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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000732
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 2
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
[1,4,5,3,2] => 0
Description
The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St001167
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[2,3,1] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[3,1,2] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,4,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[2,3,1,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,3,4,1] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,4,1,3] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1,2,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,4,1] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[3,4,2,1] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[4,1,2,3] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[4,1,3,2] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,2,1,3] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,3,1,2] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,4,5,3,2] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[2,3,1] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[3,1,2] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,4,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[2,3,1,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,3,4,1] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,4,1,3] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1,2,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,4,1] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[3,4,2,1] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[4,1,2,3] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[4,1,3,2] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,2,1,3] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,3,1,2] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,4,5,3,2] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St001066
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001483
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,1,2,3] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000366
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00239: Permutations —Corteel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,2,1] => [2,3,1] => [1,3,2] => 0
[3,1,2] => [3,1,2] => [3,2,1] => [3,2,1] => 1
[3,2,1] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => [1,2,4,3] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => [1,3,2,4] => 0
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => [1,2,4,3] => 0
[2,4,1,3] => [4,2,1,3] => [2,4,3,1] => [1,4,3,2] => 1
[2,4,3,1] => [3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 0
[3,1,2,4] => [3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,1,4,2] => [4,1,3,2] => [3,4,2,1] => [1,4,3,2] => 1
[3,2,1,4] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[3,2,4,1] => [2,4,3,1] => [3,4,1,2] => [2,4,1,3] => 0
[3,4,1,2] => [4,3,2,1] => [3,2,4,1] => [2,1,4,3] => 0
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => [4,1,3,2] => 0
[4,1,2,3] => [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 2
[4,1,3,2] => [3,1,4,2] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,1,3] => [2,4,1,3] => [4,3,1,2] => [4,3,1,2] => 1
[4,2,3,1] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 0
[4,3,2,1] => [3,4,1,2] => [3,1,4,2] => [2,1,4,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => 0
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => [1,2,3,5,4] => 0
[1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => [1,2,5,4,3] => 1
[1,3,5,4,2] => [1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,5,2,4,3] => [1,4,5,3,2] => [1,2,5,4,3] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,3,5,2] => [1,3,5,4,2] => [1,4,5,2,3] => [1,3,5,2,4] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,3,2,5,4] => 0
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,3,4,2] => [1,5,2,4,3] => 0
[1,4,2,7,5,3,6] => [1,5,2,4,7,3,6] => [1,4,7,6,3,2,5] => [1,4,7,6,3,2,5] => ? = 2
[1,4,2,7,5,6,3] => [1,5,2,4,6,7,3] => [1,4,7,3,2,5,6] => [1,4,7,3,2,5,6] => ? = 1
[1,4,2,7,6,3,5] => [1,6,2,4,7,5,3] => [1,4,7,3,2,6,5] => [1,4,7,3,2,6,5] => ? = 1
[1,4,3,7,5,2,6] => [1,3,5,4,7,2,6] => [1,4,7,6,2,3,5] => [1,4,7,6,2,3,5] => ? = 1
[1,4,3,7,6,2,5] => [1,3,6,4,7,5,2] => [1,4,7,2,3,6,5] => [1,4,7,2,3,6,5] => ? = 0
[1,4,5,3,2,6,7] => [1,5,4,2,3,6,7] => [1,5,3,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 0
[1,4,5,3,2,7,6] => [1,5,4,2,3,7,6] => [1,5,3,4,2,7,6] => [1,5,2,4,3,7,6] => ? = 0
[1,4,5,3,7,6,2] => [1,6,4,2,5,7,3] => [1,5,7,3,4,2,6] => [1,5,7,2,4,3,6] => ? = 0
[1,4,6,3,2,5,7] => [1,6,4,2,3,5,7] => [1,6,5,3,4,2,7] => [1,6,5,2,4,3,7] => ? = 1
[1,4,6,3,5,2,7] => [1,5,4,2,6,3,7] => [1,6,3,4,2,5,7] => [1,6,2,4,3,5,7] => ? = 0
[1,4,6,3,5,7,2] => [1,5,4,2,7,6,3] => [1,6,7,3,4,2,5] => [1,5,7,2,4,3,6] => ? = 0
[1,4,6,3,7,5,2] => [1,7,4,2,6,3,5] => [1,7,5,6,3,4,2] => [1,7,3,6,2,5,4] => ? = 0
[1,4,6,5,3,2,7] => [1,6,4,5,2,3,7] => [1,6,3,4,5,2,7] => [1,6,2,3,5,4,7] => ? = 0
[1,4,6,5,3,7,2] => [1,7,4,5,2,6,3] => [1,6,7,3,4,5,2] => [1,4,7,2,3,6,5] => ? = 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000731
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,2,1] => [3,2,1] => [3,1,2] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [2,3,1] => 1
[3,2,1] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 0
[2,3,4,1] => [4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 0
[2,4,1,3] => [4,2,1,3] => [4,2,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [3,4,2,1] => [1,4,3,2] => [1,4,2,3] => 0
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[3,1,4,2] => [4,3,1,2] => [4,3,1,2] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,2,4,1] => [2,4,3,1] => [1,4,3,2] => [1,4,2,3] => 0
[3,4,1,2] => [4,1,3,2] => [4,1,3,2] => [3,4,2,1] => 0
[3,4,2,1] => [4,2,3,1] => [4,1,3,2] => [3,4,2,1] => 0
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 2
[4,1,3,2] => [3,4,1,2] => [2,4,1,3] => [3,2,4,1] => 1
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => [3,2,4,1] => 1
[4,2,3,1] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,1,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[4,3,2,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,3,4] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,3,4] => 0
[1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => [1,4,5,3,2] => 0
[1,4,5,3,2] => [1,5,3,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 0
[1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => ? = 0
[1,3,4,5,6,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => ? = 0
[1,3,4,6,2,5,7] => [1,6,4,3,2,5,7] => [1,6,4,3,2,5,7] => [1,5,2,3,6,4,7] => ? = 1
[1,3,4,6,2,7,5] => [1,7,6,4,3,2,5] => [1,7,6,4,3,2,5] => [1,5,2,3,7,4,6] => ? = 1
[1,3,4,6,7,2,5] => [1,7,4,3,2,6,5] => [1,7,4,3,2,6,5] => [1,6,2,3,7,5,4] => ? = 0
[1,3,4,6,7,5,2] => [1,7,5,6,4,3,2] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 0
[1,3,4,7,2,5,6] => [1,7,4,3,2,5,6] => [1,7,4,3,2,5,6] => [1,5,2,3,6,7,4] => ? = 2
[1,3,4,7,2,6,5] => [1,6,7,4,3,2,5] => [1,5,7,4,3,2,6] => [1,6,2,3,5,7,4] => ? = 1
[1,3,4,7,5,2,6] => [1,5,7,4,3,2,6] => [1,5,7,4,3,2,6] => [1,6,2,3,5,7,4] => ? = 1
[1,3,5,2,7,6,4] => [1,6,7,5,3,2,4] => [1,4,7,6,3,2,5] => [1,5,2,4,7,3,6] => ? = 1
[1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => [1,6,3,2,5,4,7] => [1,5,2,6,4,3,7] => ? = 0
[1,3,5,6,2,7,4] => [1,7,6,3,2,5,4] => [1,7,6,3,2,5,4] => [1,5,2,7,4,3,6] => ? = 0
[1,3,5,6,4,2,7] => [1,6,4,5,3,2,7] => [1,6,2,5,4,3,7] => [1,5,6,3,4,2,7] => ? = 0
[1,3,5,6,4,7,2] => [1,7,6,4,5,3,2] => [1,7,6,2,5,4,3] => [1,5,7,3,4,2,6] => ? = 0
[1,3,5,6,7,2,4] => [1,7,3,2,6,5,4] => [1,7,3,2,6,5,4] => [1,6,2,7,4,5,3] => ? = 0
[1,3,5,6,7,4,2] => [1,7,4,6,5,3,2] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 0
[1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => [1,7,3,2,5,4,6] => [1,5,2,6,4,7,3] => ? = 1
[1,3,5,7,2,6,4] => [1,6,7,3,2,5,4] => [1,4,7,3,2,6,5] => [1,6,2,4,7,5,3] => ? = 0
[1,3,5,7,4,2,6] => [1,7,4,5,3,2,6] => [1,7,2,5,4,3,6] => [1,5,6,3,4,7,2] => ? = 1
[1,3,5,7,4,6,2] => [1,6,7,4,5,3,2] => [1,3,7,2,6,5,4] => [1,6,3,7,4,5,2] => ? = 0
[1,3,6,2,5,4,7] => [1,5,6,3,2,4,7] => [1,4,6,3,2,5,7] => [1,5,2,4,6,3,7] => ? = 1
[1,3,6,2,5,7,4] => [1,5,7,6,3,2,4] => [1,4,7,6,3,2,5] => [1,5,2,4,7,3,6] => ? = 1
[1,3,6,2,7,5,4] => [1,7,5,6,3,2,4] => [1,7,4,6,3,2,5] => [1,5,2,6,7,3,4] => ? = 1
[1,3,6,4,2,5,7] => [1,4,6,3,2,5,7] => [1,4,6,3,2,5,7] => [1,5,2,4,6,3,7] => ? = 1
[1,3,6,4,2,7,5] => [1,4,7,6,3,2,5] => [1,4,7,6,3,2,5] => [1,5,2,4,7,3,6] => ? = 1
[1,3,6,4,7,2,5] => [1,4,7,3,2,6,5] => [1,4,7,3,2,6,5] => [1,6,2,4,7,5,3] => ? = 0
[1,3,6,5,7,2,4] => [1,7,5,3,2,6,4] => [1,7,4,3,2,6,5] => [1,6,2,3,7,5,4] => ? = 0
[1,3,6,5,7,4,2] => [1,7,5,4,6,3,2] => [1,7,3,2,6,5,4] => [1,6,2,7,4,5,3] => ? = 0
[1,3,6,7,2,4,5] => [1,7,3,2,6,4,5] => [1,7,3,2,6,4,5] => [1,6,2,5,7,4,3] => ? = 1
[1,3,6,7,4,2,5] => [1,7,4,6,3,2,5] => [1,7,4,6,3,2,5] => [1,5,2,6,7,3,4] => ? = 1
[1,3,6,7,5,2,4] => [1,5,7,3,2,6,4] => [1,4,7,3,2,6,5] => [1,6,2,4,7,5,3] => ? = 0
[1,3,6,7,5,4,2] => [1,5,7,4,6,3,2] => [1,3,7,2,6,5,4] => [1,6,3,7,4,5,2] => ? = 0
[1,3,7,2,6,5,4] => [1,6,5,7,3,2,4] => [1,5,4,7,3,2,6] => [1,6,2,5,4,7,3] => ? = 1
[1,3,7,4,2,5,6] => [1,4,7,3,2,5,6] => [1,4,7,3,2,5,6] => [1,5,2,4,6,7,3] => ? = 2
[1,3,7,5,2,4,6] => [1,5,3,2,7,4,6] => [1,5,3,2,7,4,6] => [1,5,2,6,3,7,4] => ? = 1
[1,3,7,5,4,2,6] => [1,5,4,7,3,2,6] => [1,5,4,7,3,2,6] => [1,6,2,5,4,7,3] => ? = 1
[1,3,7,6,2,4,5] => [1,6,3,2,7,4,5] => [1,5,3,2,7,4,6] => [1,5,2,6,3,7,4] => ? = 1
[1,3,7,6,4,2,5] => [1,6,4,7,3,2,5] => [1,5,4,7,3,2,6] => [1,6,2,5,4,7,3] => ? = 1
[1,4,2,6,7,5,3] => [1,7,5,6,4,2,3] => [1,7,3,6,5,2,4] => [1,4,6,7,2,5,3] => ? = 1
[1,4,2,7,6,5,3] => [1,6,5,7,4,2,3] => [1,4,3,7,6,2,5] => [1,5,4,3,7,2,6] => ? = 1
[1,4,5,2,6,7,3] => [1,7,6,5,2,4,3] => [1,7,6,5,2,4,3] => [1,4,7,3,2,5,6] => ? = 0
[1,4,5,2,7,6,3] => [1,6,7,5,2,4,3] => [1,3,7,6,2,5,4] => [1,5,3,7,4,2,6] => ? = 0
[1,4,5,3,6,7,2] => [1,7,6,5,3,4,2] => [1,7,6,5,2,4,3] => [1,4,7,3,2,5,6] => ? = 0
[1,4,5,3,7,6,2] => [1,6,7,5,3,4,2] => [1,3,7,6,2,5,4] => [1,5,3,7,4,2,6] => ? = 0
[1,4,5,6,2,3,7] => [1,6,2,5,4,3,7] => [1,6,2,5,4,3,7] => [1,5,6,3,4,2,7] => ? = 0
[1,4,5,6,2,7,3] => [1,7,6,2,5,4,3] => [1,7,6,2,5,4,3] => [1,5,7,3,4,2,6] => ? = 0
[1,4,5,6,3,2,7] => [1,6,3,5,4,2,7] => [1,6,2,5,4,3,7] => [1,5,6,3,4,2,7] => ? = 0
[1,4,5,6,3,7,2] => [1,7,6,3,5,4,2] => [1,7,6,2,5,4,3] => [1,5,7,3,4,2,6] => ? = 0
[1,4,5,6,7,2,3] => [1,7,2,6,5,4,3] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 0
[1,4,5,6,7,3,2] => [1,7,3,6,5,4,2] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 0
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000365
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [3,2,1] => [1,2,3] => [1,2,3] => 1
[3,2,1] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 0
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 0
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => [2,1,4,3] => 0
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 1
[1,4,3,2] => [1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 0
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 0
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 0
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[2,4,1,3] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 1
[2,4,3,1] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,1,2,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 1
[3,1,4,2] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 1
[3,2,1,4] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[3,2,4,1] => [2,4,1,3] => [3,1,4,2] => [2,1,4,3] => 0
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,4,2,1] => [4,1,3,2] => [2,3,1,4] => [1,3,2,4] => 0
[4,1,2,3] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 2
[4,1,3,2] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,2,1,3] => [2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 1
[4,2,3,1] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,3,1,2] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,2,1] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => [2,1,5,4,3] => 0
[1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,2,1,4,3] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => [3,2,1,5,4] => 0
[1,3,5,2,4] => [1,5,4,2,3] => [3,2,4,5,1] => [2,1,3,5,4] => 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2,5,4,1] => [2,1,5,4,3] => 0
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => 1
[1,4,2,5,3] => [1,5,3,2,4] => [4,2,3,5,1] => [3,1,2,5,4] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => 0
[1,4,3,5,2] => [1,3,5,2,4] => [4,2,5,3,1] => [2,1,5,4,3] => 0
[1,4,5,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => [2,5,1,4,3] => 0
[1,4,5,3,2] => [1,5,2,4,3] => [3,4,2,5,1] => [1,3,2,5,4] => 0
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 0
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 0
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => ? = 0
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,2,1,6,5,4,3] => ? = 0
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => ? = 0
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [2,1,7,6,5,4,3] => ? = 0
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,1,7,6,5,4,3] => ? = 0
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [2,7,1,6,5,4,3] => ? = 0
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => ? = 0
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 0
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 0
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => ? = 0
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [3,2,7,1,6,5,4] => ? = 0
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 0
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => ? = 0
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [3,7,2,1,6,5,4] => ? = 0
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [7,5,4,3,6,2,1] => [7,3,2,1,6,5,4] => ? = 0
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => [4,3,2,1,7,6,5] => ? = 0
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [5,4,3,6,7,2,1] => [3,2,1,4,7,6,5] => ? = 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [5,4,3,7,6,2,1] => [3,2,1,7,6,5,4] => ? = 0
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [7,4,3,5,6,2,1] => [7,2,1,3,6,5,4] => ? = 1
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [6,4,3,5,7,2,1] => [4,2,1,3,7,6,5] => ? = 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => [7,2,1,6,5,4,3] => ? = 0
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [3,2,1,7,6,5,4] => ? = 0
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [3,7,2,1,6,5,4] => ? = 0
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [4,3,5,6,7,2,1] => [2,1,3,4,7,6,5] => ? = 2
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [4,3,5,7,6,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [4,3,6,7,5,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [2,1,7,6,5,4,3] => ? = 0
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [4,3,6,5,7,2,1] => [2,1,4,3,7,6,5] => ? = 0
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [4,3,7,5,6,2,1] => [2,1,7,3,6,5,4] => ? = 0
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => ? = 1
[1,2,5,3,6,4,7] => [1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => [7,3,1,2,6,5,4] => ? = 1
[1,2,5,3,6,7,4] => [1,2,7,4,3,5,6] => [6,5,3,4,7,2,1] => [4,3,1,2,7,6,5] => ? = 1
[1,2,5,3,7,4,6] => [1,2,7,6,4,3,5] => [5,3,4,6,7,2,1] => [3,1,2,4,7,6,5] => ? = 2
[1,2,5,3,7,6,4] => [1,2,6,7,4,3,5] => [5,3,4,7,6,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 0
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => ? = 0
[1,2,5,4,6,3,7] => [1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,2,1,6,5,4,3] => ? = 0
[1,2,5,4,6,7,3] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [3,2,1,7,6,5,4] => ? = 0
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St001238
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001238: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001238: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,3,2] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,2,1,3] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,6,7,4,5] => [1,2,3,6,4,7,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 1 + 1
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 0 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,4,5,7,6,3] => [1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,6,5,7,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,6,7,3,5] => [1,2,6,3,4,7,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,6,7,5,3] => [1,2,7,3,4,6,5] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,2,4,7,5,6,3] => [1,2,5,6,7,3,4] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[1,2,4,7,6,3,5] => [1,2,7,5,6,3,4] => [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[1,2,4,7,6,5,3] => [1,2,6,5,7,3,4] => [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0 + 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
Description
The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S.
Matching statistic: St001651
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001651: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 40%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001651: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 40%
Values
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? = 0
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(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)
=> 0
[2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(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)
=> 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(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)
=> 0
[3,4,2,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 2
[4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[4,2,1,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(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)
=> 0
[4,3,1,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[4,3,2,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(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)
=> 0
[1,3,5,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 0
[1,4,5,3,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? = 2
[1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[1,5,3,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(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)
=> 0
[1,5,4,2,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 0
[1,5,4,3,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(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)
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(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)
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(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)
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(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)
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 1
[2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 0
[2,4,1,3,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 1
[2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[2,4,3,1,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 0
[2,4,3,5,1] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 0
[2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[2,4,5,3,1] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[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)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
=> ? = 2
[2,5,1,4,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 1
[2,5,3,1,4] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 1
[2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 0
[2,5,4,1,3] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 0
[2,5,4,3,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[3,1,4,2,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[3,1,4,5,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
[3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 2
[3,1,5,4,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(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)
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0
[3,2,5,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 1
[3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 0
[3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(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)
=> 0
[3,4,2,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 0
[3,4,2,5,1] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[3,4,5,1,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 0
[3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 0
[3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 1
[3,5,2,1,4] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 1
[3,5,2,4,1] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 0
Description
The Frankl number of a lattice.
For a lattice $L$ on at least two elements, this is
$$
\max_x(|L|-2|[x, 1]|),
$$
where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001845The number of join irreducibles minus the rank of a lattice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1.
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