [3] => ([],3)
=> ? ∊ {0,0,0}
[[1,2],[3]]
=> [3,1,2] => [3] => ([],3)
=> ? ∊ {0,0,0}
[[1],[2],[3]]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[[1,2,3,4]]
=> [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,3,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,3,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,3,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,3,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,3,3}
[[1,2],[3,4]]
=> [3,4,1,2] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,3,3}
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,3,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,3,3,3,3,3,3,4,4,4,4,4,6,6,6}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1],[2],[3],[4],[5]]
=> [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)
=> 2
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,11}
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000264The girth of a graph, which is not a tree. St001902The number of potential covers of a poset. St001472The permanent of the Coxeter matrix of the poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001545The second Elser number of a connected graph. St001095The number of non-isomorphic posets with precisely one further covering relation.
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