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Your data matches 52 different statistics following compositions of up to 3 maps.
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Matching statistic: St000058
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1
[[2,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 2
[[1,3]]
=> [1,2] => 1
[[2,3]]
=> [1,2] => 1
[[3,3]]
=> [1,2] => 1
[[1],[3]]
=> [2,1] => 2
[[2],[3]]
=> [2,1] => 2
[[1,1,2]]
=> [1,2,3] => 1
[[1,2,2]]
=> [1,2,3] => 1
[[2,2,2]]
=> [1,2,3] => 1
[[1,1],[2]]
=> [3,1,2] => 3
[[1,2],[2]]
=> [2,1,3] => 2
[[1,4]]
=> [1,2] => 1
[[2,4]]
=> [1,2] => 1
[[3,4]]
=> [1,2] => 1
[[4,4]]
=> [1,2] => 1
[[1],[4]]
=> [2,1] => 2
[[2],[4]]
=> [2,1] => 2
[[3],[4]]
=> [2,1] => 2
[[1,1,3]]
=> [1,2,3] => 1
[[1,2,3]]
=> [1,2,3] => 1
[[1,3,3]]
=> [1,2,3] => 1
[[2,2,3]]
=> [1,2,3] => 1
[[2,3,3]]
=> [1,2,3] => 1
[[3,3,3]]
=> [1,2,3] => 1
[[1,1],[3]]
=> [3,1,2] => 3
[[1,2],[3]]
=> [3,1,2] => 3
[[1,3],[2]]
=> [2,1,3] => 2
[[1,3],[3]]
=> [2,1,3] => 2
[[2,2],[3]]
=> [3,1,2] => 3
[[2,3],[3]]
=> [2,1,3] => 2
[[1],[2],[3]]
=> [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => 1
[[1,1,2,2]]
=> [1,2,3,4] => 1
[[1,2,2,2]]
=> [1,2,3,4] => 1
[[2,2,2,2]]
=> [1,2,3,4] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => 4
[[1,1,2],[2]]
=> [3,1,2,4] => 3
[[1,2,2],[2]]
=> [2,1,3,4] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => 2
[[1,5]]
=> [1,2] => 1
[[2,5]]
=> [1,2] => 1
[[3,5]]
=> [1,2] => 1
[[4,5]]
=> [1,2] => 1
[[5,5]]
=> [1,2] => 1
[[1],[5]]
=> [2,1] => 2
[[2],[5]]
=> [2,1] => 2
[[3],[5]]
=> [2,1] => 2
[[4],[5]]
=> [2,1] => 2
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000668
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,1]
=> 1
[[2,2]]
=> [1,2] => [1,1]
=> 1
[[1],[2]]
=> [2,1] => [2]
=> 2
[[1,3]]
=> [1,2] => [1,1]
=> 1
[[2,3]]
=> [1,2] => [1,1]
=> 1
[[3,3]]
=> [1,2] => [1,1]
=> 1
[[1],[3]]
=> [2,1] => [2]
=> 2
[[2],[3]]
=> [2,1] => [2]
=> 2
[[1,1,2]]
=> [1,2,3] => [1,1,1]
=> 1
[[1,2,2]]
=> [1,2,3] => [1,1,1]
=> 1
[[2,2,2]]
=> [1,2,3] => [1,1,1]
=> 1
[[1,1],[2]]
=> [3,1,2] => [3]
=> 3
[[1,2],[2]]
=> [2,1,3] => [2,1]
=> 2
[[1,4]]
=> [1,2] => [1,1]
=> 1
[[2,4]]
=> [1,2] => [1,1]
=> 1
[[3,4]]
=> [1,2] => [1,1]
=> 1
[[4,4]]
=> [1,2] => [1,1]
=> 1
[[1],[4]]
=> [2,1] => [2]
=> 2
[[2],[4]]
=> [2,1] => [2]
=> 2
[[3],[4]]
=> [2,1] => [2]
=> 2
[[1,1,3]]
=> [1,2,3] => [1,1,1]
=> 1
[[1,2,3]]
=> [1,2,3] => [1,1,1]
=> 1
[[1,3,3]]
=> [1,2,3] => [1,1,1]
=> 1
[[2,2,3]]
=> [1,2,3] => [1,1,1]
=> 1
[[2,3,3]]
=> [1,2,3] => [1,1,1]
=> 1
[[3,3,3]]
=> [1,2,3] => [1,1,1]
=> 1
[[1,1],[3]]
=> [3,1,2] => [3]
=> 3
[[1,2],[3]]
=> [3,1,2] => [3]
=> 3
[[1,3],[2]]
=> [2,1,3] => [2,1]
=> 2
[[1,3],[3]]
=> [2,1,3] => [2,1]
=> 2
[[2,2],[3]]
=> [3,1,2] => [3]
=> 3
[[2,3],[3]]
=> [2,1,3] => [2,1]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [2,1]
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4]
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1]
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,1]
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [2,2]
=> 2
[[1,5]]
=> [1,2] => [1,1]
=> 1
[[2,5]]
=> [1,2] => [1,1]
=> 1
[[3,5]]
=> [1,2] => [1,1]
=> 1
[[4,5]]
=> [1,2] => [1,1]
=> 1
[[5,5]]
=> [1,2] => [1,1]
=> 1
[[1],[5]]
=> [2,1] => [2]
=> 2
[[2],[5]]
=> [2,1] => [2]
=> 2
[[3],[5]]
=> [2,1] => [2]
=> 2
[[4],[5]]
=> [2,1] => [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St001555
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 86%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 86%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 1
[[2,2]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,3]]
=> [1,2] => [1,2] => [1,2] => 1
[[2,3]]
=> [1,2] => [1,2] => [1,2] => 1
[[3,3]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[3]]
=> [2,1] => [2,1] => [2,1] => 2
[[2],[3]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[1,2],[2]]
=> [2,1,3] => [2,3,1] => [2,3,1] => 3
[[1,4]]
=> [1,2] => [1,2] => [1,2] => 1
[[2,4]]
=> [1,2] => [1,2] => [1,2] => 1
[[3,4]]
=> [1,2] => [1,2] => [1,2] => 1
[[4,4]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[4]]
=> [2,1] => [2,1] => [2,1] => 2
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 2
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => [2,3,1] => 3
[[1,3],[3]]
=> [2,1,3] => [2,3,1] => [2,3,1] => 3
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[2,3],[3]]
=> [2,1,3] => [2,3,1] => [2,3,1] => 3
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,4,2] => [1,3,4,2] => 3
[[1,2,2],[2]]
=> [2,1,3,4] => [2,3,4,1] => [2,3,4,1] => 4
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[[1,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[2,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[3,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[4,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[5,5]]
=> [1,2] => [1,2] => [1,2] => 1
[[1],[5]]
=> [2,1] => [2,1] => [2,1] => 2
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 2
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 2
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 2
[[1,1,1,1,1,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[2,2,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [1,4,5,6,2,3] => [1,4,5,6,2,3] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? ∊ {1,1,1,1,1,1,2,2,2,3,3,4,5,5,6}
[[1,1,1,1,1,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,1,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,1,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[2,2,2,2,2,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[2,2,2,2,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[2,2,2,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[2,2,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[2,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[3,3,3,3,3,3]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,1,1],[3]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,1,2],[3]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,1,3],[2]]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,1,3],[3]]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,2,2],[3]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,2,3],[2]]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,2,3],[3]]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,3,3],[2]]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,1,3,3],[3]]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,2,2,2],[3]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,2,2,3],[2]]
=> [3,1,2,4,5,6] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,2,2,3],[3]]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,2,3,3],[2]]
=> [3,1,2,4,5,6] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,1,2,3,3],[3]]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The order of a signed permutation.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[1,1],[2]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1
[[1,1],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[[2,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[1,1,2],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,5]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[2,5]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[3,5]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[4,5]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[5,5]]
=> [1,2] => [1,2] => ([],2)
=> 1
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2
[[1,2],[2,3]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4}
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4}
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,4,4}
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[[1,2],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[2,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,4}
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,4,4,5,5,5,5,5,5,5,5,6,6,6}
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [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)
=> ? ∊ {2,3,5}
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,3,5}
[[1,1,1],[2,2,2]]
=> [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)
=> ? ∊ {2,3,5}
[[1,2],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[2,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[2,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[2,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[3,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,1],[2],[5]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,1],[3],[5]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,1],[4],[5]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2],[2],[5]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2],[3],[5]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000454
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[2,2],[3]]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 2 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[[1,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,2] => ([],2)
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2} - 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2} - 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2} - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,4,4} - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,4,4} - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,4,4} - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,4,4} - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,4,4} - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,4,4} - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,4,4} - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,3,4,4} - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,5} - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,5} - 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {2,2,2,2,2,2} - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,3,4,4,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,2,2,3,4,4,5,5,5,5,5,5,5,5,6,6,6} - 1
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: St001207
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 57%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 57%
Values
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[3]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[1,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1,2],[3]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[2,2],[3]]
=> [3,1,2] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [1,3,2] => 1 = 2 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => [2,1,4,3] => 1 = 2 - 1
[[1,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[2,2,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => [5,2,1,4,3] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [2,1,4,3,5] => ? ∊ {1,1,1,1,1,2,2,3,4,5,5} - 1
[[1,1,1,1,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,2,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,2,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,2,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[2,2,2,2,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[2,2,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[2,2,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[2,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[3,3,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1,1],[3]]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1,2],[3]]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,2,2],[3]]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,2,3],[2]]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,3,3],[2]]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,3,3],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,2,2,2],[3]]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,2,3,3],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[2,2,2,2],[3]]
=> [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[2,2,3,3],[3]]
=> [3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => [5,2,1,4,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => [5,2,1,4,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => [2,1,5,4,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [2,1,4,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6} - 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001877
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 43%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2 - 1
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,4} - 1
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 3 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,4} - 1
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2} - 1
[[1],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2} - 1
[[2],[3],[4]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2} - 1
[[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[2,2,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {2,2,2,2,2,3,4,4,4,4,4,4} - 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,3,4,5,5} - 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {2,3,4,5,5} - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {2,3,4,5,5} - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,3,4,5,5} - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {2,3,4,5,5} - 1
[[1],[2],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2,2,2,2} - 1
[[1],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2,2,2,2} - 1
[[1],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2,2,2,2} - 1
[[2],[3],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2,2,2,2} - 1
[[2],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2,2,2,2} - 1
[[3],[4],[5]]
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,2,2,2,2} - 1
[[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,1,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[2,2,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[2,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[2,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[3,3,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,1],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,1],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[2,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[2,2],[4,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4} - 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[3]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[1,2],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3],[4]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2} - 1
[[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2} - 1
[[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2} - 1
[[2,3],[3]]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2} - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3} - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,3} - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3} - 1
[[1,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[2],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4],[5]]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[1,1,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,4,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,4]]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[1,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[1,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[2,2],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[2,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[3,3],[4]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2} - 1
[[1,1,1],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,1,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,1,3],[2]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,1,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[2,2,2],[3]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[2,2,3],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} - 1
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,4,5} - 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,3,4,5} - 1
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {2,2,3,4,5} - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,4,5} - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,3,4,5} - 1
[[1,1],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[1,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[1,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[1,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[2,2],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[2,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[2,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[3,3],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[3,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[4,4],[5]]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
[[1],[4],[5]]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001060
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 29%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,2}
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,2}
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,2}
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2}
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2}
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,2,2}
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,2,2}
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,2,2}
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,3}
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,3}
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2}
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2}
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2}
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,2,2,2}
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2}
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2}
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,2}
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,3,3}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,4}
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,4}
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,4}
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,4}
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,4}
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,4}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,4}
[[1,1],[2,2]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[2,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[3,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[4,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[5,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2,2,2,2}
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3}
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3}
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[2,3]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[2,3]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[2,2],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1],[2,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[3,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[2,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[2,2],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[2,3],[3,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[2,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[3,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(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)
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 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
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000259
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2}
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2}
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[2,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[3,3]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[1],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,3}
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,3}
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,3}
[[1,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2}
[[2,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2}
[[3,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2}
[[4,4]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2}
[[1],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[4]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,2,4}
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,2,4}
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,2,4}
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,2,2,4}
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,4}
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,4}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,4}
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[2,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[3,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[4,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[5,5]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {1,2,2,2,2}
[[1],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[5]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,2,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[2,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,3,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[3,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[4,4,4]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,1],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1,2],[4]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[4],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[5],[6]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,2],[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
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 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
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[1,1,1],[2,2,2]]
=> [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
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001568The smallest positive integer that does not appear twice in the partition. St001118The acyclic chromatic index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001624The breadth of a lattice. St000264The girth of a graph, which is not a tree. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001281The normalized isoperimetric number of a graph. St001545The second Elser number of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001645The pebbling number of a connected graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000103The sum of the entries of a semistandard tableau. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
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