Processing math: 0%

Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000567
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000567: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,3,2] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[2,1,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[2,3,1] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,3,2,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,3,4,2] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,4,2,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[2,1,4,3] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[2,3,1,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[2,4,1,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[3,1,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[3,2,1,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,5,3,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,3,5,2,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,2,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,2,5,3] => [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,3,5,2] => [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,5,3,2] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,5,2,3,4] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,5,2,4,3] => [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,5,3,4,2] => [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,5,4,3,2] => [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 6
Description
The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2} for a partition \lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n, see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Mp00254: Permutations Inverse fireworks mapPermutations
Mp00170: Permutations to signed permutationSigned permutations
Mp00168: Signed permutations Kreweras complementSigned permutations
St001817: Signed permutations ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 7%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,-1] => 5 = 1 + 4
[1,3,2] => [1,3,2] => [1,3,2] => [2,-1,3] => 5 = 1 + 4
[2,1,3] => [2,1,3] => [2,1,3] => [3,2,-1] => 5 = 1 + 4
[2,3,1] => [1,3,2] => [1,3,2] => [2,-1,3] => 5 = 1 + 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 7 = 3 + 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 7 = 3 + 4
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 7 = 3 + 4
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 7 = 3 + 4
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [2,4,-1,3] => 5 = 1 + 4
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 5 = 1 + 4
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,2,4,-1] => 7 = 3 + 4
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,2,-1,4] => 7 = 3 + 4
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 7 = 3 + 4
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 7 = 3 + 4
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [4,2,-1,3] => 5 = 1 + 4
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 5 = 1 + 4
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [3,4,2,-1] => 5 = 1 + 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,-1] => 5 = 1 + 4
[3,4,1,2] => [2,4,1,3] => [2,4,1,3] => [4,2,-1,3] => 5 = 1 + 4
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 5 = 1 + 4
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [3,4,-1,2] => 5 = 1 + 4
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => [4,3,-1,2] => 5 = 1 + 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 6 + 4
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 6 + 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,-1] => ? = 6 + 4
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 6 + 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,-1,4] => ? = 3 + 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => ? = 3 + 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,-1] => ? = 6 + 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,-1,5] => ? = 6 + 4
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,-1] => ? = 6 + 4
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 6 + 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,-1,4] => ? = 3 + 4
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => ? = 3 + 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,-1] => ? = 3 + 4
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,-1,5] => ? = 1 + 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,-1] => ? = 3 + 4
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,-1,5] => ? = 1 + 4
[1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,-1,4] => ? = 3 + 4
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => ? = 3 + 4
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,-1,3] => ? = 3 + 4
[1,5,2,4,3] => [1,5,2,4,3] => [1,5,2,4,3] => [2,4,-1,5,3] => ? = 1 + 4
[1,5,3,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => [2,5,4,-1,3] => ? = 3 + 4
[1,5,3,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => [2,4,-1,5,3] => ? = 1 + 4
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => [2,5,-1,4,3] => ? = 1 + 4
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 1 + 4
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,-1] => ? = 6 + 4
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,-1,5] => ? = 6 + 4
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,-1] => ? = 6 + 4
[2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,-1,5] => ? = 6 + 4
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,-1,4] => ? = 3 + 4
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [3,2,-1,5,4] => ? = 3 + 4
[2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,-1] => ? = 6 + 4
[2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,-1,5] => ? = 6 + 4
[2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,-1] => ? = 6 + 4
[2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 6 + 4
[2,3,5,1,4] => [1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,-1,4] => ? = 3 + 4
[2,3,5,4,1] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => ? = 3 + 4
[2,4,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => [4,2,5,3,-1] => ? = 3 + 4
[2,4,1,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,-1,5] => ? = 1 + 4
[2,4,3,1,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,-1] => ? = 3 + 4
[2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,-1,5] => ? = 1 + 4
[2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,-1,4] => ? = 3 + 4
[2,4,5,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => ? = 3 + 4
[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => [4,2,5,-1,3] => ? = 3 + 4
[2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => [4,2,-1,5,3] => ? = 1 + 4
[2,5,3,1,4] => [1,5,3,2,4] => [1,5,3,2,4] => [2,5,4,-1,3] => ? = 3 + 4
[2,5,3,4,1] => [1,5,2,4,3] => [1,5,2,4,3] => [2,4,-1,5,3] => ? = 1 + 4
[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [5,2,-1,4,3] => ? = 1 + 4
[2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 1 + 4
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,-1] => ? = 3 + 4
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,-1,5] => ? = 3 + 4
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 5 = 1 + 4
Description
The number of flag weak exceedances of a signed permutation. This is the number of negative entries plus twice the number of weak exceedances of the signed permutation.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00310: Permutations toric promotionPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
St001582: Permutations ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 7%
Values
[1,2,3] => [1,2,3] => [3,2,1] => [2,3,1] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [2,3,1] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [2,3,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[1,2,4,3] => [1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[1,3,2,4] => [1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[1,3,4,2] => [1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [4,2,3,1] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [4,2,3,1] => 1
[2,1,3,4] => [1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[2,1,4,3] => [1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[2,3,1,4] => [1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[2,3,4,1] => [1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [4,2,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [4,2,3,1] => 1
[3,1,2,4] => [1,3,2,4] => [2,4,1,3] => [4,3,1,2] => 1
[3,2,1,4] => [1,3,2,4] => [2,4,1,3] => [4,3,1,2] => 1
[3,4,1,2] => [1,3,2,4] => [2,4,1,3] => [4,3,1,2] => 1
[3,4,2,1] => [1,3,2,4] => [2,4,1,3] => [4,3,1,2] => 1
[4,1,2,3] => [1,4,3,2] => [3,1,2,4] => [3,2,1,4] => 1
[4,2,1,3] => [1,4,3,2] => [3,1,2,4] => [3,2,1,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[1,2,3,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[1,2,4,3,5] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[1,2,4,5,3] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[1,2,5,3,4] => [1,2,3,5,4] => [5,2,4,1,3] => [2,5,3,4,1] => ? = 3
[1,2,5,4,3] => [1,2,3,5,4] => [5,2,4,1,3] => [2,5,3,4,1] => ? = 3
[1,3,2,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[1,3,2,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[1,3,4,2,5] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[1,3,4,5,2] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[1,3,5,2,4] => [1,2,3,5,4] => [5,2,4,1,3] => [2,5,3,4,1] => ? = 3
[1,3,5,4,2] => [1,2,3,5,4] => [5,2,4,1,3] => [2,5,3,4,1] => ? = 3
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,1,2,4] => [5,4,2,3,1] => ? = 3
[1,4,2,5,3] => [1,2,4,5,3] => [5,3,4,1,2] => [5,2,3,4,1] => ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,1,2,4] => [5,4,2,3,1] => ? = 3
[1,4,3,5,2] => [1,2,4,5,3] => [5,3,4,1,2] => [5,2,3,4,1] => ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,1,2,4] => [5,4,2,3,1] => ? = 3
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,1,2,4] => [5,4,2,3,1] => ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,3,2] => [5,2,4,3,1] => ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => [5,4,1,2,3] => [4,2,5,3,1] => ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,3,2] => [5,2,4,3,1] => ? = 3
[1,5,3,4,2] => [1,2,5,3,4] => [5,4,1,2,3] => [4,2,5,3,1] => ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => [5,4,1,2,3] => [4,2,5,3,1] => ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => [5,4,1,2,3] => [4,2,5,3,1] => ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[2,1,3,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[2,1,4,3,5] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[2,1,4,5,3] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[2,1,5,3,4] => [1,2,3,5,4] => [5,2,4,1,3] => [2,5,3,4,1] => ? = 3
[2,1,5,4,3] => [1,2,3,5,4] => [5,2,4,1,3] => [2,5,3,4,1] => ? = 3
[2,3,1,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[2,3,1,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[2,3,4,1,5] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[2,3,4,5,1] => [1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => ? = 6
[2,3,5,1,4] => [1,2,3,5,4] => [5,2,4,1,3] => [2,5,3,4,1] => ? = 3
[2,3,5,4,1] => [1,2,3,5,4] => [5,2,4,1,3] => [2,5,3,4,1] => ? = 3
[2,4,1,3,5] => [1,2,4,3,5] => [5,3,1,2,4] => [5,4,2,3,1] => ? = 3
[2,4,1,5,3] => [1,2,4,5,3] => [5,3,4,1,2] => [5,2,3,4,1] => ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => [5,3,1,2,4] => [5,4,2,3,1] => ? = 3
[2,4,3,5,1] => [1,2,4,5,3] => [5,3,4,1,2] => [5,2,3,4,1] => ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => [5,3,1,2,4] => [5,4,2,3,1] => ? = 3
[2,4,5,3,1] => [1,2,4,3,5] => [5,3,1,2,4] => [5,4,2,3,1] => ? = 3
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,3,2] => [5,2,4,3,1] => ? = 3
[2,5,1,4,3] => [1,2,5,3,4] => [5,4,1,2,3] => [4,2,5,3,1] => ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,3,2] => [5,2,4,3,1] => ? = 3
[2,5,3,4,1] => [1,2,5,3,4] => [5,4,1,2,3] => [4,2,5,3,1] => ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => [5,4,1,2,3] => [4,2,5,3,1] => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => [5,4,1,2,3] => [4,2,5,3,1] => ? = 1
[3,1,2,4,5] => [1,3,2,4,5] => [2,5,1,3,4] => [5,4,3,1,2] => ? = 3
[3,1,2,5,4] => [1,3,2,4,5] => [2,5,1,3,4] => [5,4,3,1,2] => ? = 3
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001613
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00208: Permutations lattice of intervalsLattices
St001613: Lattices ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 3%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[3,5,6,1,4,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,2,4,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,4,1,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,4,2,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
Description
The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001719
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00208: Permutations lattice of intervalsLattices
St001719: Lattices ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 3%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[3,5,6,1,4,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,2,4,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,4,1,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,4,2,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval [a, b] in a lattice is small if b is a join of elements covering a.
Matching statistic: St001881
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00208: Permutations lattice of intervalsLattices
St001881: Lattices ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 3%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3
[3,5,6,1,4,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,2,4,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,4,1,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[3,5,6,4,2,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
Description
The number of factors of a lattice as a Cartesian product of lattices. Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St001616
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00208: Permutations lattice of intervalsLattices
St001616: Lattices ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 3%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[3,5,6,1,4,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[3,5,6,2,4,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[3,5,6,4,1,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[3,5,6,4,2,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
Description
The number of neutral elements in a lattice. An element e of the lattice L is neutral if the sublattice generated by e, x and y is distributive for all x, y \in L.
Matching statistic: St001720
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00208: Permutations lattice of intervalsLattices
St001720: Lattices ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 3%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 3 + 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 3 + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 6 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 3 + 1
[3,5,6,1,4,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[3,5,6,2,4,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[3,5,6,4,1,2] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[3,5,6,4,2,1] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
Description
The minimal length of a chain of small intervals in a lattice. An interval [a, b] is small if b is a join of elements covering a.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001892: Signed permutations ⟶ ℤResult quality: 0% values known / values provided: 0%distinct values known / distinct values provided: 7%
Values
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[1,3,2] => [1,2,3] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[2,1,3] => [1,2,3] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 6 = 3 + 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 6 = 3 + 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 6 = 3 + 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 6 = 3 + 3
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 4 = 1 + 3
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 4 = 1 + 3
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 6 = 3 + 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 6 = 3 + 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 6 = 3 + 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 6 = 3 + 3
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 4 = 1 + 3
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 4 = 1 + 3
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 4 = 1 + 3
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 4 = 1 + 3
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 4 = 1 + 3
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 4 = 1 + 3
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 4 = 1 + 3
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 4 = 1 + 3
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 3
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 3
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 3
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 3
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 3
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 + 3
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 3
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 + 3
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 3
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 3
[1,5,2,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 3 + 3
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 3
[1,5,3,2,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 3 + 3
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 3
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 3
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 3
[2,1,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[2,1,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[2,1,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[2,1,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[2,1,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 3
[2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 3
[2,3,1,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[2,3,1,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[2,3,4,1,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 6 + 3
[2,3,5,1,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 3
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 3
[2,4,1,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 3
[2,4,1,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 + 3
[2,4,3,1,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 3
[2,4,3,5,1] => [1,2,4,5,3] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1 + 3
[2,4,5,1,3] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 3
[2,4,5,3,1] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 3
[2,5,1,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 3 + 3
[2,5,1,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 3
[2,5,3,1,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 3 + 3
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 3
[2,5,4,1,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 3
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 1 + 3
[3,1,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 3 + 3
[3,1,2,5,4] => [1,3,2,4,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 3 + 3
Description
The flag excedance statistic of a signed permutation. This is the number of negative entries plus twice the number of excedances of the signed permutation. That is, fexc(\sigma) = 2exc(\sigma) + neg(\sigma), where exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}| neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}| It has the same distribution as the flag descent statistic.