Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001118
Mp00160: Permutations graph of inversionsGraphs
Mp00156: Graphs line graphGraphs
Mp00111: Graphs complementGraphs
St001118: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 3
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
Description
The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Mp00170: Permutations to signed permutationSigned permutations
St001771: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => [2,4,3,1] => 0 = 1 - 1
[3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,5,3] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => 0 = 1 - 1
[1,4,5,3,2] => [1,4,5,3,2] => 0 = 1 - 1
[1,5,2,4,3] => [1,5,2,4,3] => 0 = 1 - 1
[1,5,3,2,4] => [1,5,3,2,4] => 0 = 1 - 1
[1,5,3,4,2] => [1,5,3,4,2] => 0 = 1 - 1
[1,5,4,2,3] => [1,5,4,2,3] => 0 = 1 - 1
[1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ? = 1 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ? = 1 - 1
[2,1,4,5,3] => [2,1,4,5,3] => ? = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => ? = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => ? = 2 - 1
[2,3,5,1,4] => [2,3,5,1,4] => ? = 2 - 1
[2,3,5,4,1] => [2,3,5,4,1] => ? = 2 - 1
[2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[2,4,1,5,3] => [2,4,1,5,3] => ? = 2 - 1
[2,4,3,1,5] => [2,4,3,1,5] => ? = 1 - 1
[2,4,3,5,1] => [2,4,3,5,1] => ? = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[2,4,5,3,1] => [2,4,5,3,1] => ? = 2 - 1
[2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[2,5,1,4,3] => [2,5,1,4,3] => ? = 3 - 1
[2,5,3,1,4] => [2,5,3,1,4] => ? = 2 - 1
[2,5,3,4,1] => [2,5,3,4,1] => ? = 2 - 1
[2,5,4,1,3] => [2,5,4,1,3] => ? = 3 - 1
[2,5,4,3,1] => [2,5,4,3,1] => ? = 3 - 1
[3,1,2,5,4] => [3,1,2,5,4] => ? = 2 - 1
[3,1,4,2,5] => [3,1,4,2,5] => ? = 1 - 1
[3,1,4,5,2] => [3,1,4,5,2] => ? = 2 - 1
[3,1,5,2,4] => [3,1,5,2,4] => ? = 2 - 1
[3,1,5,4,2] => [3,1,5,4,2] => ? = 3 - 1
[3,2,1,5,4] => [3,2,1,5,4] => ? = 3 - 1
[3,2,4,1,5] => [3,2,4,1,5] => ? = 1 - 1
[3,2,4,5,1] => [3,2,4,5,1] => ? = 2 - 1
[3,2,5,1,4] => [3,2,5,1,4] => ? = 3 - 1
[3,2,5,4,1] => [3,2,5,4,1] => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => ? = 1 - 1
[3,4,1,5,2] => [3,4,1,5,2] => ? = 2 - 1
[3,4,2,1,5] => [3,4,2,1,5] => ? = 1 - 1
[3,4,2,5,1] => [3,4,2,5,1] => ? = 2 - 1
[3,4,5,1,2] => [3,4,5,1,2] => ? = 3 - 1
[3,4,5,2,1] => [3,4,5,2,1] => ? = 3 - 1
[3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => ? = 3 - 1
[3,5,2,1,4] => [3,5,2,1,4] => ? = 3 - 1
[3,5,2,4,1] => [3,5,2,4,1] => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => ? = 3 - 1
[4,1,2,5,3] => [4,1,2,5,3] => ? = 2 - 1
[4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[4,1,3,5,2] => [4,1,3,5,2] => ? = 2 - 1
[4,1,5,2,3] => [4,1,5,2,3] => ? = 2 - 1
[4,1,5,3,2] => [4,1,5,3,2] => ? = 3 - 1
[4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[4,2,1,5,3] => [4,2,1,5,3] => ? = 3 - 1
[4,2,3,1,5] => [4,2,3,1,5] => ? = 1 - 1
[4,2,3,5,1] => [4,2,3,5,1] => ? = 2 - 1
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation. This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Mp00170: Permutations to signed permutationSigned permutations
St001870: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => [2,4,3,1] => 0 = 1 - 1
[3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,5,3] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => 0 = 1 - 1
[1,4,5,3,2] => [1,4,5,3,2] => 0 = 1 - 1
[1,5,2,4,3] => [1,5,2,4,3] => 0 = 1 - 1
[1,5,3,2,4] => [1,5,3,2,4] => 0 = 1 - 1
[1,5,3,4,2] => [1,5,3,4,2] => 0 = 1 - 1
[1,5,4,2,3] => [1,5,4,2,3] => 0 = 1 - 1
[1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ? = 1 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ? = 1 - 1
[2,1,4,5,3] => [2,1,4,5,3] => ? = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => ? = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => ? = 2 - 1
[2,3,5,1,4] => [2,3,5,1,4] => ? = 2 - 1
[2,3,5,4,1] => [2,3,5,4,1] => ? = 2 - 1
[2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[2,4,1,5,3] => [2,4,1,5,3] => ? = 2 - 1
[2,4,3,1,5] => [2,4,3,1,5] => ? = 1 - 1
[2,4,3,5,1] => [2,4,3,5,1] => ? = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[2,4,5,3,1] => [2,4,5,3,1] => ? = 2 - 1
[2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[2,5,1,4,3] => [2,5,1,4,3] => ? = 3 - 1
[2,5,3,1,4] => [2,5,3,1,4] => ? = 2 - 1
[2,5,3,4,1] => [2,5,3,4,1] => ? = 2 - 1
[2,5,4,1,3] => [2,5,4,1,3] => ? = 3 - 1
[2,5,4,3,1] => [2,5,4,3,1] => ? = 3 - 1
[3,1,2,5,4] => [3,1,2,5,4] => ? = 2 - 1
[3,1,4,2,5] => [3,1,4,2,5] => ? = 1 - 1
[3,1,4,5,2] => [3,1,4,5,2] => ? = 2 - 1
[3,1,5,2,4] => [3,1,5,2,4] => ? = 2 - 1
[3,1,5,4,2] => [3,1,5,4,2] => ? = 3 - 1
[3,2,1,5,4] => [3,2,1,5,4] => ? = 3 - 1
[3,2,4,1,5] => [3,2,4,1,5] => ? = 1 - 1
[3,2,4,5,1] => [3,2,4,5,1] => ? = 2 - 1
[3,2,5,1,4] => [3,2,5,1,4] => ? = 3 - 1
[3,2,5,4,1] => [3,2,5,4,1] => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => ? = 1 - 1
[3,4,1,5,2] => [3,4,1,5,2] => ? = 2 - 1
[3,4,2,1,5] => [3,4,2,1,5] => ? = 1 - 1
[3,4,2,5,1] => [3,4,2,5,1] => ? = 2 - 1
[3,4,5,1,2] => [3,4,5,1,2] => ? = 3 - 1
[3,4,5,2,1] => [3,4,5,2,1] => ? = 3 - 1
[3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => ? = 3 - 1
[3,5,2,1,4] => [3,5,2,1,4] => ? = 3 - 1
[3,5,2,4,1] => [3,5,2,4,1] => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => ? = 3 - 1
[4,1,2,5,3] => [4,1,2,5,3] => ? = 2 - 1
[4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[4,1,3,5,2] => [4,1,3,5,2] => ? = 2 - 1
[4,1,5,2,3] => [4,1,5,2,3] => ? = 2 - 1
[4,1,5,3,2] => [4,1,5,3,2] => ? = 3 - 1
[4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[4,2,1,5,3] => [4,2,1,5,3] => ? = 3 - 1
[4,2,3,1,5] => [4,2,3,1,5] => ? = 1 - 1
[4,2,3,5,1] => [4,2,3,5,1] => ? = 2 - 1
Description
The number of positive entries followed by a negative entry in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Mp00170: Permutations to signed permutationSigned permutations
St001895: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,4,1,3] => [2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => [2,4,3,1] => 0 = 1 - 1
[3,1,4,2] => [3,1,4,2] => 0 = 1 - 1
[3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => 0 = 1 - 1
[3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
[4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,5,3] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => 0 = 1 - 1
[1,4,5,3,2] => [1,4,5,3,2] => 0 = 1 - 1
[1,5,2,4,3] => [1,5,2,4,3] => 0 = 1 - 1
[1,5,3,2,4] => [1,5,3,2,4] => 0 = 1 - 1
[1,5,3,4,2] => [1,5,3,4,2] => 0 = 1 - 1
[1,5,4,2,3] => [1,5,4,2,3] => 0 = 1 - 1
[1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => ? = 1 - 1
[2,1,4,3,5] => [2,1,4,3,5] => ? = 1 - 1
[2,1,4,5,3] => [2,1,4,5,3] => ? = 2 - 1
[2,1,5,3,4] => [2,1,5,3,4] => ? = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => ? = 2 - 1
[2,3,5,1,4] => [2,3,5,1,4] => ? = 2 - 1
[2,3,5,4,1] => [2,3,5,4,1] => ? = 2 - 1
[2,4,1,3,5] => [2,4,1,3,5] => ? = 1 - 1
[2,4,1,5,3] => [2,4,1,5,3] => ? = 2 - 1
[2,4,3,1,5] => [2,4,3,1,5] => ? = 1 - 1
[2,4,3,5,1] => [2,4,3,5,1] => ? = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => ? = 2 - 1
[2,4,5,3,1] => [2,4,5,3,1] => ? = 2 - 1
[2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[2,5,1,4,3] => [2,5,1,4,3] => ? = 3 - 1
[2,5,3,1,4] => [2,5,3,1,4] => ? = 2 - 1
[2,5,3,4,1] => [2,5,3,4,1] => ? = 2 - 1
[2,5,4,1,3] => [2,5,4,1,3] => ? = 3 - 1
[2,5,4,3,1] => [2,5,4,3,1] => ? = 3 - 1
[3,1,2,5,4] => [3,1,2,5,4] => ? = 2 - 1
[3,1,4,2,5] => [3,1,4,2,5] => ? = 1 - 1
[3,1,4,5,2] => [3,1,4,5,2] => ? = 2 - 1
[3,1,5,2,4] => [3,1,5,2,4] => ? = 2 - 1
[3,1,5,4,2] => [3,1,5,4,2] => ? = 3 - 1
[3,2,1,5,4] => [3,2,1,5,4] => ? = 3 - 1
[3,2,4,1,5] => [3,2,4,1,5] => ? = 1 - 1
[3,2,4,5,1] => [3,2,4,5,1] => ? = 2 - 1
[3,2,5,1,4] => [3,2,5,1,4] => ? = 3 - 1
[3,2,5,4,1] => [3,2,5,4,1] => ? = 3 - 1
[3,4,1,2,5] => [3,4,1,2,5] => ? = 1 - 1
[3,4,1,5,2] => [3,4,1,5,2] => ? = 2 - 1
[3,4,2,1,5] => [3,4,2,1,5] => ? = 1 - 1
[3,4,2,5,1] => [3,4,2,5,1] => ? = 2 - 1
[3,4,5,1,2] => [3,4,5,1,2] => ? = 3 - 1
[3,4,5,2,1] => [3,4,5,2,1] => ? = 3 - 1
[3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => ? = 3 - 1
[3,5,2,1,4] => [3,5,2,1,4] => ? = 3 - 1
[3,5,2,4,1] => [3,5,2,4,1] => ? = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => ? = 3 - 1
[4,1,2,5,3] => [4,1,2,5,3] => ? = 2 - 1
[4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[4,1,3,5,2] => [4,1,3,5,2] => ? = 2 - 1
[4,1,5,2,3] => [4,1,5,2,3] => ? = 2 - 1
[4,1,5,3,2] => [4,1,5,3,2] => ? = 3 - 1
[4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[4,2,1,5,3] => [4,2,1,5,3] => ? = 3 - 1
[4,2,3,1,5] => [4,2,3,1,5] => ? = 1 - 1
[4,2,3,5,1] => [4,2,3,5,1] => ? = 2 - 1
Description
The oddness of a signed permutation. The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$. This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00170: Permutations to signed permutationSigned permutations
St001769: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 3 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2 + 1
[2,3,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[2,4,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,4,5,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,5,1,3,4] => [3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,5,1,4,3] => [3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[2,5,3,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,5,4,1,3] => [4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2 + 1
[3,1,4,2,5] => [4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[3,1,4,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[3,1,5,4,2] => [5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 3 + 1
[3,2,4,1,5] => [4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[3,2,4,5,1] => [5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[3,2,5,1,4] => [4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,2,5,4,1] => [5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[3,4,1,5,2] => [5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[3,4,2,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,5,1,2,4] => [4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,5,2,1,4] => [4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[4,1,2,5,3] => [5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[4,1,3,2,5] => [4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[4,1,3,5,2] => [5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[4,1,5,2,3] => [5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[4,1,5,3,2] => [5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[4,2,1,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[4,2,1,5,3] => [5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[4,2,3,5,1] => [5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
Description
The reflection length of a signed permutation. This is the minimal numbers of reflections needed to express a signed permutation.
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00170: Permutations to signed permutationSigned permutations
Mp00244: Signed permutations barSigned permutations
St001772: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 0 = 1 - 1
[2,4,1,3] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 0 = 1 - 1
[2,4,3,1] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0 = 1 - 1
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0 = 1 - 1
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 0 = 1 - 1
[3,4,1,2] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 0 = 1 - 1
[3,4,2,1] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0 = 1 - 1
[4,1,3,2] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 0 = 1 - 1
[4,2,1,3] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0 = 1 - 1
[4,2,3,1] => [2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => 0 = 1 - 1
[4,3,1,2] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0 = 1 - 1
[4,3,2,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0 = 1 - 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => 0 = 1 - 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 0 = 1 - 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => 0 = 1 - 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0 = 1 - 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => 0 = 1 - 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => 0 = 1 - 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0 = 1 - 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 0 = 1 - 1
[1,5,4,2,3] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 0 = 1 - 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 1 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 1 - 1
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => ? = 2 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 2 - 1
[2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 3 - 1
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => ? = 2 - 1
[2,3,5,1,4] => [5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => ? = 2 - 1
[2,3,5,4,1] => [4,5,1,2,3] => [4,5,1,2,3] => [-4,-5,-1,-2,-3] => ? = 2 - 1
[2,4,1,3,5] => [4,3,1,2,5] => [4,3,1,2,5] => [-4,-3,-1,-2,-5] => ? = 1 - 1
[2,4,1,5,3] => [5,3,1,2,4] => [5,3,1,2,4] => [-5,-3,-1,-2,-4] => ? = 2 - 1
[2,4,3,1,5] => [3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => ? = 1 - 1
[2,4,3,5,1] => [3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => ? = 2 - 1
[2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => [-4,-1,-2,-5,-3] => ? = 2 - 1
[2,4,5,3,1] => [5,1,2,4,3] => [5,1,2,4,3] => [-5,-1,-2,-4,-3] => ? = 2 - 1
[2,5,1,3,4] => [5,4,3,1,2] => [5,4,3,1,2] => [-5,-4,-3,-1,-2] => ? = 2 - 1
[2,5,1,4,3] => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => ? = 3 - 1
[2,5,3,1,4] => [3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => ? = 2 - 1
[2,5,3,4,1] => [3,4,5,1,2] => [3,4,5,1,2] => [-3,-4,-5,-1,-2] => ? = 2 - 1
[2,5,4,1,3] => [5,3,4,1,2] => [5,3,4,1,2] => [-5,-3,-4,-1,-2] => ? = 3 - 1
[2,5,4,3,1] => [4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => ? = 3 - 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => ? = 2 - 1
[3,1,4,2,5] => [4,2,1,3,5] => [4,2,1,3,5] => [-4,-2,-1,-3,-5] => ? = 1 - 1
[3,1,4,5,2] => [5,2,1,3,4] => [5,2,1,3,4] => [-5,-2,-1,-3,-4] => ? = 2 - 1
[3,1,5,2,4] => [5,4,2,1,3] => [5,4,2,1,3] => [-5,-4,-2,-1,-3] => ? = 2 - 1
[3,1,5,4,2] => [4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => ? = 3 - 1
[3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 3 - 1
[3,2,4,1,5] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 1 - 1
[3,2,4,5,1] => [2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => ? = 2 - 1
[3,2,5,1,4] => [2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,-4,-1,-3] => ? = 3 - 1
[3,2,5,4,1] => [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 3 - 1
[3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 1 - 1
[3,4,1,5,2] => [3,1,5,2,4] => [3,1,5,2,4] => [-3,-1,-5,-2,-4] => ? = 2 - 1
[3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => [-4,-1,-3,-2,-5] => ? = 1 - 1
[3,4,2,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => [-5,-1,-3,-2,-4] => ? = 2 - 1
[3,4,5,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => [-5,-2,-4,-1,-3] => ? = 3 - 1
[3,4,5,2,1] => [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => ? = 3 - 1
[3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 2 - 1
[3,5,1,4,2] => [3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => ? = 3 - 1
[3,5,2,1,4] => [5,4,1,3,2] => [5,4,1,3,2] => [-5,-4,-1,-3,-2] => ? = 3 - 1
[3,5,2,4,1] => [4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => ? = 3 - 1
[3,5,4,1,2] => [4,1,3,5,2] => [4,1,3,5,2] => [-4,-1,-3,-5,-2] => ? = 3 - 1
[4,1,2,5,3] => [5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => ? = 2 - 1
[4,1,3,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => [-3,-4,-2,-1,-5] => ? = 1 - 1
[4,1,3,5,2] => [3,5,2,1,4] => [3,5,2,1,4] => [-3,-5,-2,-1,-4] => ? = 2 - 1
[4,1,5,2,3] => [4,2,1,5,3] => [4,2,1,5,3] => [-4,-2,-1,-5,-3] => ? = 2 - 1
[4,1,5,3,2] => [5,2,1,4,3] => [5,2,1,4,3] => [-5,-2,-1,-4,-3] => ? = 3 - 1
[4,2,1,3,5] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 1 - 1
[4,2,1,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 3 - 1
[4,2,3,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => ? = 1 - 1
[4,2,3,5,1] => [2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => ? = 2 - 1
Description
The number of occurrences of the signed pattern 12 in a signed permutation. This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < \pi(j)$.
Mp00149: Permutations Lehmer code rotationPermutations
Mp00170: Permutations to signed permutationSigned permutations
Mp00167: Signed permutations inverse Kreweras complementSigned permutations
St001851: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => 0 = 1 - 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => 0 = 1 - 1
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 0 = 1 - 1
[3,2,4,1] => [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 0 = 1 - 1
[3,4,1,2] => [4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => 0 = 1 - 1
[3,4,2,1] => [4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => 0 = 1 - 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 0 = 1 - 1
[1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => 0 = 1 - 1
[1,3,5,2,4] => [2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => 0 = 1 - 1
[1,3,5,4,2] => [2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => 0 = 1 - 1
[1,4,2,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => 0 = 1 - 1
[1,4,3,5,2] => [2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => 0 = 1 - 1
[1,4,5,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => 0 = 1 - 1
[1,4,5,3,2] => [2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => 0 = 1 - 1
[1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => 0 = 1 - 1
[1,5,3,2,4] => [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => 0 = 1 - 1
[1,5,3,4,2] => [2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => 0 = 1 - 1
[1,5,4,2,3] => [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => 0 = 1 - 1
[1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => 0 = 1 - 1
[2,1,3,5,4] => [3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => ? = 1 - 1
[2,1,4,3,5] => [3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,-5] => ? = 1 - 1
[2,1,4,5,3] => [3,2,5,1,4] => [3,2,5,1,4] => [2,1,5,3,-4] => ? = 2 - 1
[2,1,5,3,4] => [3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => ? = 2 - 1
[2,1,5,4,3] => [3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => ? = 3 - 1
[2,3,1,5,4] => [3,4,2,1,5] => [3,4,2,1,5] => [3,1,2,5,-4] => ? = 2 - 1
[2,3,5,1,4] => [3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => ? = 2 - 1
[2,3,5,4,1] => [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => ? = 2 - 1
[2,4,1,3,5] => [3,5,2,4,1] => [3,5,2,4,1] => [3,1,4,2,-5] => ? = 1 - 1
[2,4,1,5,3] => [3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => ? = 2 - 1
[2,4,3,1,5] => [3,5,4,2,1] => [3,5,4,2,1] => [4,1,3,2,-5] => ? = 1 - 1
[2,4,3,5,1] => [3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => ? = 2 - 1
[2,4,5,1,3] => [3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => ? = 2 - 1
[2,4,5,3,1] => [3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => ? = 2 - 1
[2,5,1,3,4] => [3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => ? = 2 - 1
[2,5,1,4,3] => [3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => ? = 3 - 1
[2,5,3,1,4] => [3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => ? = 2 - 1
[2,5,3,4,1] => [3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => ? = 2 - 1
[2,5,4,1,3] => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => ? = 3 - 1
[2,5,4,3,1] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => ? = 3 - 1
[3,1,2,5,4] => [4,2,3,1,5] => [4,2,3,1,5] => [2,3,1,5,-4] => ? = 2 - 1
[3,1,4,2,5] => [4,2,5,3,1] => [4,2,5,3,1] => [2,4,1,3,-5] => ? = 1 - 1
[3,1,4,5,2] => [4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => ? = 2 - 1
[3,1,5,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => [2,5,1,4,-3] => ? = 2 - 1
[3,1,5,4,2] => [4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,5,-3] => ? = 3 - 1
[3,2,1,5,4] => [4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,-4] => ? = 3 - 1
[3,2,4,1,5] => [4,3,5,2,1] => [4,3,5,2,1] => [4,2,1,3,-5] => ? = 1 - 1
[3,2,4,5,1] => [4,3,5,1,2] => [4,3,5,1,2] => [5,2,1,3,-4] => ? = 2 - 1
[3,2,5,1,4] => [4,3,1,5,2] => [4,3,1,5,2] => [5,2,1,4,-3] => ? = 3 - 1
[3,2,5,4,1] => [4,3,1,2,5] => [4,3,1,2,5] => [4,2,1,5,-3] => ? = 3 - 1
[3,4,1,2,5] => [4,5,2,3,1] => [4,5,2,3,1] => [3,4,1,2,-5] => ? = 1 - 1
[3,4,1,5,2] => [4,5,2,1,3] => [4,5,2,1,3] => [3,5,1,2,-4] => ? = 2 - 1
[3,4,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => [4,3,1,2,-5] => ? = 1 - 1
[3,4,2,5,1] => [4,5,3,1,2] => [4,5,3,1,2] => [5,3,1,2,-4] => ? = 2 - 1
[3,4,5,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => [5,4,1,2,-3] => ? = 3 - 1
[3,4,5,2,1] => [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => ? = 3 - 1
[3,5,1,2,4] => [4,1,3,5,2] => [4,1,3,5,2] => [5,3,1,4,-2] => ? = 2 - 1
[3,5,1,4,2] => [4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => ? = 3 - 1
[3,5,2,1,4] => [4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => ? = 3 - 1
[3,5,2,4,1] => [4,1,5,2,3] => [4,1,5,2,3] => [4,5,1,3,-2] => ? = 3 - 1
[3,5,4,1,2] => [4,1,2,5,3] => [4,1,2,5,3] => [3,5,1,4,-2] => ? = 3 - 1
[4,1,2,5,3] => [5,2,3,1,4] => [5,2,3,1,4] => [2,3,5,1,-4] => ? = 2 - 1
[4,1,3,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [2,4,3,1,-5] => ? = 1 - 1
[4,1,3,5,2] => [5,2,4,1,3] => [5,2,4,1,3] => [2,5,3,1,-4] => ? = 2 - 1
[4,1,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => [2,5,4,1,-3] => ? = 2 - 1
[4,1,5,3,2] => [5,2,1,3,4] => [5,2,1,3,4] => [2,4,5,1,-3] => ? = 3 - 1
[4,2,1,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,1,-5] => ? = 1 - 1
[4,2,1,5,3] => [5,3,2,1,4] => [5,3,2,1,4] => [3,2,5,1,-4] => ? = 3 - 1
[4,2,3,1,5] => [5,3,4,2,1] => [5,3,4,2,1] => [4,2,3,1,-5] => ? = 1 - 1
[4,2,3,5,1] => [5,3,4,1,2] => [5,3,4,1,2] => [5,2,3,1,-4] => ? = 2 - 1
Description
The number of Hecke atoms of a signed permutation. For a signed permutation $z\in\mathfrak H_n$, this is the cardinality of the set $$ \{ w\in\mathfrak H_n | w^{-1} \star w = z\}, $$ where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
Mp00277: Permutations catalanizationPermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00170: Permutations to signed permutationSigned permutations
St001862: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[2,4,1,3] => [4,3,1,2] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[2,4,3,1] => [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[3,1,4,2] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[3,2,4,1] => [3,2,4,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[3,4,2,1] => [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,1,3,2] => [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,2,1,3] => [3,2,4,1] => [4,2,3,1] => [4,2,3,1] => 0 = 1 - 1
[4,2,3,1] => [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,3,1,2] => [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,4,2,5,3] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0 = 1 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,4,5,3,2] => [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,5,2,4,3] => [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,5,3,2,4] => [1,4,3,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0 = 1 - 1
[1,5,3,4,2] => [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,5,4,2,3] => [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2 - 1
[2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 3 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2 - 1
[2,3,5,1,4] => [2,5,4,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2 - 1
[2,3,5,4,1] => [2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2 - 1
[2,4,1,3,5] => [4,3,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[2,4,1,5,3] => [4,3,1,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
[2,4,3,1,5] => [2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[2,4,3,5,1] => [2,4,3,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
[2,4,5,1,3] => [3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
[2,4,5,3,1] => [2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
[2,5,1,3,4] => [4,3,1,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
[2,5,1,4,3] => [5,3,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[2,5,3,1,4] => [5,4,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
[2,5,3,4,1] => [2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
[2,5,4,1,3] => [3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[3,1,2,5,4] => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2 - 1
[3,1,4,2,5] => [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1 - 1
[3,1,4,5,2] => [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2 - 1
[3,1,5,2,4] => [2,5,4,1,3] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 2 - 1
[3,1,5,4,2] => [2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 3 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 3 - 1
[3,2,4,1,5] => [3,2,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1 - 1
[3,2,4,5,1] => [3,2,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2 - 1
[3,2,5,1,4] => [5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[3,2,5,4,1] => [3,2,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 3 - 1
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[3,4,1,5,2] => [4,3,2,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
[3,4,2,1,5] => [3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[3,4,2,5,1] => [3,4,2,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
[3,4,5,1,2] => [3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[3,4,5,2,1] => [3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[3,5,1,2,4] => [4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
[3,5,1,4,2] => [5,3,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[3,5,2,1,4] => [3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[3,5,2,4,1] => [5,4,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[3,5,4,1,2] => [3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[4,1,2,5,3] => [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2 - 1
[4,1,3,2,5] => [2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[4,1,3,5,2] => [2,4,3,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
[4,1,5,2,3] => [2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 - 1
[4,1,5,3,2] => [2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 - 1
[4,2,1,3,5] => [3,2,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1 - 1
[4,2,1,5,3] => [3,2,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 - 1
[4,2,3,1,5] => [3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[4,2,3,5,1] => [3,4,2,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
Description
The number of crossings of a signed permutation. A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that * $i < j \leq \pi(i) < \pi(j)$, or * $-i < j \leq -\pi(i) < \pi(j)$, or * $i > j > \pi(i) > \pi(j)$.
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00170: Permutations to signed permutationSigned permutations
Mp00244: Signed permutations barSigned permutations
St001863: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 0 = 1 - 1
[2,4,1,3] => [4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 0 = 1 - 1
[2,4,3,1] => [3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 0 = 1 - 1
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 0 = 1 - 1
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 0 = 1 - 1
[3,4,1,2] => [3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 0 = 1 - 1
[3,4,2,1] => [4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 0 = 1 - 1
[4,1,3,2] => [3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 0 = 1 - 1
[4,2,1,3] => [2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 0 = 1 - 1
[4,2,3,1] => [2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => 0 = 1 - 1
[4,3,1,2] => [4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 0 = 1 - 1
[4,3,2,1] => [3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 0 = 1 - 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => 0 = 1 - 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 0 = 1 - 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => 0 = 1 - 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 0 = 1 - 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => 0 = 1 - 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => 0 = 1 - 1
[1,5,3,2,4] => [1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 0 = 1 - 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 0 = 1 - 1
[1,5,4,2,3] => [1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 0 = 1 - 1
[1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => ? = 1 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => ? = 1 - 1
[2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => ? = 2 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => ? = 2 - 1
[2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => ? = 3 - 1
[2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => ? = 2 - 1
[2,3,5,1,4] => [5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => ? = 2 - 1
[2,3,5,4,1] => [4,5,1,2,3] => [4,5,1,2,3] => [-4,-5,-1,-2,-3] => ? = 2 - 1
[2,4,1,3,5] => [4,3,1,2,5] => [4,3,1,2,5] => [-4,-3,-1,-2,-5] => ? = 1 - 1
[2,4,1,5,3] => [5,3,1,2,4] => [5,3,1,2,4] => [-5,-3,-1,-2,-4] => ? = 2 - 1
[2,4,3,1,5] => [3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => ? = 1 - 1
[2,4,3,5,1] => [3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => ? = 2 - 1
[2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => [-4,-1,-2,-5,-3] => ? = 2 - 1
[2,4,5,3,1] => [5,1,2,4,3] => [5,1,2,4,3] => [-5,-1,-2,-4,-3] => ? = 2 - 1
[2,5,1,3,4] => [5,4,3,1,2] => [5,4,3,1,2] => [-5,-4,-3,-1,-2] => ? = 2 - 1
[2,5,1,4,3] => [4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => ? = 3 - 1
[2,5,3,1,4] => [3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => ? = 2 - 1
[2,5,3,4,1] => [3,4,5,1,2] => [3,4,5,1,2] => [-3,-4,-5,-1,-2] => ? = 2 - 1
[2,5,4,1,3] => [5,3,4,1,2] => [5,3,4,1,2] => [-5,-3,-4,-1,-2] => ? = 3 - 1
[2,5,4,3,1] => [4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => ? = 3 - 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => ? = 2 - 1
[3,1,4,2,5] => [4,2,1,3,5] => [4,2,1,3,5] => [-4,-2,-1,-3,-5] => ? = 1 - 1
[3,1,4,5,2] => [5,2,1,3,4] => [5,2,1,3,4] => [-5,-2,-1,-3,-4] => ? = 2 - 1
[3,1,5,2,4] => [5,4,2,1,3] => [5,4,2,1,3] => [-5,-4,-2,-1,-3] => ? = 2 - 1
[3,1,5,4,2] => [4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => ? = 3 - 1
[3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => ? = 3 - 1
[3,2,4,1,5] => [2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => ? = 1 - 1
[3,2,4,5,1] => [2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => ? = 2 - 1
[3,2,5,1,4] => [2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,-4,-1,-3] => ? = 3 - 1
[3,2,5,4,1] => [2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => ? = 3 - 1
[3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => ? = 1 - 1
[3,4,1,5,2] => [3,1,5,2,4] => [3,1,5,2,4] => [-3,-1,-5,-2,-4] => ? = 2 - 1
[3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => [-4,-1,-3,-2,-5] => ? = 1 - 1
[3,4,2,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => [-5,-1,-3,-2,-4] => ? = 2 - 1
[3,4,5,1,2] => [5,2,4,1,3] => [5,2,4,1,3] => [-5,-2,-4,-1,-3] => ? = 3 - 1
[3,4,5,2,1] => [4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => ? = 3 - 1
[3,5,1,2,4] => [3,1,5,4,2] => [3,1,5,4,2] => [-3,-1,-5,-4,-2] => ? = 2 - 1
[3,5,1,4,2] => [3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => ? = 3 - 1
[3,5,2,1,4] => [5,4,1,3,2] => [5,4,1,3,2] => [-5,-4,-1,-3,-2] => ? = 3 - 1
[3,5,2,4,1] => [4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => ? = 3 - 1
[3,5,4,1,2] => [4,1,3,5,2] => [4,1,3,5,2] => [-4,-1,-3,-5,-2] => ? = 3 - 1
[4,1,2,5,3] => [5,3,2,1,4] => [5,3,2,1,4] => [-5,-3,-2,-1,-4] => ? = 2 - 1
[4,1,3,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => [-3,-4,-2,-1,-5] => ? = 1 - 1
[4,1,3,5,2] => [3,5,2,1,4] => [3,5,2,1,4] => [-3,-5,-2,-1,-4] => ? = 2 - 1
[4,1,5,2,3] => [4,2,1,5,3] => [4,2,1,5,3] => [-4,-2,-1,-5,-3] => ? = 2 - 1
[4,1,5,3,2] => [5,2,1,4,3] => [5,2,1,4,3] => [-5,-2,-1,-4,-3] => ? = 3 - 1
[4,2,1,3,5] => [2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => ? = 1 - 1
[4,2,1,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => ? = 3 - 1
[4,2,3,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => ? = 1 - 1
[4,2,3,5,1] => [2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => ? = 2 - 1
Description
The number of weak excedances of a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
Mp00159: Permutations Demazure product with inversePermutations
Mp00241: Permutations invert Laguerre heapPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001864: Signed permutations ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 17%
Values
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[2,4,1,3] => [3,4,1,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[3,1,4,2] => [4,2,3,1] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[3,2,4,1] => [4,2,3,1] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,1,3,2] => [4,2,3,1] => [3,1,4,2] => [3,1,4,2] => 2 = 1 + 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,3,5,2,4] => [1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,4,2,5,3] => [1,4,2,5,3] => 2 = 1 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,4,2,5,3] => [1,4,2,5,3] => 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,4,2,5,3] => [1,4,2,5,3] => 2 = 1 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 3 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2 + 1
[2,3,5,1,4] => [4,2,5,1,3] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2 + 1
[2,3,5,4,1] => [5,2,4,3,1] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[2,4,3,5,1] => [5,3,2,4,1] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2 + 1
[2,4,5,1,3] => [4,5,3,1,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 2 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,5,1,3,4] => [3,5,1,4,2] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2 + 1
[2,5,1,4,3] => [3,5,1,4,2] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 3 + 1
[2,5,3,1,4] => [4,5,3,1,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 2 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
[2,5,4,1,3] => [4,5,3,1,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 3 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2 + 1
[3,1,4,2,5] => [4,2,3,1,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[3,1,4,5,2] => [5,2,3,4,1] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 2 + 1
[3,1,5,2,4] => [4,2,5,1,3] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2 + 1
[3,1,5,4,2] => [5,2,4,3,1] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 3 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 3 + 1
[3,2,4,1,5] => [4,2,3,1,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[3,2,4,5,1] => [5,2,3,4,1] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 2 + 1
[3,2,5,1,4] => [4,2,5,1,3] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 3 + 1
[3,2,5,4,1] => [5,2,4,3,1] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 3 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[3,4,1,5,2] => [5,3,2,4,1] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2 + 1
[3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[3,4,2,5,1] => [5,3,2,4,1] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,5,1,2,4] => [4,5,3,1,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 2 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,5,2,1,4] => [4,5,3,1,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 3 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 3 + 1
[4,1,2,5,3] => [5,2,3,4,1] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 2 + 1
[4,1,3,2,5] => [4,2,3,1,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 1 + 1
[4,1,3,5,2] => [5,2,3,4,1] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 2 + 1
[4,1,5,2,3] => [5,2,4,3,1] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 2 + 1
[4,1,5,3,2] => [5,2,4,3,1] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 3 + 1
[4,2,1,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[4,2,1,5,3] => [5,3,2,4,1] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 3 + 1
[4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 + 1
[4,2,3,5,1] => [5,3,2,4,1] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2 + 1
Description
The number of excedances of a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001625The Möbius invariant of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice.