Your data matches 201 different statistics following compositions of up to 3 maps.
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Mp00223: Permutations runsortPermutations
Mp00067: Permutations Foata bijectionPermutations
Mp00160: Permutations graph of inversionsGraphs
St000422: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [1,2] => [1,2] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Mp00223: Permutations runsortPermutations
Mp00067: Permutations Foata bijectionPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St001726: Permutations ⟶ ℤResult quality: 85% values known / values provided: 85%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [1,5,4,3,2] => 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,5,4,3,2,6] => 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,2,6,5,4,3] => 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,2,6,5,4,3] => 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,5,4,3,2,6] => 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,2,6,5,4,3] => 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,2,6,5,4,3] => 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => 6
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [1,4,5,6,3,2] => 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[2,3,5,7,4,6,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[2,3,5,7,6,4,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[3,5,4,6,7,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[3,5,7,4,6,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[3,5,7,4,6,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[3,5,7,6,4,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,1,2,3,5,7,6] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,2,3,5,7,6,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,3,5,7,6,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,3,5,7,6,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,6,2,3,5,7,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,6,3,5,7,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,6,3,5,7,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[4,6,7,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[4,6,7,2,3,5,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[4,6,7,3,5,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[4,6,7,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[6,1,2,3,5,7,4] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[6,2,3,5,7,4,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[6,3,5,7,4,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[6,4,1,2,3,5,7] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[6,4,2,3,5,7,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[6,4,3,5,7,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[6,4,3,5,7,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [1,5,4,3,2,6,7] => ? = 6
[6,7,1,2,3,5,4] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[6,7,2,3,5,4,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[6,7,3,5,4,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[6,7,4,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[6,7,4,2,3,5,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[6,7,4,3,5,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,1,2,3,5,4,6] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,2,3,5,4,6,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,3,5,4,6,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,3,5,4,6,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,4,6,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,4,6,2,3,5,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,4,6,3,5,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,4,6,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,6,1,2,3,5,4] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
[7,6,2,3,5,4,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 4
Description
The number of visible inversions of a permutation. A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$.
Mp00223: Permutations runsortPermutations
Mp00067: Permutations Foata bijectionPermutations
Mp00086: Permutations first fundamental transformationPermutations
St000957: Permutations ⟶ ℤResult quality: 83% values known / values provided: 83%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [2,3,4,5,1,6] => 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,3,4,5,6,2] => 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,3,4,5,6,2] => 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [2,4,6,1,5,3] => 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [2,3,4,5,1,6] => 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,3,4,5,6,2] => 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,3,4,5,6,2] => 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [2,4,6,1,5,3] => 6
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [2,4,6,1,5,3] => 6
[3,2,4,5,6,1] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [2,4,6,1,5,3] => 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [2,4,6,1,5,3,7] => ? = 6
[2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[2,3,5,7,4,6,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[2,3,5,7,6,4,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[2,4,5,6,3,7,1] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [2,4,6,1,5,3,7] => ? = 6
[3,5,4,6,7,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[3,5,7,4,6,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[3,5,7,4,6,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[3,5,7,6,4,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[3,7,1,2,4,5,6] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [2,4,6,1,5,3,7] => ? = 6
[3,7,2,4,5,6,1] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [2,4,6,1,5,3,7] => ? = 6
[4,1,2,3,5,7,6] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[4,2,3,5,7,6,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[4,3,5,7,6,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[4,3,5,7,6,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[4,6,2,3,5,7,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[4,6,3,5,7,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[4,6,3,5,7,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[4,6,7,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[4,6,7,2,3,5,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[4,6,7,3,5,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[4,6,7,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[6,1,2,3,5,7,4] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[6,2,3,5,7,4,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[6,3,5,7,4,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[6,4,1,2,3,5,7] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[6,4,2,3,5,7,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[6,4,3,5,7,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[6,4,3,5,7,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 6
[6,7,1,2,3,5,4] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[6,7,2,3,5,4,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[6,7,3,5,4,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[6,7,4,1,2,3,5] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[6,7,4,2,3,5,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[6,7,4,3,5,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[7,1,2,3,5,4,6] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[7,1,2,4,5,6,3] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [2,4,6,1,5,3,7] => ? = 6
[7,2,3,5,4,6,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 4
[7,2,4,5,6,3,1] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [2,4,6,1,5,3,7] => ? = 6
[7,3,1,2,4,5,6] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [2,4,6,1,5,3,7] => ? = 6
Description
The number of Bruhat lower covers of a permutation. This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$. This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Mp00223: Permutations runsortPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00067: Permutations Foata bijectionPermutations
St001464: Permutations ⟶ ℤResult quality: 67% values known / values provided: 67%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 5 = 4 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 5 = 4 + 1
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[3,5,4,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 5 = 4 + 1
[3,5,4,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 5 = 4 + 1
[4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 5 = 4 + 1
[4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 5 = 4 + 1
[4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 5 = 4 + 1
[4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 5 = 4 + 1
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 5 = 4 + 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 5 = 4 + 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 5 = 4 + 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [6,1,4,2,5,3] => 7 = 6 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1 = 0 + 1
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 5 = 4 + 1
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 5 = 4 + 1
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 5 = 4 + 1
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [6,1,4,2,5,3] => 7 = 6 + 1
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [6,1,4,2,5,3] => 7 = 6 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [6,1,4,2,5,3,7] => ? = 6 + 1
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4 + 1
[2,3,5,7,4,6,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[2,3,5,7,6,4,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[2,3,6,4,5,7,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[2,3,6,5,7,4,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[2,4,5,6,3,7,1] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [6,1,4,2,5,3,7] => ? = 6 + 1
[2,5,6,7,3,4,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[2,5,6,7,4,3,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[3,1,2,5,6,7,4] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[3,2,5,6,7,4,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[3,5,4,6,7,1,2] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4 + 1
[3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4 + 1
[3,5,7,4,6,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[3,5,7,4,6,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[3,5,7,6,4,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[3,6,4,5,7,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[3,6,4,5,7,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[3,6,5,7,4,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[3,7,1,2,4,5,6] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [6,1,4,2,5,3,7] => ? = 6 + 1
[3,7,2,4,5,6,1] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [6,1,4,2,5,3,7] => ? = 6 + 1
[4,1,2,3,5,7,6] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[4,1,2,3,6,5,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[4,1,2,5,6,7,3] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[4,2,3,5,7,6,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[4,2,3,6,5,7,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[4,2,5,6,7,3,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[4,3,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[4,3,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [7,1,2,5,3,6,4] => ? = 6 + 1
[4,3,5,7,6,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[4,3,5,7,6,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[4,3,6,5,7,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[4,3,6,5,7,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[4,5,7,1,2,3,6] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[4,5,7,2,3,6,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[4,5,7,3,6,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[4,5,7,3,6,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4 + 1
[4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
[4,6,2,3,5,7,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [5,7,1,2,3,4,6] => ? = 6 + 1
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Mp00223: Permutations runsortPermutations
Mp00067: Permutations Foata bijectionPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
St000673: Permutations ⟶ ℤResult quality: 67% values known / values provided: 67%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [5,4,3,2,1] => 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,4,3,2,1,6] => 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,5,4,3,2] => 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,5,4,3,2] => 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,4,3,2,1,6] => 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,5,4,3,2] => 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,5,4,3,2] => 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => 6
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => 6
[3,2,4,5,6,1] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [5,4,3,2,1,6,7] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [6,3,2,1,4,5,7] => ? = 6
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [5,4,3,2,1,6,7] => ? = 4
[2,3,5,7,4,6,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[2,3,5,7,6,4,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[2,3,6,4,5,7,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[2,3,6,5,7,4,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[2,4,5,6,3,7,1] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [6,3,2,1,4,5,7] => ? = 6
[2,5,6,7,3,4,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[2,5,6,7,4,3,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[3,1,2,5,6,7,4] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[3,2,5,6,7,4,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[3,5,4,6,7,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [5,4,3,2,1,6,7] => ? = 4
[3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [5,4,3,2,1,6,7] => ? = 4
[3,5,7,4,6,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[3,5,7,4,6,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[3,5,7,6,4,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[3,6,4,5,7,1,2] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[3,6,4,5,7,2,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[3,6,5,7,4,1,2] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[3,7,1,2,4,5,6] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [6,3,2,1,4,5,7] => ? = 6
[3,7,2,4,5,6,1] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [6,3,2,1,4,5,7] => ? = 6
[4,1,2,3,5,7,6] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[4,1,2,3,6,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[4,1,2,5,6,7,3] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[4,2,3,5,7,6,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[4,2,3,6,5,7,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[4,2,5,6,7,3,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[4,3,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[4,3,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 6
[4,3,5,7,6,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[4,3,5,7,6,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
[4,3,6,5,7,1,2] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[4,3,6,5,7,2,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[4,5,7,1,2,3,6] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[4,5,7,2,3,6,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[4,5,7,3,6,1,2] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[4,5,7,3,6,2,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
[4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,4,3,2,1,7,6] => ? = 6
Description
The number of non-fixed points of a permutation. In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Mp00223: Permutations runsortPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
St000795: Permutations ⟶ ℤResult quality: 67% values known / values provided: 67%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[3,5,4,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => 6
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => 6
[3,2,4,5,6,1] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [4,6,5,1,2,3] => 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => ? = 6
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[2,3,5,7,4,6,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[2,3,5,7,6,4,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[2,3,6,4,5,7,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[2,3,6,5,7,4,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[2,4,5,6,3,7,1] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => ? = 6
[2,5,6,7,3,4,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[2,5,6,7,4,3,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[3,1,2,5,6,7,4] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[3,2,5,6,7,4,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[3,5,4,6,7,1,2] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[3,5,7,4,6,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[3,5,7,4,6,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[3,5,7,6,4,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[3,6,4,5,7,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,4,5,7,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,5,7,4,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,7,1,2,4,5,6] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => ? = 6
[3,7,2,4,5,6,1] => [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => ? = 6
[4,1,2,3,5,7,6] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[4,1,2,3,6,5,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,1,2,5,6,7,3] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[4,2,3,5,7,6,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[4,2,3,6,5,7,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,2,5,6,7,3,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[4,3,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[4,3,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [1,7,5,6,2,3,4] => ? = 6
[4,3,5,7,6,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[4,3,5,7,6,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[4,3,6,5,7,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,3,6,5,7,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,1,2,3,6] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,2,3,6,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,3,6,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,3,6,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
Description
The mad of a permutation. According to [1], this is the sum of twice the number of occurrences of the vincular pattern of $(2\underline{31})$ plus the number of occurrences of the vincular patterns $(\underline{31}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Mp00223: Permutations runsortPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00175: Permutations inverse Foata bijectionPermutations
St000809: Permutations ⟶ ℤResult quality: 67% values known / values provided: 67%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[3,5,4,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 6
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 6
[3,2,4,5,6,1] => [1,2,4,5,6,3] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 6
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[2,3,5,7,4,6,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[2,3,5,7,6,4,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[2,3,6,4,5,7,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[2,3,6,5,7,4,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[2,4,5,6,3,7,1] => [1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 6
[2,5,6,7,3,4,1] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[2,5,6,7,4,3,1] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[3,1,2,5,6,7,4] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[3,2,5,6,7,4,1] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[3,5,4,6,7,1,2] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[3,5,7,4,6,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[3,5,7,4,6,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[3,5,7,6,4,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[3,6,4,5,7,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,4,5,7,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,5,7,4,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,7,1,2,4,5,6] => [1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 6
[3,7,2,4,5,6,1] => [1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 6
[4,1,2,3,5,7,6] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[4,1,2,3,6,5,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,1,2,5,6,7,3] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[4,2,3,5,7,6,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[4,2,3,6,5,7,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,2,5,6,7,3,1] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[4,3,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[4,3,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => ? = 6
[4,3,5,7,6,1,2] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[4,3,5,7,6,2,1] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
[4,3,6,5,7,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,3,6,5,7,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,1,2,3,6] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,2,3,6,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,3,6,1,2] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,3,6,2,1] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [7,1,5,2,3,4,6] => ? = 6
Description
The reduced reflection length of the permutation. Let $T$ be the set of reflections in a Coxeter group and let $\ell(w)$ be the usual length function. Then the reduced reflection length of $w$ is $$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$$ In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].
Matching statistic: St000866
Mp00223: Permutations runsortPermutations
Mp00067: Permutations Foata bijectionPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
St000866: Permutations ⟶ ℤResult quality: 67% values known / values provided: 67%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [4,1,6,2,5,3] => 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [4,1,6,2,5,3] => 6
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [4,1,6,2,5,3] => 6
[3,2,4,5,6,1] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [4,1,6,2,5,3] => 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [4,1,6,2,5,3,7] => ? = 6
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[2,3,5,7,4,6,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[2,3,5,7,6,4,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[2,3,6,4,5,7,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[2,3,6,5,7,4,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[2,4,5,6,3,7,1] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [4,1,6,2,5,3,7] => ? = 6
[2,5,6,7,3,4,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[2,5,6,7,4,3,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[3,1,2,5,6,7,4] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[3,2,5,6,7,4,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[3,5,4,6,7,1,2] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[3,5,7,4,6,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[3,5,7,4,6,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[3,5,7,6,4,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[3,6,4,5,7,1,2] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,4,5,7,2,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,5,7,4,1,2] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[3,7,1,2,4,5,6] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [4,1,6,2,5,3,7] => ? = 6
[3,7,2,4,5,6,1] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => [4,1,6,2,5,3,7] => ? = 6
[4,1,2,3,5,7,6] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[4,1,2,3,6,5,7] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,1,2,5,6,7,3] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[4,2,3,5,7,6,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[4,2,3,6,5,7,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,2,5,6,7,3,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[4,3,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[4,3,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,5,2,3,6,7,4] => [1,5,2,7,3,6,4] => ? = 6
[4,3,5,7,6,1,2] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[4,3,5,7,6,2,1] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
[4,3,6,5,7,1,2] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,3,6,5,7,2,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,1,2,3,6] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,2,3,6,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,3,6,1,2] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,5,7,3,6,2,1] => [1,2,3,6,4,5,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 4
[4,6,1,2,3,5,7] => [1,2,3,5,7,4,6] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 6
Description
The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. An admissible inversion of a permutation $\sigma$ is a pair $(\sigma_i,\sigma_j)$ such that 1. $i < j$ and $\sigma_i > \sigma_j$ and 2. either $\sigma_j < \sigma_{j+1}$ or there exists a $i < k < j$ with $\sigma_k < \sigma_j$. This version was introduced by John Shareshian and Michelle L. Wachs in [1], for a closely related version, see [[St000463]].
Matching statistic: St000455
Mp00223: Permutations runsortPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00160: Permutations graph of inversionsGraphs
St000455: Graphs ⟶ ℤResult quality: 33% values known / values provided: 53%distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 4
[1,2] => [1,2] => [1,2] => ([],2)
=> ? = 0 - 4
[2,1] => [1,2] => [1,2] => ([],2)
=> ? = 0 - 4
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 - 4
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 - 4
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 - 4
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 - 4
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 - 4
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 - 4
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 - 4
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 - 4
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 - 4
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 - 4
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 - 4
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 0 - 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[3,5,4,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[3,5,4,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 0 - 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 6 - 4
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 6 - 4
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 6 - 4
[3,2,4,5,6,1] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 6 - 4
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[3,5,4,6,1,2] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,5,4,6,2,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[3,6,4,5,1,2] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[3,6,4,5,2,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[3,6,5,4,1,2] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[3,6,5,4,2,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,1,2,3,6,5] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,2,3,6,5,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,3,6,5,1,2] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,3,6,5,2,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,5,1,2,3,6] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,5,2,3,6,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,5,3,6,1,2] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,5,3,6,2,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[4,6,1,2,3,5] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[4,6,2,3,5,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[4,6,3,5,1,2] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[4,6,3,5,2,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[5,1,2,3,6,4] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[5,2,3,6,4,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[5,3,6,4,1,2] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[5,3,6,4,2,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[5,4,1,2,3,6] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[5,4,2,3,6,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[5,4,3,6,1,2] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[5,4,3,6,2,1] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 4 - 4
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 0 - 4
[6,1,2,3,5,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[6,2,3,5,4,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[6,3,5,4,1,2] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[6,3,5,4,2,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[6,4,1,2,3,5] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[6,4,2,3,5,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[6,4,3,5,1,2] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[6,4,3,5,2,1] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 0 = 4 - 4
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000796
Mp00223: Permutations runsortPermutations
Mp00066: Permutations inversePermutations
Mp00175: Permutations inverse Foata bijectionPermutations
St000796: Permutations ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,5,4,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[3,5,4,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => 4
[2,4,5,6,3,1] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 6
[3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 6
[3,2,4,5,6,1] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[1,2,3,5,7,6,4] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
[1,2,5,6,7,4,3] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
[2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[2,3,5,7,4,6,1] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[2,3,5,7,6,4,1] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[2,3,6,4,5,7,1] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[2,3,6,5,7,4,1] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[2,3,7,4,5,6,1] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[2,3,7,5,6,4,1] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[2,3,7,6,4,5,1] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[2,3,7,6,5,4,1] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[2,5,6,7,3,4,1] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
[2,5,6,7,4,3,1] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
[3,1,2,5,6,7,4] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
[3,2,5,6,7,4,1] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
[3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
[3,4,2,5,6,7,1] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
[3,5,4,6,7,1,2] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 4
[3,5,7,4,6,1,2] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[3,5,7,4,6,2,1] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[3,5,7,6,4,1,2] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[3,6,4,5,7,1,2] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[3,6,4,5,7,2,1] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[3,6,5,7,4,1,2] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[3,7,4,5,6,1,2] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[3,7,4,5,6,2,1] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[3,7,5,6,4,1,2] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[3,7,6,4,5,1,2] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[3,7,6,4,5,2,1] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[3,7,6,5,4,1,2] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[4,1,2,3,5,7,6] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,6,7,2,3,4,5] => ? = 6
[4,1,2,3,6,5,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [5,6,1,2,3,4,7] => ? = 4
[4,1,2,3,7,5,6] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[4,1,2,3,7,6,5] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ? = 4
[4,1,2,5,6,7,3] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,6,2,7,3,4,5] => ? = 6
Description
The stat' of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns $(\underline{13}2)$, $(\underline{31}2)$, $(\underline{32}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
The following 191 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001519The pinnacle sum of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000632The jump number of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000030The sum of the descent differences of a permutations. St001511The minimal number of transpositions needed to sort a permutation in either direction. St000029The depth of a permutation. St000154The sum of the descent bottoms of a permutation. St000224The sorting index of a permutation. St000311The number of vertices of odd degree in a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000794The mak of a permutation. St000264The girth of a graph, which is not a tree. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St001964The interval resolution global dimension of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001428The number of B-inversions of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001060The distinguishing index of a graph. St001684The reduced word complexity of a permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000462The major index minus the number of excedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001703The villainy of a graph. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001848The atomic length of a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000653The last descent of a permutation. St001497The position of the largest weak excedence of a permutation. St001555The order of a signed permutation. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St000635The number of strictly order preserving maps of a poset into itself. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St001769The reflection length of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001894The depth of a signed permutation. St001895The oddness of a signed permutation. St001896The number of right descents of a signed permutations. St001851The number of Hecke atoms of a signed permutation. St001852The size of the conjugacy class of the signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001890The maximum magnitude of the Möbius function of a poset. St000068The number of minimal elements in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001624The breadth of a lattice. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St000322The skewness of a graph. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph.