Processing math: 66%

Your data matches 125 different statistics following compositions of up to 3 maps.
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Matching statistic: St001604
St001604: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3]
=> 1
[2,1]
=> 0
[1,1,1]
=> 0
[4]
=> 1
[3,1]
=> 0
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 1
[4,1]
=> 0
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 1
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 0
[4,2]
=> 2
[4,1,1]
=> 0
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 1
[2,2,2]
=> 2
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 1
[6,1]
=> 0
[5,2]
=> 2
[5,1,1]
=> 0
[4,3]
=> 1
[4,2,1]
=> 3
[4,1,1,1]
=> 1
[3,3,1]
=> 0
[3,2,2]
=> 3
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 0
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 1
[7,1]
=> 0
[6,2]
=> 3
[6,1,1]
=> 0
[5,3]
=> 1
[5,2,1]
=> 4
[5,1,1,1]
=> 2
[4,4]
=> 3
[4,3,1]
=> 3
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00121: Dyck paths Cori-Le Borgne involutionDyck paths
Mp00201: Dyck paths RingelPermutations
St000871: Permutations ⟶ ℤResult quality: 1% values known / values provided: 6%distinct values known / distinct values provided: 1%
Values
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? ∊ {0,1,2}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? ∊ {0,1,2}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? ∊ {0,1,2}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? ∊ {0,0,0,1,3,3,3}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ? ∊ {0,0,0,1,3,3,3}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? ∊ {0,0,0,1,3,3,3}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? ∊ {0,0,0,1,3,3,3}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? ∊ {0,0,0,1,3,3,3}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? ∊ {0,0,0,1,3,3,3}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => ? ∊ {0,0,0,1,3,3,3}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 0
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,8,2,3,4,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,8,5,1,2,3,4,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,2,8,3,4,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [8,5,4,1,2,3,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [8,3,4,1,2,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => 0
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,7,8,1,2,3,4,5,6] => 0
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,10,2,3,4,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,7,9,1,2,3,4,5,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,9,3,4,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,1,6,5,2,3,4,7] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,8,6,5,1,2,3,4] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,9,5,1,2,3,4,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,3,9,4,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,1,4,5,2,3,6,7] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,8,4,5,1,2,3,6] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [9,5,4,1,2,3,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [7,3,4,8,1,2,5,6] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [9,3,4,1,2,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,9,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [2,3,7,8,1,4,5,6] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,9,1,4,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [10,1,2,3,4,5,6,9,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => 0
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,10,1,2,3,4,5,6,7,8,9] => ? ∊ {0,0,1,2,2,2,3,4,4,6,6,6,6,8,8,8,8,9,10,10,12,13,14,14,14,16,18,20,20,20,22,22,22,26,26,29,31,38}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [10,9,8,1,2,3,4,5,6,7] => 0
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => 1
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [7,8,9,10,1,2,3,4,5,6] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 0
Description
The number of very big ascents of a permutation. A very big ascent of a permutation π is an index i such that πi+1πi>2. For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General r-ascents were for example be studied in [1, Section 2].
Matching statistic: St001083
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00121: Dyck paths Cori-Le Borgne involutionDyck paths
Mp00201: Dyck paths RingelPermutations
St001083: Permutations ⟶ ℤResult quality: 1% values known / values provided: 6%distinct values known / distinct values provided: 1%
Values
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? ∊ {1,2,2}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? ∊ {1,2,2}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? ∊ {1,2,2}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? ∊ {0,0,1,1,3,3,3}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ? ∊ {0,0,1,1,3,3,3}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? ∊ {0,0,1,1,3,3,3}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? ∊ {0,0,1,1,3,3,3}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => 0
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? ∊ {0,0,1,1,3,3,3}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? ∊ {0,0,1,1,3,3,3}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => ? ∊ {0,0,1,1,3,3,3}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 0
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,8,2,3,4,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,8,5,1,2,3,4,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,2,8,3,4,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [8,5,4,1,2,3,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [8,3,4,1,2,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => ? ∊ {0,2,2,3,3,3,3,4,4,4,4,4,5,7}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => 0
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,7,8,1,2,3,4,5,6] => 0
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,10,2,3,4,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,7,9,1,2,3,4,5,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,9,3,4,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [8,1,6,5,2,3,4,7] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,8,6,5,1,2,3,4] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,9,5,1,2,3,4,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,3,9,4,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [8,1,4,5,2,3,6,7] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [7,8,4,5,1,2,3,6] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [9,5,4,1,2,3,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [7,3,4,8,1,2,5,6] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [9,3,4,1,2,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,9,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [2,3,7,8,1,4,5,6] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,9,1,4,5,6,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [10,1,2,3,4,5,6,9,7,8] => ? ∊ {0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => 0
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [11,10,1,2,3,4,5,6,7,8,9] => ? ∊ {0,0,1,1,2,2,3,4,4,6,6,6,6,8,8,8,8,9,10,10,12,13,14,14,14,16,18,20,20,20,22,22,22,26,26,29,31,38}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [10,9,8,1,2,3,4,5,6,7] => 0
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => 1
[7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [7,8,9,10,1,2,3,4,5,6] => 0
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 0
Description
The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern 132 such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St001816
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 0% values known / values provided: 6%distinct values known / distinct values provided: 0%
Values
[3]
=> []
=> []
=> ?
=> ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[4]
=> []
=> []
=> ?
=> ? ∊ {0,1}
[3,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? ∊ {0,1}
[5]
=> []
=> []
=> ?
=> ? ∊ {1,1,1}
[4,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? ∊ {1,1,1}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? ∊ {1,1,1}
[6]
=> []
=> []
=> ?
=> ? ∊ {0,0,1,1,1,2,2}
[5,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? ∊ {0,0,1,1,1,2,2}
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? ∊ {0,0,1,1,1,2,2}
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? ∊ {0,0,1,1,1,2,2}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? ∊ {0,0,1,1,1,2,2}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? ∊ {0,0,1,1,1,2,2}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? ∊ {0,0,1,1,1,2,2}
[7]
=> []
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[6,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? ∊ {0,0,0,1,1,1,2,2,3,3,3}
[8]
=> []
=> []
=> ?
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[7,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? ∊ {0,0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[9]
=> []
=> []
=> ?
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[8,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[9,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[10,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[11,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[12,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[11,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[13,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[12,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[12,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[14,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 1
[13,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[15,1]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
Description
Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition λ has dimension equal to the number of standard tableaux of shape λ. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape λ; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St000245
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
St000245: Permutations ⟶ ℤResult quality: 0% values known / values provided: 5%distinct values known / distinct values provided: 0%
Values
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,3,2] => 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,4,2] => 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,1,4,3,2] => 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => ? ∊ {1,1} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6,3,5,1,4,2] => 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? ∊ {1,1} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,8,7,6,5,4,1,3] => ? ∊ {0,1,2,2} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [7,3,6,5,1,4,2] => ? ∊ {0,1,2,2} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,6,5,1,4,3] => 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,5,2] => 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? ∊ {0,1,2,2} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,8,7,6,5,4,2] => ? ∊ {0,1,2,2} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [2,9,8,7,6,5,4,1,3] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [8,3,7,6,5,1,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,7,6,5,1,4,3] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [5,3,7,1,6,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [6,4,1,5,3,2] => 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,6,4,2] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,7,6,5,4,3,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,9,8,7,6,5,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [2,10,9,8,7,6,5,4,1,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [9,3,8,7,6,5,1,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [2,8,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [6,3,8,7,1,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,5,4,2] => 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [9,1,8,7,6,5,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,10,9,8,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [2,11,10,9,8,7,6,5,4,1,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => [10,3,9,8,7,6,5,1,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => [2,9,8,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [7,3,9,8,6,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => [2,8,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => [8,6,4,7,1,5,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [5,3,8,1,7,6,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [7,3,6,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [6,3,7,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,1,8,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1 = 0 + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [8,1,7,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,1,9,8,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
Description
The number of ascents of a permutation.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
St000834: Permutations ⟶ ℤResult quality: 0% values known / values provided: 5%distinct values known / distinct values provided: 0%
Values
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,3,2] => 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,4,2] => 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,1,4,3,2] => 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => ? ∊ {1,1} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6,3,5,1,4,2] => 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? ∊ {1,1} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,8,7,6,5,4,1,3] => ? ∊ {0,1,2,2} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [7,3,6,5,1,4,2] => ? ∊ {0,1,2,2} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,6,5,1,4,3] => 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,5,2] => 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? ∊ {0,1,2,2} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,8,7,6,5,4,2] => ? ∊ {0,1,2,2} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [2,9,8,7,6,5,4,1,3] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [8,3,7,6,5,1,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,7,6,5,1,4,3] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [5,3,7,1,6,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [6,4,1,5,3,2] => 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,6,4,2] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,7,6,5,4,3,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,9,8,7,6,5,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [2,10,9,8,7,6,5,4,1,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [9,3,8,7,6,5,1,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [2,8,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [6,3,8,7,1,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,5,4,2] => 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [9,1,8,7,6,5,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,10,9,8,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [2,11,10,9,8,7,6,5,4,1,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => [10,3,9,8,7,6,5,1,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => [2,9,8,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [7,3,9,8,6,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => [2,8,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => [8,6,4,7,1,5,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [5,3,8,1,7,6,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [7,3,6,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [6,3,7,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 2 = 1 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1 = 0 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,1,8,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 1 = 0 + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [8,1,7,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,1,9,8,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 1 = 0 + 1
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
[4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => 1 = 0 + 1
Description
The number of right outer peaks of a permutation. A right outer peak in a permutation w=[w1,...,wn] is either a position i such that wi1<wi>wi+1 or n if wn>wn1. In other words, it is a peak in the word [w1,...,wn,0].
Matching statistic: St000742
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
St000742: Permutations ⟶ ℤResult quality: 0% values known / values provided: 5%distinct values known / distinct values provided: 0%
Values
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,5,4,1,3] => 3 = 1 + 2
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,3,2] => 2 = 0 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 2 = 0 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,6,5,4,1,3] => 3 = 1 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,3,1,4,2] => 2 = 0 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,5,1,4,3] => 3 = 1 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,1,4,3,2] => 2 = 0 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 2 = 0 + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => ? ∊ {1,1} + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [6,3,5,1,4,2] => 3 = 1 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,4,3] => 3 = 1 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 2 = 0 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 2 = 0 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 2 = 0 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? ∊ {1,1} + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [2,8,7,6,5,4,1,3] => ? ∊ {0,1,2,2} + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [7,3,6,5,1,4,2] => ? ∊ {0,1,2,2} + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [2,6,5,1,4,3] => 3 = 1 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,5,2] => 2 = 0 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,6,5,1,4,3] => 3 = 1 + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 2 = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 2 = 0 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 3 = 1 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 2 = 0 + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? ∊ {0,1,2,2} + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,8,7,6,5,4,2] => ? ∊ {0,1,2,2} + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [2,9,8,7,6,5,4,1,3] => ? ∊ {0,1,1,2,2,3,3,3} + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [8,3,7,6,5,1,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [2,7,6,5,1,4,3] => ? ∊ {0,1,1,2,2,3,3,3} + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [5,3,7,1,6,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,5,1,4,3] => 3 = 1 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [6,4,1,5,3,2] => 2 = 0 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 2 = 0 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,6,4,2] => 2 = 0 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 3 = 1 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 2 = 0 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 2 = 0 + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? ∊ {0,1,1,2,2,3,3,3} + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,7,6,5,4,3,2] => ? ∊ {0,1,1,2,2,3,3,3} + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,9,8,7,6,5,4,2] => ? ∊ {0,1,1,2,2,3,3,3} + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [2,10,9,8,7,6,5,4,1,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => [9,3,8,7,6,5,1,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => [2,8,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [6,3,8,7,1,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => [2,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [2,7,6,5,1,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,5,4,2] => 2 = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 3 = 1 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 2 = 0 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 3 = 1 + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 2 = 0 + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 2 = 0 + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,1,8,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [9,1,8,7,6,5,4,3,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,10,9,8,7,6,5,4,2] => ? ∊ {1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [2,11,10,9,8,7,6,5,4,1,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => [10,3,9,8,7,6,5,1,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => [2,9,8,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [7,3,9,8,6,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => [2,8,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => [8,6,4,7,1,5,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [5,3,8,1,7,6,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => [2,7,6,5,1,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [7,3,6,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [6,3,7,1,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 3 = 1 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 2 = 0 + 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 2 = 0 + 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,1,8,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 2 = 0 + 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [8,1,7,6,5,4,3,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,1,9,8,7,6,5,4,2] => ? ∊ {0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 2 = 0 + 2
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
[4,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => 2 = 0 + 2
Description
The number of big ascents of a permutation after prepending zero. Given a permutation π of {1,,n} we set π(0)=0 and then count the number of indices i{0,,n1} such that π(i+1)π(i)>1. It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]). G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]]. [[St000646]] is the statistic without the border condition π(0)=0.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00201: Dyck paths RingelPermutations
St001085: Permutations ⟶ ℤResult quality: 0% values known / values provided: 5%distinct values known / distinct values provided: 0%
Values
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? ∊ {0,1,2,2}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? ∊ {0,1,2,2}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? ∊ {0,1,2,2}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? ∊ {0,1,2,2}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,3,4,5,8,7,1,6] => ? ∊ {0,0,0,2,2,3,3,3}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => ? ∊ {0,0,0,2,2,3,3,3}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,3,4,8,6,7,1,5] => ? ∊ {0,0,0,2,2,3,3,3}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? ∊ {0,0,0,2,2,3,3,3}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => ? ∊ {0,0,0,2,2,3,3,3}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,0,2,2,3,3,3}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,8,4,5,6,7,1,3] => ? ∊ {0,0,0,2,2,3,3,3}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? ∊ {0,0,0,2,2,3,3,3}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,9,8,1,7] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [2,3,4,8,1,7,5,6] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [2,3,8,1,6,7,4,5] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,1,5] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,8,1,5,6,7,3,4] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [2,3,9,5,6,7,8,1,4] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,9,4,5,6,7,8,1,3] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? ∊ {0,1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,9,1,7,8] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,7,10,9,1,8] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [2,3,4,5,9,1,8,6,7] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,6,10,8,9,1,7] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [2,3,8,6,1,7,4,5] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [2,3,4,9,1,7,8,5,6] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [2,3,4,5,10,7,8,9,1,6] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [2,8,5,1,6,7,3,4] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,3,8,1,4,7,5,6] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,9,1,6,7,8,4,5] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [2,3,4,10,6,7,8,9,1,5] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [8,4,1,5,6,7,2,3] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,8,1,3,6,7,4,5] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,9,1,5,6,7,8,3,4] => ? ∊ {0,0,0,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => 0
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 0
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 0
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 0
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,8,1,3,4,5,6,7] => 1
[4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => 0
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 0
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 0
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern 213, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00123: Dyck paths Barnabei-Castronuovo involutionDyck paths
Mp00201: Dyck paths RingelPermutations
St001096: Permutations ⟶ ℤResult quality: 1% values known / values provided: 5%distinct values known / distinct values provided: 1%
Values
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2 = 1 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2 = 1 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 1 = 0 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 1 = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 2 = 1 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? ∊ {0,2,2} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? ∊ {0,2,2} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? ∊ {0,2,2} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 1 = 0 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 1 = 0 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => ? ∊ {0,0,2,3,3,3} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => 2 = 1 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => ? ∊ {0,0,2,3,3,3} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 2 = 1 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ? ∊ {0,0,2,3,3,3} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => 1 = 0 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 1 = 0 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? ∊ {0,0,2,3,3,3} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [8,1,4,2,3,5,6,7] => ? ∊ {0,0,2,3,3,3} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 2 = 1 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => ? ∊ {0,0,2,3,3,3} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 1 = 0 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 1 = 0 + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,1,2,3,4,9,5,7,8] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [9,1,2,3,4,7,5,6,8] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => 1 = 0 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [7,4,1,2,3,5,8,6] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 1 = 0 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [7,3,8,6,1,2,4,5] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,8,4,1,2,3,5,6,7] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1 = 0 + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,4,1,8,3,5,6,7] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,4,1,8,2,3,5,6,7] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1 = 0 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [8,4,1,2,9,3,5,6,7] => ? ∊ {1,1,1,2,2,3,3,3,3,4,4,4,4,4,5,7} + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => 2 = 1 + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => 1 = 0 + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [10,6,1,2,3,4,9,5,7,8] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,1,2,3,4,7,9,5,8] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [10,7,1,2,3,4,9,5,6,8] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [8,4,1,2,3,7,5,6] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [8,1,2,3,4,7,5,9,6] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [9,10,1,2,3,4,8,5,6,7] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [7,3,8,5,1,2,4,6] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [6,4,1,2,3,7,8,5] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [9,7,4,1,2,3,5,6,8] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [5,10,1,2,3,4,6,7,8,9] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [8,3,1,7,6,2,4,5] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [8,3,6,5,1,2,4,7] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [9,1,4,8,2,3,5,6,7] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [10,1,5,2,3,4,6,7,8,9] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1 = 0 + 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,3,1,2,4,5,6,7] => 1 = 0 + 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,1,8,7,6,2,4,5] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,8,4,1,9,3,5,6,7] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [10,1,2,5,3,4,6,7,8,9] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [9,3,1,8,2,4,5,6,7] => ? ∊ {0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 1 = 0 + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => 1 = 0 + 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 1 + 1
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,3,1,5,6,2,4,7] => 1 = 0 + 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => 2 = 1 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1 = 0 + 1
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => 2 = 1 + 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 1 = 0 + 1
Description
The size of the overlap set of a permutation. For a permutation πSn this is the number of indices i<n such that the standardisation of π1πni equals the standardisation of πi+1πn. In particular, for n>1, the statistic is at least one, because the standardisations of π1 and πn are both 1. For example, for π=2143, the standardisations of 21 and 43 are equal, and so are the standardisations of 2 and 3. Thus, the statistic on π is 2.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00201: Dyck paths RingelPermutations
St000007: Permutations ⟶ ℤResult quality: 1% values known / values provided: 5%distinct values known / distinct values provided: 1%
Values
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 0 + 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 3 = 1 + 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 0 + 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2 = 0 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 3 = 1 + 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 3 = 1 + 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 2 = 0 + 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 3 = 1 + 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 3 = 1 + 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 3 = 1 + 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 3 = 1 + 2
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 0 + 2
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 0 + 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ? ∊ {0,1,2,2} + 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 3 = 1 + 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 3 = 1 + 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? ∊ {0,1,2,2} + 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 2 = 0 + 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 3 = 1 + 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? ∊ {0,1,2,2} + 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? ∊ {0,1,2,2} + 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 2 = 0 + 2
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => 2 = 0 + 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => 2 = 0 + 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,3,4,5,8,7,1,6] => ? ∊ {0,0,1,2,2,3,3,3} + 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 3 = 1 + 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,7,1,6,4,5] => ? ∊ {0,0,1,2,2,3,3,3} + 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,3,4,8,6,7,1,5] => ? ∊ {0,0,1,2,2,3,3,3} + 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 3 = 1 + 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? ∊ {0,0,1,2,2,3,3,3} + 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => ? ∊ {0,0,1,2,2,3,3,3} + 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 3 = 1 + 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? ∊ {0,0,1,2,2,3,3,3} + 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,8,4,5,6,7,1,3] => ? ∊ {0,0,1,2,2,3,3,3} + 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? ∊ {0,0,1,2,2,3,3,3} + 2
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 0 + 2
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => 2 = 0 + 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,9,8,1,7] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [2,3,4,8,1,7,5,6] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 4 = 2 + 2
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [2,3,8,1,6,7,4,5] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,1,5] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 2 = 0 + 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,8,1,5,6,7,3,4] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [2,3,9,5,6,7,8,1,4] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 2 = 0 + 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,9,4,5,6,7,8,1,3] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? ∊ {0,1,1,1,1,2,3,3,3,3,4,4,4,4,4,5,7} + 2
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => 2 = 0 + 2
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,9,1,7,8] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [2,3,4,5,6,7,10,9,1,8] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,4,8,7,1,5,6] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [2,3,4,5,9,1,8,6,7] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,6,10,8,9,1,7] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [2,3,8,6,1,7,4,5] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [2,3,4,9,1,7,8,5,6] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [2,3,4,5,10,7,8,9,1,6] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [2,8,5,1,6,7,3,4] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,3,8,1,4,7,5,6] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,9,1,6,7,8,4,5] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [2,3,4,10,6,7,8,9,1,5] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 2 = 0 + 2
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [8,4,1,5,6,7,2,3] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => 2 = 0 + 2
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,8,1,3,6,7,4,5] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,9,1,5,6,7,8,3,4] => ? ∊ {0,0,1,1,3,3,3,3,3,3,3,3,6,6,6,6,7,7,7,9,9,9,9,11,11,12,12} + 2
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 0 + 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 4 = 2 + 2
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => 2 = 0 + 2
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => 2 = 0 + 2
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => 4 = 2 + 2
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 3 = 1 + 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => 2 = 0 + 2
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,8,1,3,4,5,6,7] => 2 = 0 + 2
[4,4,3,3]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => 2 = 0 + 2
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => 2 = 0 + 2
[3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 3 = 1 + 2
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => 2 = 0 + 2
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern ([1],(1,1)), i.e., the upper right quadrant is shaded, see [1].
The following 115 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000035The number of left outer peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000648The number of 2-excedences of a permutation. St000366The number of double descents of a permutation. St000352The Elizalde-Pak rank of a permutation. St000990The first ascent of a permutation. St000647The number of big descents of a permutation. St001597The Frobenius rank of a skew partition. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001737The number of descents of type 2 in a permutation. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000954Number of times the corresponding LNakayama algebra has Exti(D(A),A)=0 for i>0. St001061The number of indices that are both descents and recoils of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001520The number of strict 3-descents. St001728The number of invisible descents of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001928The number of non-overlapping descents in a permutation. St000542The number of left-to-right-minima of a permutation. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000779The tier of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000534The number of 2-rises of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000633The size of the automorphism group of a poset. St000862The number of parts of the shifted shape of a permutation. St001399The distinguishing number of a poset. St000632The jump number of the poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001729The number of visible descents of a permutation. St000023The number of inner peaks of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000353The number of inner valleys of a permutation. St001174The Gorenstein dimension of the algebra A/I when I is the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001549The number of restricted non-inversions between exceedances. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a permutation. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000789The number of crossing-similar perfect matchings of a perfect matching. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of A/AfA in the corresponding Nakayama algebra A when Af is the minimal faithful projective-injective left A-module St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001487The number of inner corners of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001665The number of pure excedances of a permutation. St001732The number of peaks visible from the left. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001192The maximal dimension of Ext_A^2(S,A) for a simple module S over the corresponding Nakayama algebra A. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000326The position of the first one in a binary word after appending a 1 at the end. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000640The rank of the largest boolean interval in a poset. St000768The number of peaks in an integer composition. St000764The number of strong records in an integer composition. St001435The number of missing boxes in the first row. St000455The second largest eigenvalue of a graph if it is integral.