Your data matches 36 different statistics following compositions of up to 3 maps.
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Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00086: Permutations first fundamental transformationPermutations
St000710: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => 1
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => 0
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 0
Description
The number of big deficiencies of a permutation. A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$. This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00069: Permutations complementPermutations
Mp00252: Permutations restrictionPermutations
St000646: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1] => ? = 0
[2,1] => [1,2] => [2,1] => [1] => ? = 0
[1,2,3] => [1,2,3] => [3,2,1] => [2,1] => 0
[1,3,2] => [1,2,3] => [3,2,1] => [2,1] => 0
[2,1,3] => [1,2,3] => [3,2,1] => [2,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => [2,1] => 0
[3,1,2] => [1,3,2] => [3,1,2] => [1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => [1,2] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 0
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 0
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 0
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 0
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [3,1,2] => 0
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 0
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [2,1,3] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 0
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [2,1,3] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 0
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 0
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [1,2,3] => 0
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [4,2,1,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [4,2,1,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 0
[1,4,5,3,2] => [1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => 0
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [4,1,2,3] => 0
[1,5,2,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [4,1,3,2] => 1
Description
The number of big ascents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$. For the number of small ascents, see [[St000441]].
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00241: Permutations invert Laguerre heapPermutations
St000647: Permutations ⟶ ℤResult quality: 94% values known / values provided: 94%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 0
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[3,1,5,4,7,2,6] => [1,3,5,7,6,2,4] => [1,4,7,6,2,3,5] => ? = 1
[3,1,6,4,2,7,5] => [1,3,6,7,5,2,4] => [1,4,7,5,2,3,6] => ? = 2
[3,1,6,4,7,5,2] => [1,3,6,5,7,2,4] => [1,4,7,2,3,6,5] => ? = 1
[3,1,6,7,4,2,5] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[3,1,6,7,5,2,4] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[3,1,7,4,2,5,6] => [1,3,7,6,5,2,4] => [1,4,7,6,5,2,3] => ? = 1
[3,1,7,4,2,6,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[3,1,7,6,4,5,2] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[3,1,7,6,5,4,2] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[3,2,5,4,7,1,6] => [1,3,5,7,6,2,4] => [1,4,7,6,2,3,5] => ? = 1
[3,2,6,4,1,7,5] => [1,3,6,7,5,2,4] => [1,4,7,5,2,3,6] => ? = 2
[3,2,6,4,7,5,1] => [1,3,6,5,7,2,4] => [1,4,7,2,3,6,5] => ? = 1
[3,2,6,7,4,1,5] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[3,2,6,7,5,1,4] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[3,2,7,4,1,5,6] => [1,3,7,6,5,2,4] => [1,4,7,6,5,2,3] => ? = 1
[3,2,7,4,1,6,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[3,2,7,6,1,4,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[3,2,7,6,4,5,1] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[3,2,7,6,5,4,1] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[3,4,5,1,7,2,6] => [1,3,5,7,6,2,4] => [1,4,7,6,2,3,5] => ? = 1
[3,4,5,2,7,1,6] => [1,3,5,7,6,2,4] => [1,4,7,6,2,3,5] => ? = 1
[3,4,6,1,2,7,5] => [1,3,6,7,5,2,4] => [1,4,7,5,2,3,6] => ? = 2
[3,4,6,1,7,5,2] => [1,3,6,5,7,2,4] => [1,4,7,2,3,6,5] => ? = 1
[3,4,6,2,1,7,5] => [1,3,6,7,5,2,4] => [1,4,7,5,2,3,6] => ? = 2
[3,4,6,2,7,5,1] => [1,3,6,5,7,2,4] => [1,4,7,2,3,6,5] => ? = 1
[3,4,6,7,1,2,5] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[3,4,6,7,2,1,5] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[3,4,6,7,5,1,2] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[3,4,6,7,5,2,1] => [1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 2
[3,4,7,1,2,5,6] => [1,3,7,6,5,2,4] => [1,4,7,6,5,2,3] => ? = 1
[3,4,7,1,2,6,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[3,4,7,2,1,5,6] => [1,3,7,6,5,2,4] => [1,4,7,6,5,2,3] => ? = 1
[3,4,7,2,1,6,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[3,4,7,6,1,2,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[3,4,7,6,1,5,2] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[3,4,7,6,2,1,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2
[3,4,7,6,2,5,1] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[3,4,7,6,5,1,2] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[3,4,7,6,5,2,1] => [1,3,7,2,4,6,5] => [1,4,6,5,7,2,3] => ? = 1
[4,5,1,2,3,6,7] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 2
[4,5,1,2,3,7,6] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 2
[4,5,1,2,6,3,7] => [1,4,2,5,6,3,7] => [1,6,3,4,2,5,7] => ? = 2
[4,5,1,2,7,6,3] => [1,4,2,5,7,3,6] => [1,6,7,3,4,2,5] => ? = 2
[4,5,2,1,3,6,7] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 2
[4,5,2,1,3,7,6] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 2
[4,5,2,1,6,3,7] => [1,4,2,5,6,3,7] => [1,6,3,4,2,5,7] => ? = 2
[4,5,2,1,7,6,3] => [1,4,2,5,7,3,6] => [1,6,7,3,4,2,5] => ? = 2
[4,5,3,1,2,6,7] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 2
[4,5,3,1,2,7,6] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => ? = 2
Description
The number of big descents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$. The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].
Matching statistic: St000711
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00086: Permutations first fundamental transformationPermutations
Mp00066: Permutations inversePermutations
St000711: Permutations ⟶ ℤResult quality: 73% values known / values provided: 73%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 0
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[3,1,5,4,2,6,7] => [1,3,5,2,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[3,1,5,4,2,7,6] => [1,3,5,2,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[3,1,5,4,6,2,7] => [1,3,5,6,2,4,7] => [1,4,3,6,5,2,7] => [1,6,3,2,5,4,7] => ? = 1
[3,1,5,4,6,7,2] => [1,3,5,6,7,2,4] => [1,4,3,7,5,6,2] => [1,7,3,2,5,6,4] => ? = 1
[3,1,5,4,7,2,6] => [1,3,5,7,6,2,4] => [1,4,3,7,5,2,6] => [1,6,3,2,5,7,4] => ? = 1
[3,1,5,4,7,6,2] => [1,3,5,7,2,4,6] => [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => ? = 1
[3,1,5,6,2,4,7] => [1,3,5,2,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[3,1,5,6,2,7,4] => [1,3,5,2,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[3,1,5,6,7,4,2] => [1,3,5,7,2,4,6] => [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => ? = 1
[3,1,5,7,2,4,6] => [1,3,5,2,4,7,6] => [1,4,3,5,2,7,6] => [1,5,3,2,4,7,6] => ? = 1
[3,1,5,7,2,6,4] => [1,3,5,2,4,7,6] => [1,4,3,5,2,7,6] => [1,5,3,2,4,7,6] => ? = 1
[3,1,5,7,6,2,4] => [1,3,5,6,2,4,7] => [1,4,3,6,5,2,7] => [1,6,3,2,5,4,7] => ? = 1
[3,1,6,4,2,5,7] => [1,3,6,5,2,4,7] => [1,4,3,6,2,5,7] => [1,5,3,2,6,4,7] => ? = 1
[3,1,6,4,2,7,5] => [1,3,6,7,5,2,4] => [1,4,3,7,2,6,5] => [1,5,3,2,7,6,4] => ? = 2
[3,1,6,4,5,2,7] => [1,3,6,2,4,5,7] => [1,4,3,5,6,2,7] => [1,6,3,2,4,5,7] => ? = 1
[3,1,6,4,5,7,2] => [1,3,6,7,2,4,5] => [1,4,3,5,7,6,2] => [1,7,3,2,4,6,5] => ? = 1
[3,1,6,4,7,2,5] => [1,3,6,2,4,5,7] => [1,4,3,5,6,2,7] => [1,6,3,2,4,5,7] => ? = 1
[3,1,6,4,7,5,2] => [1,3,6,5,7,2,4] => [1,4,3,7,6,5,2] => [1,7,3,2,6,5,4] => ? = 1
[3,1,6,5,4,2,7] => [1,3,6,2,4,5,7] => [1,4,3,5,6,2,7] => [1,6,3,2,4,5,7] => ? = 1
[3,1,6,5,4,7,2] => [1,3,6,7,2,4,5] => [1,4,3,5,7,6,2] => [1,7,3,2,4,6,5] => ? = 1
[3,1,6,5,7,2,4] => [1,3,6,2,4,5,7] => [1,4,3,5,6,2,7] => [1,6,3,2,4,5,7] => ? = 1
[3,1,6,7,2,5,4] => [1,3,6,5,2,4,7] => [1,4,3,6,2,5,7] => [1,5,3,2,6,4,7] => ? = 1
[3,1,6,7,4,2,5] => [1,3,6,2,4,7,5] => [1,4,3,6,7,2,5] => [1,6,3,2,7,4,5] => ? = 2
[3,1,6,7,5,2,4] => [1,3,6,2,4,7,5] => [1,4,3,6,7,2,5] => [1,6,3,2,7,4,5] => ? = 2
[3,1,7,4,2,5,6] => [1,3,7,6,5,2,4] => [1,4,3,7,2,5,6] => [1,5,3,2,6,7,4] => ? = 1
[3,1,7,4,2,6,5] => [1,3,7,5,2,4,6] => [1,4,3,6,2,7,5] => [1,5,3,2,7,4,6] => ? = 2
[3,1,7,4,5,2,6] => [1,3,7,6,2,4,5] => [1,4,3,5,7,2,6] => [1,6,3,2,4,7,5] => ? = 1
[3,1,7,4,5,6,2] => [1,3,7,2,4,5,6] => [1,4,3,5,6,7,2] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,4,6,2,5] => [1,3,7,5,6,2,4] => [1,4,3,7,6,2,5] => [1,6,3,2,7,5,4] => ? = 2
[3,1,7,4,6,5,2] => [1,3,7,2,4,5,6] => [1,4,3,5,6,7,2] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,5,4,2,6] => [1,3,7,6,2,4,5] => [1,4,3,5,7,2,6] => [1,6,3,2,4,7,5] => ? = 1
[3,1,7,5,4,6,2] => [1,3,7,2,4,5,6] => [1,4,3,5,6,7,2] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,5,6,4,2] => [1,3,7,2,4,5,6] => [1,4,3,5,6,7,2] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,4,3,6,2,7,5] => [1,5,3,2,7,4,6] => ? = 2
[3,1,7,6,4,5,2] => [1,3,7,2,4,6,5] => [1,4,3,6,7,5,2] => [1,7,3,2,6,4,5] => ? = 1
[3,1,7,6,5,4,2] => [1,3,7,2,4,6,5] => [1,4,3,6,7,5,2] => [1,7,3,2,6,4,5] => ? = 1
[3,2,5,4,1,6,7] => [1,3,5,2,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[3,2,5,4,1,7,6] => [1,3,5,2,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[3,2,5,4,6,1,7] => [1,3,5,6,2,4,7] => [1,4,3,6,5,2,7] => [1,6,3,2,5,4,7] => ? = 1
[3,2,5,4,6,7,1] => [1,3,5,6,7,2,4] => [1,4,3,7,5,6,2] => [1,7,3,2,5,6,4] => ? = 1
[3,2,5,4,7,1,6] => [1,3,5,7,6,2,4] => [1,4,3,7,5,2,6] => [1,6,3,2,5,7,4] => ? = 1
[3,2,5,4,7,6,1] => [1,3,5,7,2,4,6] => [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => ? = 1
[3,2,5,6,1,4,7] => [1,3,5,2,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[3,2,5,6,1,7,4] => [1,3,5,2,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[3,2,5,6,7,4,1] => [1,3,5,7,2,4,6] => [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => ? = 1
[3,2,5,7,1,4,6] => [1,3,5,2,4,7,6] => [1,4,3,5,2,7,6] => [1,5,3,2,4,7,6] => ? = 1
[3,2,5,7,1,6,4] => [1,3,5,2,4,7,6] => [1,4,3,5,2,7,6] => [1,5,3,2,4,7,6] => ? = 1
[3,2,5,7,6,1,4] => [1,3,5,6,2,4,7] => [1,4,3,6,5,2,7] => [1,6,3,2,5,4,7] => ? = 1
[3,2,6,4,1,5,7] => [1,3,6,5,2,4,7] => [1,4,3,6,2,5,7] => [1,5,3,2,6,4,7] => ? = 1
[3,2,6,4,1,7,5] => [1,3,6,7,5,2,4] => [1,4,3,7,2,6,5] => [1,5,3,2,7,6,4] => ? = 2
Description
The number of big exceedences of a permutation. A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$. This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00241: Permutations invert Laguerre heapPermutations
Mp00069: Permutations complementPermutations
St000742: Permutations ⟶ ℤResult quality: 25% values known / values provided: 25%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[1,3,2] => [1,2,3] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[2,1,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [3,2,1] => 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[3,2,1] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 0 + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 0 + 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 0 + 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 0 + 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1 = 0 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [4,2,1,3] => 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [4,2,1,3] => 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [4,2,1,3] => 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [4,2,1,3] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1 = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1 = 0 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1 = 0 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1 = 0 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1 = 0 + 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,2,3,6,4,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1 + 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1 + 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,2,3,7,4,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1 + 1
[1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1 + 1
[1,2,3,7,6,4,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1 + 1
[1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1 + 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,2,4,6,3,7,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1 + 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,2,4,6,5,7,3] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 1 + 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0 + 1
[1,2,4,7,3,6,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1 + 1
[1,2,4,7,5,6,3] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1 + 1
[1,2,4,7,6,3,5] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1 + 1
[1,2,4,7,6,5,3] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 1 + 1
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,2,5,3,4,7,6] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,2,5,3,6,4,7] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1 + 1
[1,2,5,3,6,7,4] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1 + 1
[1,2,5,3,7,4,6] => [1,2,3,5,7,6,4] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1 + 1
[1,2,5,3,7,6,4] => [1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1 + 1
[1,2,5,4,3,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,2,5,4,3,7,6] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,2,5,4,6,3,7] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1 + 1
[1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 1 + 1
[1,2,5,4,7,3,6] => [1,2,3,5,7,6,4] => [1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 1 + 1
[1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1 + 1
[1,2,5,6,3,4,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,2,5,6,3,7,4] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,2,5,6,4,3,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,2,5,6,4,7,3] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0 + 1
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1 + 1
[1,2,5,6,7,4,3] => [1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 1 + 1
[1,2,5,7,6,3,4] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1 + 1
[1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 1 + 1
[1,2,6,3,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 0 + 1
[1,2,6,3,4,7,5] => [1,2,3,6,7,5,4] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 1 + 1
[1,2,6,3,5,4,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1 + 1
[1,2,6,3,5,7,4] => [1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 1 + 1
[1,2,6,3,7,4,5] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1 + 1
[1,2,6,3,7,5,4] => [1,2,3,6,5,7,4] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 1 + 1
[1,2,6,4,3,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 0 + 1
[1,2,6,4,3,7,5] => [1,2,3,6,7,5,4] => [1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 1 + 1
[1,2,6,4,5,3,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 1 + 1
[1,2,6,4,5,7,3] => [1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 1 + 1
Description
The number of big ascents of a permutation after prepending zero. Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$. It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]). G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]]. [[St000646]] is the statistic without the border condition $\pi(0) = 0$.
Matching statistic: St000259
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St000259: Graphs ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St000260: Graphs ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000302
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St000302: Graphs ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St000466: Graphs ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The Gutman (or modified Schultz) index of a connected graph. This is $$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$$ where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$. For trees on $n$ vertices, the modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].
Matching statistic: St000467
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00247: Graphs de-duplicateGraphs
St000467: Graphs ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 25%
Values
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The hyper-Wiener index of a connected graph. This is $$ \sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2. $$
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. St001964The interval resolution global dimension of a poset. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001845The number of join irreducibles minus the rank of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St001890The maximum magnitude of the Möbius function of a poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001905The number of preferred parking spots in a parking function less than the index of the car. St001490The number of connected components of a skew partition.