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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St001182
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 3
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 4
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4
Description
Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001255
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 3
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 4
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
Description
The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St000235
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => [2,1] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [1,2] => 2 = 3 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 2 = 3 - 1
[2,3,1] => [3,1,2] => [3,1,2] => [3,1,2] => 3 = 4 - 1
[3,1,2] => [2,3,1] => [3,2,1] => [2,1,3] => 2 = 3 - 1
[3,2,1] => [3,2,1] => [2,3,1] => [1,2,3] => 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 3 = 4 - 1
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 4 = 5 - 1
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [3,1,4,2] => 3 = 4 - 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 4 = 5 - 1
[2,4,1,3] => [3,4,1,2] => [4,1,3,2] => [4,3,1,2] => 3 = 4 - 1
[2,4,3,1] => [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 4 = 5 - 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [2,1,4,3] => 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [3,4,2,1] => [3,1,2,4] => 4 = 5 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 3 = 4 - 1
[3,2,4,1] => [4,1,3,2] => [4,3,1,2] => [4,2,1,3] => 4 = 5 - 1
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 3 = 4 - 1
[3,4,2,1] => [4,2,1,3] => [2,4,1,3] => [1,4,2,3] => 4 = 5 - 1
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => [2,3,1,4] => 2 = 3 - 1
[4,1,3,2] => [3,2,4,1] => [4,3,2,1] => [3,2,1,4] => 4 = 5 - 1
[4,2,1,3] => [3,4,2,1] => [2,4,3,1] => [1,3,2,4] => 3 = 4 - 1
[4,2,3,1] => [3,1,4,2] => [3,4,1,2] => [4,1,2,3] => 4 = 5 - 1
[4,3,1,2] => [2,4,3,1] => [3,2,4,1] => [2,1,3,4] => 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [1,2,3,4] => 4 = 5 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 2 = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [2,5,4,3,1] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [2,5,3,4,1] => 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 4 = 5 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => 3 = 4 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => 4 = 5 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,2,4,3] => [3,5,4,2,1] => 3 = 4 - 1
[1,3,5,4,2] => [1,5,4,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => 4 = 5 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,4,5,3,2] => [5,4,2,3,1] => 4 = 5 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 3 = 4 - 1
[1,4,3,5,2] => [1,5,2,4,3] => [1,5,4,2,3] => [4,5,3,2,1] => 4 = 5 - 1
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => [4,3,5,2,1] => 3 = 4 - 1
Description
The number of indices that are not cyclical small weak excedances.
A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
Matching statistic: St001279
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 0 = 1 - 1
[1,2] => [1,2] => [1,2] => [1,1]
=> 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [2]
=> 2 = 3 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1]
=> 2 = 3 - 1
[2,3,1] => [3,1,2] => [3,1,2] => [3]
=> 3 = 4 - 1
[3,1,2] => [2,3,1] => [3,2,1] => [2,1]
=> 2 = 3 - 1
[3,2,1] => [3,2,1] => [2,3,1] => [3]
=> 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 3 = 4 - 1
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [3,1]
=> 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,2]
=> 4 = 5 - 1
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [3,1]
=> 3 = 4 - 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [4]
=> 4 = 5 - 1
[2,4,1,3] => [3,4,1,2] => [4,1,3,2] => [3,1]
=> 3 = 4 - 1
[2,4,3,1] => [4,3,1,2] => [3,1,4,2] => [4]
=> 4 = 5 - 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [3,4,2,1] => [4]
=> 4 = 5 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,1]
=> 3 = 4 - 1
[3,2,4,1] => [4,1,3,2] => [4,3,1,2] => [4]
=> 4 = 5 - 1
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [3,1]
=> 3 = 4 - 1
[3,4,2,1] => [4,2,1,3] => [2,4,1,3] => [4]
=> 4 = 5 - 1
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 2 = 3 - 1
[4,1,3,2] => [3,2,4,1] => [4,3,2,1] => [2,2]
=> 4 = 5 - 1
[4,2,1,3] => [3,4,2,1] => [2,4,3,1] => [3,1]
=> 3 = 4 - 1
[4,2,3,1] => [3,1,4,2] => [3,4,1,2] => [2,2]
=> 4 = 5 - 1
[4,3,1,2] => [2,4,3,1] => [3,2,4,1] => [3,1]
=> 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [4]
=> 4 = 5 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [3,1,1]
=> 3 = 4 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [3,1,1]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 4 = 5 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [3,1,1]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> 4 = 5 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,2,4,3] => [3,1,1]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,5,4,2,3] => [1,4,2,5,3] => [4,1]
=> 4 = 5 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,4,5,3,2] => [4,1]
=> 4 = 5 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [3,1,1]
=> 3 = 4 - 1
[1,4,3,5,2] => [1,5,2,4,3] => [1,5,4,2,3] => [4,1]
=> 4 = 5 - 1
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => [3,1,1]
=> 3 = 4 - 1
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000673
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 1 - 1
[1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => 2 = 3 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => 2 = 3 - 1
[2,3,1] => [3,1,2] => [2,3,1] => 3 = 4 - 1
[3,1,2] => [2,3,1] => [3,2,1] => 2 = 3 - 1
[3,2,1] => [3,2,1] => [3,1,2] => 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 3 = 4 - 1
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4 = 5 - 1
[2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 3 = 4 - 1
[2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 4 = 5 - 1
[2,4,1,3] => [3,4,1,2] => [2,4,3,1] => 3 = 4 - 1
[2,4,3,1] => [4,3,1,2] => [2,4,1,3] => 4 = 5 - 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [4,3,1,2] => 4 = 5 - 1
[3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 3 = 4 - 1
[3,2,4,1] => [4,1,3,2] => [3,4,2,1] => 4 = 5 - 1
[3,4,1,2] => [2,4,1,3] => [3,2,4,1] => 3 = 4 - 1
[3,4,2,1] => [4,2,1,3] => [3,1,4,2] => 4 = 5 - 1
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 2 = 3 - 1
[4,1,3,2] => [3,2,4,1] => [4,3,2,1] => 4 = 5 - 1
[4,2,1,3] => [3,4,2,1] => [4,1,3,2] => 3 = 4 - 1
[4,2,3,1] => [3,1,4,2] => [3,4,1,2] => 4 = 5 - 1
[4,3,1,2] => [2,4,3,1] => [4,2,1,3] => 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 4 = 5 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4 = 5 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => 3 = 4 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => 4 = 5 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,3,5,4,2] => 3 = 4 - 1
[1,3,5,4,2] => [1,5,4,2,3] => [1,3,5,2,4] => 4 = 5 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,2,3] => 4 = 5 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => 3 = 4 - 1
[1,4,3,5,2] => [1,5,2,4,3] => [1,4,5,3,2] => 4 = 5 - 1
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,3,5,2] => 3 = 4 - 1
[1,4,5,3,2] => [1,5,3,2,4] => [1,4,2,5,3] => 4 = 5 - 1
Description
The number of non-fixed points of a permutation.
In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St001005
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1] => ? = 1 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2,1] => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3 = 4 - 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 4 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 5 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3 = 4 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4 = 5 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3 = 4 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 4 = 5 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 5 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3 = 4 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 4 = 5 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3 = 4 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 4 = 5 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4 = 5 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 4 = 5 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3 = 4 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4 = 5 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 4 = 5 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 4 = 5 - 1
Description
The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.
Matching statistic: St000896
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
=> ? = 1 - 1
[1,2] => [1,2] => [1,2] => [[1,0],[0,1]]
=> 0 = 1 - 1
[2,1] => [2,1] => [2,1] => [[0,1],[1,0]]
=> 2 = 3 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 3 - 1
[2,1,3] => [2,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 3 - 1
[2,3,1] => [3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 3 = 4 - 1
[3,1,2] => [2,3,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 2 = 3 - 1
[3,2,1] => [3,2,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3 = 4 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 4 - 1
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 4 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 4 = 5 - 1
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 4 - 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 4 = 5 - 1
[2,4,1,3] => [3,4,1,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 3 = 4 - 1
[2,4,3,1] => [4,3,1,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 4 = 5 - 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2 = 3 - 1
[3,1,4,2] => [4,2,3,1] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 4 = 5 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3 = 4 - 1
[3,2,4,1] => [4,1,3,2] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 3 = 4 - 1
[3,4,2,1] => [4,2,1,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 4 = 5 - 1
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2 = 3 - 1
[4,1,3,2] => [3,2,4,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 5 - 1
[4,2,1,3] => [3,4,2,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 3 = 4 - 1
[4,2,3,1] => [3,1,4,2] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 4 = 5 - 1
[4,3,1,2] => [2,4,3,1] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3 = 4 - 1
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4 = 5 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2 = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2 = 3 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 3 = 4 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 4 = 5 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 4 = 5 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,5,4,2,3] => [1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 4 = 5 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 4 = 5 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3 = 4 - 1
[1,4,3,5,2] => [1,5,2,4,3] => [1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> 4 = 5 - 1
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 3 = 4 - 1
[1,4,5,3,2] => [1,5,3,2,4] => [1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 4 = 5 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 4 - 1
[1,2,3,6,4,5] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 1
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 5 - 1
[1,2,4,6,3,5] => [1,2,5,6,3,4] => [1,2,6,3,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 4 - 1
[1,2,4,6,5,3] => [1,2,6,5,3,4] => [1,2,5,3,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 5 - 1
[1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,5,6,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 5 - 1
[1,2,5,4,6,3] => [1,2,6,3,5,4] => [1,2,6,5,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5 - 1
[1,2,5,6,3,4] => [1,2,4,6,3,5] => [1,2,6,4,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 4 - 1
[1,2,5,6,4,3] => [1,2,6,4,3,5] => [1,2,4,6,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 5 - 1
[1,2,6,3,4,5] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 3 - 1
[1,2,6,3,5,4] => [1,2,5,4,6,3] => [1,2,6,5,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5 - 1
[1,2,6,4,3,5] => [1,2,5,6,4,3] => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 4 - 1
[1,2,6,4,5,3] => [1,2,5,3,6,4] => [1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 5 - 1
[1,2,6,5,3,4] => [1,2,4,6,5,3] => [1,2,5,4,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 1
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 5 - 1
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 6 - 1
[1,3,2,6,4,5] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 5 - 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 6 - 1
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 6 - 1
[1,3,4,6,2,5] => [1,5,6,2,3,4] => [1,6,2,3,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 5 - 1
[1,3,4,6,5,2] => [1,6,5,2,3,4] => [1,5,2,3,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 6 - 1
[1,3,5,2,6,4] => [1,6,4,5,2,3] => [1,5,2,6,4,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 6 - 1
[1,3,5,4,6,2] => [1,6,2,3,5,4] => [1,6,2,5,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 6 - 1
[1,3,5,6,2,4] => [1,4,6,2,3,5] => [1,6,2,4,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 5 - 1
[1,3,5,6,4,2] => [1,6,4,2,3,5] => [1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 6 - 1
[1,3,6,2,4,5] => [1,4,5,6,2,3] => [1,6,2,4,5,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 4 - 1
[1,3,6,2,5,4] => [1,5,4,6,2,3] => [1,6,2,5,4,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 6 - 1
[1,3,6,4,2,5] => [1,5,6,4,2,3] => [1,4,2,6,5,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 5 - 1
[1,3,6,4,5,2] => [1,5,2,3,6,4] => [1,5,2,6,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 6 - 1
[1,3,6,5,2,4] => [1,4,6,5,2,3] => [1,5,2,4,6,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 5 - 1
[1,3,6,5,4,2] => [1,6,5,4,2,3] => [1,4,2,5,6,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 6 - 1
[1,4,2,5,6,3] => [1,6,3,4,2,5] => [1,4,6,3,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 6 - 1
[1,4,2,6,3,5] => [1,5,6,3,4,2] => [1,4,6,3,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 5 - 1
[1,4,2,6,5,3] => [1,6,5,3,4,2] => [1,4,5,3,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 6 - 1
[1,4,3,5,6,2] => [1,6,2,4,3,5] => [1,6,4,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 6 - 1
[1,4,3,6,2,5] => [1,5,6,2,4,3] => [1,6,4,2,5,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 5 - 1
[1,4,3,6,5,2] => [1,6,5,2,4,3] => [1,5,4,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 6 - 1
[1,4,5,2,6,3] => [1,6,3,5,2,4] => [1,5,6,2,3,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 6 - 1
[1,4,5,3,6,2] => [1,6,2,5,3,4] => [1,6,5,3,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 6 - 1
[1,4,5,6,2,3] => [1,3,6,2,4,5] => [1,6,3,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 5 - 1
[1,4,5,6,3,2] => [1,6,3,2,4,5] => [1,3,6,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 6 - 1
[1,4,6,2,3,5] => [1,3,5,6,2,4] => [1,6,3,2,5,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 4 - 1
[1,4,6,2,5,3] => [1,5,3,6,2,4] => [1,6,5,2,3,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 6 - 1
[1,4,6,3,2,5] => [1,5,6,3,2,4] => [1,3,6,2,5,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 5 - 1
[1,4,6,3,5,2] => [1,5,2,6,3,4] => [1,5,6,3,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 6 - 1
[1,4,6,5,2,3] => [1,3,6,5,2,4] => [1,5,3,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 5 - 1
[1,4,6,5,3,2] => [1,6,5,3,2,4] => [1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 6 - 1
[1,5,2,3,6,4] => [1,3,6,4,5,2] => [1,5,3,6,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 5 - 1
Description
The number of zeros on the main diagonal of an alternating sign matrix.
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