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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000693
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(load all 3 compositions to match this statistic)
St000693: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0
[[1],[2]]
=> 1
[[1,2,3]]
=> 0
[[1,3],[2]]
=> 1
[[1,2],[3]]
=> 2
[[1],[2],[3]]
=> 0
[[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> 1
[[1,2,4],[3]]
=> 2
[[1,2,3],[4]]
=> 3
[[1,3],[2,4]]
=> 0
[[1,2],[3,4]]
=> 2
[[1,4],[2],[3]]
=> 3
[[1,3],[2],[4]]
=> 0
[[1,2],[3],[4]]
=> 1
[[1],[2],[3],[4]]
=> 2
[[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> 1
[[1,2,4,5],[3]]
=> 2
[[1,2,3,5],[4]]
=> 3
[[1,2,3,4],[5]]
=> 4
[[1,3,5],[2,4]]
=> 4
[[1,2,5],[3,4]]
=> 2
[[1,3,4],[2,5]]
=> 0
[[1,2,4],[3,5]]
=> 1
[[1,2,3],[4,5]]
=> 3
[[1,4,5],[2],[3]]
=> 3
[[1,3,5],[2],[4]]
=> 4
[[1,2,5],[3],[4]]
=> 0
[[1,3,4],[2],[5]]
=> 0
[[1,2,4],[3],[5]]
=> 1
[[1,2,3],[4],[5]]
=> 2
[[1,4],[2,5],[3]]
=> 2
[[1,3],[2,5],[4]]
=> 4
[[1,2],[3,5],[4]]
=> 0
[[1,3],[2,4],[5]]
=> 3
[[1,2],[3,4],[5]]
=> 1
[[1,5],[2],[3],[4]]
=> 1
[[1,4],[2],[3],[5]]
=> 2
[[1,3],[2],[4],[5]]
=> 3
[[1,2],[3],[4],[5]]
=> 4
[[1],[2],[3],[4],[5]]
=> 0
[[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> 1
[[1,2,4,5,6],[3]]
=> 2
[[1,2,3,5,6],[4]]
=> 3
[[1,2,3,4,6],[5]]
=> 4
[[1,2,3,4,5],[6]]
=> 5
[[1,3,5,6],[2,4]]
=> 4
[[1,2,5,6],[3,4]]
=> 2
Description
The modular (standard) major index of a standard tableau.
The modular major index is the usual major index [[St000330]] modulo $n$, where $n$ is the number of boxes in the standard tableau.
Matching statistic: St001232
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(load all 7 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 88%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 88%
Values
[[1,2]]
=> [2] => [2] => [1,1,0,0]
=> 0
[[1],[2]]
=> [1,1] => [1,1] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [3] => [3] => [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> [1,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[[1,2],[3]]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[[1],[2],[3]]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> ? = 0
[[1,2,3,4]]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> [1,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[[1,2,4],[3]]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,2,3],[4]]
=> [3,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[[1,3],[2,4]]
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,2}
[[1,2],[3,4]]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [1,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,2}
[[1,3],[2],[4]]
=> [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,2}
[[1,2],[3],[4]]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,2}
[[1,2,3,4,5]]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> [1,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,4,5],[3]]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,5],[4]]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,2,3,4],[5]]
=> [4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3,5],[2,4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3,4]]
=> [2,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[1,3,4],[2,5]]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,2,4],[3,5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4,5]]
=> [3,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,5],[3],[4]]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,3,4],[2],[5]]
=> [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,2,4],[3],[5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2],[3,5],[4]]
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,2],[3,4],[5]]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2}
[[1,2,3,4,5,6]]
=> [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,4,5,6],[3]]
=> [2,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[[1,2,3,5,6],[4]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,2,3,4,6],[5]]
=> [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[[1,2,3,4,5],[6]]
=> [5,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,5,6],[3,4]]
=> [2,4] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[[1,3,4,6],[2,5]]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,4,6],[3,5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3,6],[4,5]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,3,4,5],[2,6]]
=> [1,4,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,4,5],[3,6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,3,5],[4,6]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4],[5,6]]
=> [4,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[[1,4,5,6],[2],[3]]
=> [1,1,4] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,5,6],[2],[4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,5,6],[3],[4]]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,4,6],[2],[5]]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,4,6],[3],[5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3,6],[4],[5]]
=> [3,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[[1,3,4,5],[2],[6]]
=> [1,4,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,4,5],[3],[6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,2,3,5],[4],[6]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[[1,3,5],[2,4,6]]
=> [1,2,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,5],[3,4,6]]
=> [2,3,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[[1,3,4],[2,5,6]]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,4],[3,5,6]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,2,3],[4,5,6]]
=> [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[1,4,6],[2,5],[3]]
=> [1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,6],[2,5],[4]]
=> [1,2,3] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[[1,2,6],[3,5],[4]]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,6],[2,4],[5]]
=> [1,2,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,6],[3,4],[5]]
=> [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[1,4,5],[2,6],[3]]
=> [1,1,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,5],[2,6],[4]]
=> [1,2,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,5],[3,6],[4]]
=> [2,1,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[[1,3,4],[2,6],[5]]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,5],[2,4],[6]]
=> [1,2,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,4],[2,5],[6]]
=> [1,3,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,4],[3,5],[6]]
=> [2,2,1,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,4,6],[2],[3],[5]]
=> [1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,6],[2],[4],[5]]
=> [1,2,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,6],[3],[4],[5]]
=> [2,1,1,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,4,5],[2],[3],[6]]
=> [1,1,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,5],[2],[4],[6]]
=> [1,2,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3,4],[2],[5],[6]]
=> [1,3,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,4],[3],[5],[6]]
=> [2,2,1,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,4],[2,5],[3,6]]
=> [1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3],[2,5],[4,6]]
=> [1,2,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3],[2,4],[5,6]]
=> [1,2,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,5],[2,6],[3],[4]]
=> [1,1,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,3],[2,6],[4],[5]]
=> [1,2,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
[[1,2],[3,6],[4],[5]]
=> [2,1,1,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 75%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 75%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,2}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,2}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,2}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,1,2,2,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3}
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,1,2,2,3}
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,5,3,2,4,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,3,1,4,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [5,2,1,6,3,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,1,6,3,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,3,2,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,2,1,3,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [6,4,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,6,2,1,5] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,6,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [6,2,1,5,4,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(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)
=> 1
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [7,2,1,4,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [3,1,7,4,2,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> 4
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [7,3,5,2,1,4,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [7,2,4,1,5,3,6] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 2
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [4,7,3,1,5,2,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [5,7,2,1,4,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[[1,2,6],[3,4,7],[5]]
=> [5,3,4,7,1,2,6] => [3,5,1,7,4,2,6] => ([(0,5),(1,2),(1,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 4
[[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [4,6,2,1,7,3,5] => ([(0,1),(0,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 3
[[1,2,5],[3,6,7],[4]]
=> [4,3,6,7,1,2,5] => [4,6,1,7,3,2,5] => ([(0,3),(0,6),(1,2),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3
[[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [5,6,2,1,3,7,4] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[[1,2,4],[3,6,7],[5]]
=> [5,3,6,7,1,2,4] => [3,5,1,6,2,7,4] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4
[[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [5,4,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [4,6,2,1,3,7,5] => ([(0,2),(1,5),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000454
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 88%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 88%
Values
[[1,2]]
=> [2] => ([],2)
=> ([],2)
=> 0
[[1],[2]]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,3]]
=> [3] => ([],3)
=> ([],3)
=> 0
[[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0
[[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,2,3,4]]
=> [4] => ([],4)
=> ([],4)
=> 0
[[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2}
[[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2}
[[1,3],[2,4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2}
[[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2}
[[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2}
[[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ([],5)
=> 0
[[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,3,5],[2,4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,3,4],[2,5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[[1,3,5],[2],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,4],[2],[5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2,5],[3]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,2],[3,5],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ([],6)
=> 0
[[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ([(4,5)],6)
=> 1
[[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,3,5,6],[2,4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,3,4,6],[2,5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,3,6],[4,5]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,3,4,5],[2,6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[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)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5,6],[2],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,5,6],[3],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,4,6],[2],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,4,5],[2],[6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[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)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[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)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,3,5],[2,4,6]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,5],[3,4,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,3,4],[2,5,6]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,4],[3,5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,4,6],[2,5],[3]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,6],[3,5],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,6],[2,4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,6],[3,4],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,4,5],[2,6],[3]]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,3,5],[2,6],[4]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[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)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,3,4],[2,6],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5}
[[1,2,3],[4,6],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3,6],[2],[4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2,6],[3],[4],[5]]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,3],[2,4],[5,6]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,3],[2,6],[4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2],[3,6],[4],[5]]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,4],[2,5],[3],[6]]
=> [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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,3],[2,4],[5],[6]]
=> [1,2,1,1,1] => ([(0,3),(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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,2],[3,4],[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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,4],[2],[3],[5],[6]]
=> [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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,3],[2],[4],[5],[6]]
=> [1,2,1,1,1] => ([(0,3),(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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => ([(0,2),(0,3),(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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(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)
=> ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ([],7)
=> 0
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: St000772
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Values
[[1,2]]
=> [2] => ([],2)
=> ? ∊ {0,1}
[[1],[2]]
=> [2] => ([],2)
=> ? ∊ {0,1}
[[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {0,0,2}
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3] => ([],3)
=> ? ∊ {0,0,2}
[[1],[2],[3]]
=> [3] => ([],3)
=> ? ∊ {0,0,2}
[[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,3,3}
[[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,3,3}
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,3,3}
[[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,3,3}
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,3,3}
[[1,2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,3,3}
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,1,1,3,3}
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,3,4,5],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,5],[3,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,3,4],[2,5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,4,5],[2],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3,4],[2],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,4],[2,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,4],[2],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,3],[2],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,4,4,4,4,4}
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,5,6],[3,4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,6],[4,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,4,5,6],[2],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,3,4,5],[2],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,6],[3,4],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,4],[5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,6],[2],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000260
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 38%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 38%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,2}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,2}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,2}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,1,2,2,3,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,1,2,2,3,3}
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3,3}
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3,3}
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,3,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,3,3,3,3,3,4,4,4,4,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5}
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,6,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(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] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(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] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(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] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(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)
=> 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 3
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 3
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [3,2,6,1,4,7,5] => ([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> 2
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 2
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> 3
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> 3
[[1,5,6],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6] => [4,3,7,2,1,5,6] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [3,2,7,6,1,4,5] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [3,5,2,1,7,6,4] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> 2
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> 2
[[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,4,6,1,7,5,3] => ([(0,5),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> 2
[[1,3],[2,5],[4,6],[7]]
=> [7,4,6,2,5,1,3] => [2,4,1,7,6,5,3] => ([(0,2),(1,2),(1,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [4,6,3,2,1,7,5] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4],[2,7],[3],[5],[6]]
=> [6,5,3,2,7,1,4] => [3,6,5,2,1,7,4] => ([(0,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3],[2,7],[4],[5],[6]]
=> [6,5,4,2,7,1,3] => [2,6,5,4,1,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,5],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5] => [4,3,7,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4],[2,6],[3],[5],[7]]
=> [7,5,3,2,6,1,4] => [3,5,7,2,1,6,4] => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3],[2,6],[4],[5],[7]]
=> [7,5,4,2,6,1,3] => [2,5,7,4,1,6,3] => ([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,4],[2,5],[3],[6],[7]]
=> [7,6,3,2,5,1,4] => [3,2,7,6,1,5,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3],[2,5],[4],[6],[7]]
=> [7,6,4,2,5,1,3] => [2,4,7,6,1,5,3] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [5,7,4,3,2,1,6] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [4,7,6,3,2,1,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [3,7,6,5,2,1,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [2,7,6,5,4,1,3] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000771
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 75%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 75%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,1,2,2,2,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,2,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,2,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,2,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,1,2,2,2,3}
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,2,3}
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,2,3}
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,2,2,2,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5}
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,2,6,1,4,5] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,5,2,1,6,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,6,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(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] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(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] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(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] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(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)
=> 5
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [3,2,6,1,4,7,5] => ([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> 1
[[1,5,6],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6] => [4,3,7,2,1,5,6] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [3,2,7,6,1,4,5] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [3,5,2,1,7,6,4] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> 1
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ([(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> 1
[[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,4,6,1,7,5,3] => ([(0,5),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> 1
[[1,3],[2,5],[4,6],[7]]
=> [7,4,6,2,5,1,3] => [2,4,1,7,6,5,3] => ([(0,2),(1,2),(1,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [4,6,3,2,1,7,5] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,4],[2,7],[3],[5],[6]]
=> [6,5,3,2,7,1,4] => [3,6,5,2,1,7,4] => ([(0,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,3],[2,7],[4],[5],[6]]
=> [6,5,4,2,7,1,3] => [2,6,5,4,1,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[[1,5],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5] => [4,3,7,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1
[[1,4],[2,6],[3],[5],[7]]
=> [7,5,3,2,6,1,4] => [3,5,7,2,1,6,4] => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 1
[[1,3],[2,6],[4],[5],[7]]
=> [7,5,4,2,6,1,3] => [2,5,7,4,1,6,3] => ([(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
[[1,4],[2,5],[3],[6],[7]]
=> [7,6,3,2,5,1,4] => [3,2,7,6,1,5,4] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 1
[[1,3],[2,5],[4],[6],[7]]
=> [7,6,4,2,5,1,3] => [2,4,7,6,1,5,3] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 1
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [5,7,4,3,2,1,6] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [4,7,6,3,2,1,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [3,7,6,5,2,1,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [2,7,6,5,4,1,3] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St001645
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 88%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 88%
Values
[[1,2]]
=> [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[[1],[2]]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,2,3]]
=> [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 1
[[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,2,3,4]]
=> [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,1,2,2,3} + 1
[[1,2,4],[3]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,1,2,2,3} + 1
[[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2,3} + 1
[[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,1,2,2,3} + 1
[[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2,3} + 1
[[1,3],[2],[4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,2,2,3} + 1
[[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,4,5],[3]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,3,5],[4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,3,5],[2,4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,3,4],[2,5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,4],[3,5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,3,5],[2],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,5],[3],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,3,4],[2],[5]]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,4],[3],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2],[3,5],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4} + 1
[[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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,4,5,6],[3]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,3,5,6],[4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,3,4,6],[5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,3,4,6],[2,5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,4,6],[3,5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,3,6],[4,5]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,3,4,5],[2,6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,4,5],[3,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,3,5,6],[2],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,5,6],[3],[4]]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,3,4,6],[2],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,4,6],[3],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,3,6],[4],[5]]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,3,4,5],[2],[6]]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[1,2,4,5],[3],[6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5} + 1
[[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)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,3],[2,4],[5],[6]]
=> [1,2,1,1,1] => ([(0,3),(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)
=> ([(0,3),(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)
=> 6 = 5 + 1
[[1,3],[2],[4],[5],[6]]
=> [1,2,1,1,1] => ([(0,3),(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)
=> ([(0,3),(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)
=> 6 = 5 + 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => ([(0,2),(0,3),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(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)
=> ([(0,1),(0,2),(0,3),(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)
=> 6 = 5 + 1
[[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ([],1)
=> 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,2,3,4,5],[6],[7]]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,3,4],[2,5],[6],[7]]
=> [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(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)
=> 6 = 5 + 1
[[1,2,4],[3,5],[6],[7]]
=> [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(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)
=> 6 = 5 + 1
[[1,3,4],[2],[5],[6],[7]]
=> [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(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)
=> 6 = 5 + 1
[[1,2,4],[3],[5],[6],[7]]
=> [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(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)
=> 6 = 5 + 1
[[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[[1,4],[2,5],[3],[6],[7]]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,3],[2,4],[5],[6],[7]]
=> [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(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)
=> 6 = 5 + 1
[[1,4],[2],[3],[5],[6],[7]]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,3],[2],[4],[5],[6],[7]]
=> [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(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)
=> 6 = 5 + 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
Description
The pebbling number of a connected graph.
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