Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
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Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 6
[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[3,2,4,1,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[3,2,5,1,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[3,4,2,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 6
[3,5,1,2,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[3,5,2,1,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[4,1,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[4,2,1,3,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[4,2,3,1,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[4,2,3,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4
[4,2,5,1,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001861
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Values
[1] => [1] => [1] => [-1] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => [-2,1] => 2 = 1 + 1
[2,1] => [2,1] => [2,1] => [1,-2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [-3,1,2] => 3 = 2 + 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,-3,2] => 2 = 1 + 1
[2,3,1] => [3,1,2] => [3,1,2] => [2,-3,1] => 3 = 2 + 1
[3,1,2] => [2,3,1] => [2,3,1] => [2,1,-3] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,2,-3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 4 = 3 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,-4,2,3] => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [2,-4,1,3] => 4 = 3 + 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [3,-4,1,2] => 5 = 4 + 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [2,-4,3,1] => 4 = 3 + 1
[3,1,2,4] => [2,3,1,4] => [2,3,1,4] => [2,1,-4,3] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 1 + 1
[3,2,4,1] => [4,2,1,3] => [4,2,1,3] => [1,3,-4,2] => 3 = 2 + 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [3,2,-4,1] => 4 = 3 + 1
[3,4,2,1] => [4,3,1,2] => [4,3,1,2] => [2,3,-4,1] => 4 = 3 + 1
[4,1,2,3] => [2,3,4,1] => [2,3,4,1] => [3,1,2,-4] => 3 = 2 + 1
[4,2,1,3] => [3,2,4,1] => [3,2,4,1] => [1,3,2,-4] => 2 = 1 + 1
[4,2,3,1] => [4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 3 = 2 + 1
[4,3,1,2] => [3,4,2,1] => [3,4,2,1] => [2,1,3,-4] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [-5,1,2,3,4] => ? = 4 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [1,-5,2,3,4] => 4 = 3 + 1
[2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [2,-5,1,3,4] => ? = 5 + 1
[2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => [3,-5,1,2,4] => ? = 5 + 1
[2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => [4,-5,1,2,3] => ? = 6 + 1
[2,3,5,1,4] => [4,1,2,5,3] => [4,1,2,5,3] => [3,-5,1,4,2] => ? = 5 + 1
[2,4,1,3,5] => [3,1,4,2,5] => [3,1,4,2,5] => [2,-5,3,1,4] => ? = 5 + 1
[2,4,5,1,3] => [4,1,5,2,3] => [4,1,5,2,3] => [3,-5,4,1,2] => ? = 5 + 1
[2,5,1,3,4] => [3,1,4,5,2] => [3,1,4,5,2] => [2,-5,3,4,1] => ? = 5 + 1
[3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => [2,1,-5,3,4] => ? = 3 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 3 = 2 + 1
[3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 4 = 3 + 1
[3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => [1,4,-5,2,3] => 5 = 4 + 1
[3,2,5,1,4] => [4,2,1,5,3] => [4,2,1,5,3] => [1,3,-5,4,2] => 4 = 3 + 1
[3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => [3,2,-5,1,4] => ? = 5 + 1
[3,4,2,1,5] => [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 4 + 1
[3,4,2,5,1] => [5,3,1,2,4] => [5,3,1,2,4] => [2,4,-5,1,3] => ? = 4 + 1
[3,4,5,1,2] => [4,5,1,2,3] => [4,5,1,2,3] => [4,3,-5,1,2] => ? = 5 + 1
[3,4,5,2,1] => [5,4,1,2,3] => [5,4,1,2,3] => [3,4,-5,1,2] => ? = 6 + 1
[3,5,1,2,4] => [3,4,1,5,2] => [3,4,1,5,2] => [3,2,-5,4,1] => ? = 5 + 1
[3,5,2,1,4] => [4,3,1,5,2] => [4,3,1,5,2] => [2,3,-5,4,1] => ? = 4 + 1
[3,5,2,4,1] => [5,3,1,4,2] => [5,3,1,4,2] => [2,4,-5,3,1] => ? = 4 + 1
[4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => [3,1,2,-5,4] => ? = 3 + 1
[4,2,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => [1,3,2,-5,4] => 3 = 2 + 1
[4,2,3,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => [2,3,1,-5,4] => ? = 3 + 1
[4,2,3,5,1] => [5,2,3,1,4] => [5,2,3,1,4] => [2,4,1,-5,3] => ? = 4 + 1
[4,2,5,1,3] => [4,2,5,1,3] => [4,2,5,1,3] => [1,4,3,-5,2] => 4 = 3 + 1
[4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => [2,1,3,-5,4] => ? = 2 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => 2 = 1 + 1
[4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => 3 = 2 + 1
[4,3,5,1,2] => [4,5,2,1,3] => [4,5,2,1,3] => [3,1,4,-5,2] => ? = 3 + 1
[4,3,5,2,1] => [5,4,2,1,3] => [5,4,2,1,3] => [1,3,4,-5,2] => 4 = 3 + 1
[4,5,1,2,3] => [3,4,5,1,2] => [3,4,5,1,2] => [4,2,3,-5,1] => ? = 5 + 1
[4,5,2,1,3] => [4,3,5,1,2] => [4,3,5,1,2] => [2,4,3,-5,1] => ? = 4 + 1
[4,5,2,3,1] => [5,3,4,1,2] => [5,3,4,1,2] => [3,4,2,-5,1] => ? = 4 + 1
[4,5,3,1,2] => [4,5,3,1,2] => [4,5,3,1,2] => [3,2,4,-5,1] => ? = 4 + 1
[4,5,3,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 4 + 1
[5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => [4,1,2,3,-5] => ? = 3 + 1
[5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => [1,4,2,3,-5] => 3 = 2 + 1
[5,2,3,1,4] => [4,2,3,5,1] => [4,2,3,5,1] => [2,4,1,3,-5] => ? = 3 + 1
[5,2,3,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => [3,4,1,2,-5] => ? = 4 + 1
[5,2,4,1,3] => [4,2,5,3,1] => [4,2,5,3,1] => [2,4,3,1,-5] => ? = 3 + 1
[5,3,1,2,4] => [3,4,2,5,1] => [3,4,2,5,1] => [2,1,4,3,-5] => ? = 2 + 1
[5,3,2,1,4] => [4,3,2,5,1] => [4,3,2,5,1] => [1,2,4,3,-5] => 2 = 1 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,-5] => 3 = 2 + 1
[5,3,4,1,2] => [4,5,2,3,1] => [4,5,2,3,1] => [3,2,4,1,-5] => ? = 3 + 1
[5,3,4,2,1] => [5,4,2,3,1] => [5,4,2,3,1] => [2,3,4,1,-5] => ? = 3 + 1
[5,4,1,2,3] => [3,4,5,2,1] => [3,4,5,2,1] => [3,1,2,4,-5] => ? = 2 + 1
[5,4,2,1,3] => [4,3,5,2,1] => [4,3,5,2,1] => [1,3,2,4,-5] => 2 = 1 + 1
[5,4,2,3,1] => [5,3,4,2,1] => [5,3,4,2,1] => [2,3,1,4,-5] => ? = 2 + 1
[5,4,3,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => [2,1,3,4,-5] => ? = 1 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,-5] => 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [-6,1,2,3,4,5] => ? = 5 + 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,-6,2,3,4,5] => ? = 4 + 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => [2,-6,1,3,4,5] => ? = 7 + 1
[2,3,4,1,5,6] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => [3,-6,1,2,4,5] => ? = 8 + 1
[2,3,4,5,1,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => [4,-6,1,2,3,5] => ? = 7 + 1
[2,3,4,5,6,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => [5,-6,1,2,3,4] => ? = 8 + 1
[2,3,4,6,1,5] => [5,1,2,3,6,4] => [5,1,2,3,6,4] => [4,-6,1,2,5,3] => ? = 7 + 1
[2,3,5,1,4,6] => [4,1,2,5,3,6] => [4,1,2,5,3,6] => [3,-6,1,4,2,5] => ? = 8 + 1
[2,3,5,6,1,4] => [5,1,2,6,3,4] => [5,1,2,6,3,4] => [4,-6,1,5,2,3] => ? = 7 + 1
[2,3,6,1,4,5] => [4,1,2,5,6,3] => [4,1,2,5,6,3] => [3,-6,1,4,5,2] => ? = 8 + 1
[2,4,1,3,5,6] => [3,1,4,2,5,6] => [3,1,4,2,5,6] => [2,-6,3,1,4,5] => ? = 7 + 1
[2,4,5,1,3,6] => [4,1,5,2,3,6] => [4,1,5,2,3,6] => [3,-6,4,1,2,5] => ? = 8 + 1
[2,4,5,6,1,3] => [5,1,6,2,3,4] => [5,1,6,2,3,4] => [4,-6,5,1,2,3] => ? = 7 + 1
Description
The number of Bruhat lower covers of a permutation. This is, for a signed permutation $\pi$, the number of signed permutations $\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from $\pi$.