Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000260
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
 => 0
[-] => [1] => ([],1)
 => 0
[-,+] => [2,1] => ([(0,1)],2)
 => 1
[-,+,+] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[-,-,+] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[-,+,+,+] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,-,+,+] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[-,-,-,+] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[4,-,+,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[-,+,+,+,+] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[-,-,+,+,+] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,+,-,+,+] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[-,+,+,-,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[-,-,-,+,+] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,-,+,-,+] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[-,+,-,-,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[-,-,-,-,+] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[-,+,4,3,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[-,-,4,3,+] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[-,3,2,+,+] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[-,3,2,-,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[-,3,4,2,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[-,4,2,3,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[-,4,+,2,+] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[-,4,-,2,+] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[-,5,-,2,4] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[-,5,-,+,2] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,+,1,+] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[2,5,-,+,1] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[4,-,+,1,+] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[5,-,+,1,4] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[5,-,+,+,1] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[5,+,-,+,1] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[5,-,-,+,1] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[5,-,+,-,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[5,-,4,3,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[5,3,2,+,1] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[-,+,+,+,+,+] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[-,-,+,+,+,+] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[-,+,-,+,+,+] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[-,+,+,-,+,+] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[-,+,+,+,-,+] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2
[-,-,-,+,+,+] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 2
[-,-,+,-,+,+] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[-,-,+,+,-,+] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[-,+,-,-,+,+] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[-,+,-,+,-,+] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2
[-,+,+,-,-,+] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St001498
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 49%●distinct values known / distinct values provided: 25%
Values
[+] => [1] => [1] => [1,0]
 => ? = 0 - 2
[-] => [1] => [1] => [1,0]
 => ? = 0 - 2
[-,+] => [2,1] => [2,1] => [1,1,0,0]
 => ? = 1 - 2
[-,+,+] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => ? = 1 - 2
[-,-,+] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 1 - 2
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => ? = 1 - 2
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 0 = 2 - 2
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? = 2 - 2
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 - 2
[-,3,2,+] => [2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? = 2 - 2
[4,-,+,1] => [3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? = 2 - 2
[-,+,+,+,+] => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 - 2
[-,-,+,+,+] => [3,4,5,1,2] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,+,-,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 2 - 2
[-,+,+,-,+] => [2,3,5,1,4] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,-,-,+,+] => [4,5,1,2,3] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[-,+,-,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,-,-,-,+] => [5,1,2,3,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 - 2
[-,+,4,3,+] => [2,3,5,1,4] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,-,4,3,+] => [3,5,1,2,4] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[-,3,2,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 2 - 2
[-,3,2,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,3,4,2,+] => [2,5,1,3,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,4,2,3,+] => [2,3,5,1,4] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,4,+,2,+] => [3,2,5,1,4] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 2 - 2
[-,4,-,2,+] => [2,5,1,4,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 2 - 2
[-,5,-,2,4] => [2,4,1,5,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,5,-,+,2] => [4,2,1,5,3] => [2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 2 - 2
[2,4,+,1,+] => [3,1,5,2,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[2,5,-,+,1] => [4,1,2,5,3] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[4,-,+,1,+] => [3,1,5,4,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 2 - 2
[5,-,+,1,4] => [3,1,4,5,2] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[5,-,+,+,1] => [3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[5,+,-,+,1] => [2,4,1,5,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[5,-,-,+,1] => [4,1,5,2,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 0 = 2 - 2
[5,-,+,-,1] => [3,1,5,2,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[5,-,4,3,1] => [3,1,5,2,4] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[5,3,2,+,1] => [2,4,1,5,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 2
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 - 2
[-,-,+,+,+,+] => [3,4,5,6,1,2] => [5,1,3,6,2,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 0 = 2 - 2
[-,+,-,+,+,+] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,+,+,-,+,+] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0 = 2 - 2
[-,+,+,+,-,+] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,-,+,+,+] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 0 = 2 - 2
[-,-,+,-,+,+] => [3,5,6,1,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0 = 2 - 2
[-,-,+,+,-,+] => [3,4,6,1,2,5] => [6,5,2,4,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,+,-,-,+,+] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,+,-,+,-,+] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[-,+,+,-,-,+] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,-,-,+,+] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 0 = 2 - 2
[-,-,-,+,-,+] => [4,6,1,2,3,5] => [6,5,3,1,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,+,-,-,+] => [3,6,1,2,4,5] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0 = 2 - 2
[-,+,-,-,-,+] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,-,-,-,+] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 - 2
[-,+,+,5,4,+] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,+,5,4,+] => [3,4,6,1,2,5] => [6,5,2,4,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,+,-,5,4,+] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[-,-,-,5,4,+] => [4,6,1,2,3,5] => [6,5,3,1,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,+,4,3,+,+] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0 = 2 - 2
[-,-,4,3,+,+] => [3,5,6,1,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0 = 2 - 2
[-,+,4,3,-,+] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,4,3,-,+] => [3,6,1,2,4,5] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0 = 2 - 2
[-,+,4,5,3,+] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,4,5,3,+] => [3,6,1,2,4,5] => [3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0 = 2 - 2
[-,+,5,3,4,+] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,5,3,4,+] => [3,4,6,1,2,5] => [6,5,2,4,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,+,5,+,3,+] => [2,4,3,6,1,5] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,-,5,+,3,+] => [4,3,6,1,2,5] => [4,1,6,5,2,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 0 = 2 - 2
[-,+,5,-,3,+] => [2,3,6,1,5,4] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,-,5,-,3,+] => [3,6,1,2,5,4] => [3,1,5,6,4,2] => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 0 = 2 - 2
[-,+,6,-,3,5] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,-,6,-,3,5] => [3,5,1,2,6,4] => [3,1,6,4,2,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0 = 2 - 2
[-,+,6,-,+,3] => [2,5,3,1,6,4] => [3,6,4,1,2,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,-,6,-,+,3] => [5,3,1,2,6,4] => [6,4,2,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,3,2,+,+,+] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,3,2,-,+,+] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,3,2,+,-,+] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[-,3,2,-,-,+] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,3,2,5,4,+] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[-,3,4,2,+,+] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 2
[-,3,5,2,4,+] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[-,3,5,+,2,+] => [4,2,6,1,3,5] => [2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 0 = 2 - 2
[-,3,5,-,2,+] => [2,6,1,3,5,4] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,3,6,-,+,2] => [5,2,1,3,6,4] => [2,6,4,3,1,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,4,2,3,+,+] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0 = 2 - 2
[-,4,+,2,+,+] => [3,2,5,6,1,4] => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0 = 2 - 2
[-,4,-,2,+,+] => [2,5,6,1,4,3] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0 = 2 - 2
[-,4,+,2,-,+] => [3,2,6,1,4,5] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,4,-,2,-,+] => [2,6,1,4,3,5] => [4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,4,+,5,2,+] => [3,2,6,1,4,5] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,4,-,5,2,+] => [2,6,1,4,3,5] => [4,6,5,3,1,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,4,5,3,2,+] => [3,2,6,1,4,5] => [2,6,5,4,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,5,2,+,3,+] => [2,4,3,6,1,5] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,5,2,-,3,+] => [2,3,6,1,5,4] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,5,+,2,4,+] => [3,2,4,6,1,5] => [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0 = 2 - 2
[-,5,-,2,4,+] => [2,4,6,1,5,3] => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 0 = 2 - 2
[-,5,-,+,2,+] => [4,2,6,1,5,3] => [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 0 = 2 - 2
[-,5,+,-,2,+] => [3,2,6,1,5,4] => [2,5,6,4,1,3] => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 0 = 2 - 2
[-,5,4,2,3,+] => [2,3,6,1,5,4] => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0 = 2 - 2
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001569
(load all 100 compositions to match this statistic)
Mapping
Mp00255: Decorated permutations —lower permutation⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => ? = 0
[-] => [1] => ? = 0
[-,+] => [2,1] => 1
[-,+,+] => [2,3,1] => 1
[-,-,+] => [3,1,2] => 1
[-,+,+,+] => [2,3,4,1] => 1
[-,-,+,+] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => 2
[-,-,-,+] => [4,1,2,3] => 1
[-,3,2,+] => [2,4,1,3] => 2
[4,-,+,1] => [3,1,4,2] => 2
[-,+,+,+,+] => [2,3,4,5,1] => 1
[-,-,+,+,+] => [3,4,5,1,2] => 2
[-,+,-,+,+] => [2,4,5,1,3] => 2
[-,+,+,-,+] => [2,3,5,1,4] => 2
[-,-,-,+,+] => [4,5,1,2,3] => 2
[-,-,+,-,+] => [3,5,1,2,4] => 2
[-,+,-,-,+] => [2,5,1,3,4] => 2
[-,-,-,-,+] => [5,1,2,3,4] => 1
[-,+,4,3,+] => [2,3,5,1,4] => 2
[-,-,4,3,+] => [3,5,1,2,4] => 2
[-,3,2,+,+] => [2,4,5,1,3] => 2
[-,3,2,-,+] => [2,5,1,3,4] => 2
[-,3,4,2,+] => [2,5,1,3,4] => 2
[-,4,2,3,+] => [2,3,5,1,4] => 2
[-,4,+,2,+] => [3,2,5,1,4] => 2
[-,4,-,2,+] => [2,5,1,4,3] => 2
[-,5,-,2,4] => [2,4,1,5,3] => 2
[-,5,-,+,2] => [4,2,1,5,3] => 2
[2,4,+,1,+] => [3,1,5,2,4] => 2
[2,5,-,+,1] => [4,1,2,5,3] => 2
[4,-,+,1,+] => [3,1,5,4,2] => 2
[5,-,+,1,4] => [3,1,4,5,2] => 2
[5,-,+,+,1] => [3,4,1,5,2] => 2
[5,+,-,+,1] => [2,4,1,5,3] => 2
[5,-,-,+,1] => [4,1,5,2,3] => 2
[5,-,+,-,1] => [3,1,5,2,4] => 2
[5,-,4,3,1] => [3,1,5,2,4] => 2
[5,3,2,+,1] => [2,4,1,5,3] => 2
[-,+,+,+,+,+] => [2,3,4,5,6,1] => ? = 1
[-,-,+,+,+,+] => [3,4,5,6,1,2] => ? = 2
[-,+,-,+,+,+] => [2,4,5,6,1,3] => ? = 2
[-,+,+,-,+,+] => [2,3,5,6,1,4] => ? = 2
[-,+,+,+,-,+] => [2,3,4,6,1,5] => ? = 2
[-,-,-,+,+,+] => [4,5,6,1,2,3] => ? = 2
[-,-,+,-,+,+] => [3,5,6,1,2,4] => ? = 2
[-,-,+,+,-,+] => [3,4,6,1,2,5] => ? = 2
[-,+,-,-,+,+] => [2,5,6,1,3,4] => ? = 2
[-,+,-,+,-,+] => [2,4,6,1,3,5] => ? = 2
[-,+,+,-,-,+] => [2,3,6,1,4,5] => ? = 2
[-,-,-,-,+,+] => [5,6,1,2,3,4] => ? = 2
[-,-,-,+,-,+] => [4,6,1,2,3,5] => ? = 2
[-,-,+,-,-,+] => [3,6,1,2,4,5] => ? = 2
[-,+,-,-,-,+] => [2,6,1,3,4,5] => ? = 2
[-,-,-,-,-,+] => [6,1,2,3,4,5] => ? = 1
[-,+,+,5,4,+] => [2,3,4,6,1,5] => ? = 2
[-,-,+,5,4,+] => [3,4,6,1,2,5] => ? = 2
[-,+,-,5,4,+] => [2,4,6,1,3,5] => ? = 2
[-,-,-,5,4,+] => [4,6,1,2,3,5] => ? = 2
[-,+,4,3,+,+] => [2,3,5,6,1,4] => ? = 2
[-,-,4,3,+,+] => [3,5,6,1,2,4] => ? = 2
[-,+,4,3,-,+] => [2,3,6,1,4,5] => ? = 2
[-,-,4,3,-,+] => [3,6,1,2,4,5] => ? = 2
[-,+,4,5,3,+] => [2,3,6,1,4,5] => ? = 2
[-,-,4,5,3,+] => [3,6,1,2,4,5] => ? = 2
[-,+,5,3,4,+] => [2,3,4,6,1,5] => ? = 2
[-,-,5,3,4,+] => [3,4,6,1,2,5] => ? = 2
[-,+,5,+,3,+] => [2,4,3,6,1,5] => ? = 2
[-,-,5,+,3,+] => [4,3,6,1,2,5] => ? = 2
[-,+,5,-,3,+] => [2,3,6,1,5,4] => ? = 2
[-,-,5,-,3,+] => [3,6,1,2,5,4] => ? = 2
[-,+,6,-,3,5] => [2,3,5,1,6,4] => ? = 2
[-,-,6,-,3,5] => [3,5,1,2,6,4] => ? = 2
[-,+,6,-,+,3] => [2,5,3,1,6,4] => ? = 2
[-,-,6,-,+,3] => [5,3,1,2,6,4] => ? = 2
[-,3,2,+,+,+] => [2,4,5,6,1,3] => ? = 2
[-,3,2,-,+,+] => [2,5,6,1,3,4] => ? = 2
[-,3,2,+,-,+] => [2,4,6,1,3,5] => ? = 2
[-,3,2,-,-,+] => [2,6,1,3,4,5] => ? = 2
[-,3,2,5,4,+] => [2,4,6,1,3,5] => ? = 2
[-,3,4,2,+,+] => [2,5,6,1,3,4] => ? = 2
[-,3,4,2,-,+] => [2,6,1,3,4,5] => ? = 2
[-,3,4,5,2,+] => [2,6,1,3,4,5] => ? = 2
[-,3,5,2,4,+] => [2,4,6,1,3,5] => ? = 2
[-,3,5,+,2,+] => [4,2,6,1,3,5] => ? = 2
[-,3,5,-,2,+] => [2,6,1,3,5,4] => ? = 2
[-,3,6,-,2,5] => [2,5,1,3,6,4] => ? = 2
Description
The maximal modular displacement of a permutation. This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Matching statistic: St001668
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1] => ([],1)
 => ? = 0
[-] => [1] => [1] => ([],1)
 => ? = 0
[-,+] => [2,1] => [1,2] => ([(0,1)],2)
 => 1
[-,+,+] => [2,3,1] => [2,1,3] => ([(0,2),(1,2)],3)
 => 1
[-,-,+] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => 1
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 2
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 1
[-,3,2,+] => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[4,-,+,1] => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 2
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 1
[-,+,4,3,+] => [2,4,5,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,-,4,3,+] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[-,3,2,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,3,2,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[-,3,4,2,+] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[-,4,2,3,+] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,4,+,2,+] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,4,-,2,+] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[-,5,-,2,4] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[-,5,-,+,2] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[2,4,+,1,+] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[2,5,-,+,1] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[4,-,+,1,+] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[5,-,+,1,4] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[5,-,+,+,1] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[5,+,-,+,1] => [2,4,5,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[5,-,-,+,1] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[5,-,+,-,1] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[5,-,4,3,1] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2
[5,3,2,+,1] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 1
[-,-,+,+,+,+] => [3,4,5,6,1,2] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,+,-,+,+,+] => [2,4,5,6,1,3] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,+,+,-,+,+] => [2,3,5,6,1,4] => [5,4,2,1,6,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,+,+,+,-,+] => [2,3,4,6,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,-,-,+,+,+] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,+,-,+,+] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,+,+,-,+] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,+,-,-,+,+] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,+,-,+,-,+] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,+,+,-,-,+] => [2,3,6,1,4,5] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,-,-,+,+] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,-,-,+,-,+] => [4,6,1,2,3,5] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,-,+,-,-,+] => [3,6,1,2,4,5] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,+,-,-,-,+] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,-,-,-,-,+] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ? = 1
[-,+,+,5,4,+] => [2,3,5,6,1,4] => [5,4,2,1,6,3] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,-,+,5,4,+] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,+,-,5,4,+] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,-,5,4,+] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,+,4,3,+,+] => [2,4,5,6,1,3] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,-,4,3,+,+] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,+,4,3,-,+] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,4,3,-,+] => [4,6,1,2,3,5] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,+,4,5,3,+] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,4,5,3,+] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,+,5,3,4,+] => [2,4,5,6,1,3] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,-,5,3,4,+] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,+,5,+,3,+] => [2,4,5,6,1,3] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,-,5,+,3,+] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,+,5,-,3,+] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,5,-,3,+] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,+,6,-,3,5] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,6,-,3,5] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,+,6,-,+,3] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,-,6,-,+,3] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,3,2,+,+,+] => [3,4,5,6,1,2] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2
[-,3,2,-,+,+] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,3,2,+,-,+] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,3,2,-,-,+] => [3,6,1,2,4,5] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,3,2,5,4,+] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,3,4,2,+,+] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,3,4,2,-,+] => [4,6,1,2,3,5] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,3,4,5,2,+] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,3,5,2,4,+] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,3,5,+,2,+] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 2
[-,3,5,-,2,+] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
[-,3,6,-,2,5] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
 => ? = 2
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001520
(load all 2 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1] => [1] => ? = 0 - 1
[-] => [1] => [1] => [1] => ? = 0 - 1
[-,+] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [2,4,3,1] => 1 = 2 - 1
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [3,2,4,1] => 1 = 2 - 1
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[-,3,2,+] => [3,4,1,2] => [3,4,1,2] => [2,4,3,1] => 1 = 2 - 1
[4,-,+,1] => [3,4,1,2] => [3,4,1,2] => [2,4,3,1] => 1 = 2 - 1
[-,+,+,+,+] => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [3,2,4,1,5] => 1 = 2 - 1
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,1,3,4] => [3,2,4,5,1] => 1 = 2 - 1
[-,-,-,+,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[-,-,+,-,+] => [3,5,1,2,4] => [1,3,5,2,4] => [1,4,3,5,2] => 1 = 2 - 1
[-,+,-,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [4,2,3,5,1] => 1 = 2 - 1
[-,-,-,-,+] => [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[-,+,4,3,+] => [2,4,5,1,3] => [2,4,1,3,5] => [3,2,4,1,5] => 1 = 2 - 1
[-,-,4,3,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[-,3,2,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[-,3,2,-,+] => [3,5,1,2,4] => [1,3,5,2,4] => [1,4,3,5,2] => 1 = 2 - 1
[-,3,4,2,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[-,4,2,3,+] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[-,4,+,2,+] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[-,4,-,2,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[-,5,-,2,4] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[-,5,-,+,2] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[2,4,+,1,+] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[2,5,-,+,1] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[4,-,+,1,+] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[5,-,+,1,4] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[5,-,+,+,1] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[5,+,-,+,1] => [2,4,5,1,3] => [2,4,1,3,5] => [3,2,4,1,5] => 1 = 2 - 1
[5,-,-,+,1] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[5,-,+,-,1] => [3,5,1,2,4] => [1,3,5,2,4] => [1,4,3,5,2] => 1 = 2 - 1
[5,-,4,3,1] => [4,5,1,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => 1 = 2 - 1
[5,3,2,+,1] => [3,4,5,1,2] => [3,4,1,2,5] => [2,4,3,1,5] => 1 = 2 - 1
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1 - 1
[-,-,+,+,+,+] => [3,4,5,6,1,2] => [3,4,1,2,5,6] => [2,4,3,1,5,6] => ? = 2 - 1
[-,+,-,+,+,+] => [2,4,5,6,1,3] => [2,4,1,3,5,6] => [3,2,4,1,5,6] => ? = 2 - 1
[-,+,+,-,+,+] => [2,3,5,6,1,4] => [2,5,1,3,4,6] => [3,2,4,5,1,6] => ? = 2 - 1
[-,+,+,+,-,+] => [2,3,4,6,1,5] => [2,6,1,3,4,5] => [3,2,4,5,6,1] => ? = 2 - 1
[-,-,-,+,+,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [2,3,6,4,5,1] => ? = 2 - 1
[-,-,+,-,+,+] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => [2,4,3,6,5,1] => ? = 2 - 1
[-,-,+,+,-,+] => [3,4,6,1,2,5] => [3,4,6,1,2,5] => [2,5,3,4,6,1] => ? = 2 - 1
[-,+,-,-,+,+] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [3,2,4,6,5,1] => ? = 2 - 1
[-,+,-,+,-,+] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => [3,2,5,4,6,1] => ? = 2 - 1
[-,+,+,-,-,+] => [2,3,6,1,4,5] => [2,3,6,1,4,5] => [4,2,3,5,6,1] => ? = 2 - 1
[-,-,-,-,+,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
[-,-,-,+,-,+] => [4,6,1,2,3,5] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => ? = 2 - 1
[-,-,+,-,-,+] => [3,6,1,2,4,5] => [1,3,4,6,2,5] => [1,5,3,4,6,2] => ? = 2 - 1
[-,+,-,-,-,+] => [2,6,1,3,4,5] => [2,3,4,6,1,5] => [5,2,3,4,6,1] => ? = 2 - 1
[-,-,-,-,-,+] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 - 1
[-,+,+,5,4,+] => [2,3,5,6,1,4] => [2,5,1,3,4,6] => [3,2,4,5,1,6] => ? = 2 - 1
[-,-,+,5,4,+] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => [2,4,3,6,5,1] => ? = 2 - 1
[-,+,-,5,4,+] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [3,2,4,6,5,1] => ? = 2 - 1
[-,-,-,5,4,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
[-,+,4,3,+,+] => [2,4,5,6,1,3] => [2,4,1,3,5,6] => [3,2,4,1,5,6] => ? = 2 - 1
[-,-,4,3,+,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [2,3,6,4,5,1] => ? = 2 - 1
[-,+,4,3,-,+] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => [3,2,5,4,6,1] => ? = 2 - 1
[-,-,4,3,-,+] => [4,6,1,2,3,5] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => ? = 2 - 1
[-,+,4,5,3,+] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [3,2,4,6,5,1] => ? = 2 - 1
[-,-,4,5,3,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
[-,+,5,3,4,+] => [2,4,5,6,1,3] => [2,4,1,3,5,6] => [3,2,4,1,5,6] => ? = 2 - 1
[-,-,5,3,4,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [2,3,6,4,5,1] => ? = 2 - 1
[-,+,5,+,3,+] => [2,4,5,6,1,3] => [2,4,1,3,5,6] => [3,2,4,1,5,6] => ? = 2 - 1
[-,-,5,+,3,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [2,3,6,4,5,1] => ? = 2 - 1
[-,+,5,-,3,+] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [3,2,4,6,5,1] => ? = 2 - 1
[-,-,5,-,3,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
[-,+,6,-,3,5] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [3,2,4,6,5,1] => ? = 2 - 1
[-,-,6,-,3,5] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
[-,+,6,-,+,3] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [3,2,4,6,5,1] => ? = 2 - 1
[-,-,6,-,+,3] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
[-,3,2,+,+,+] => [3,4,5,6,1,2] => [3,4,1,2,5,6] => [2,4,3,1,5,6] => ? = 2 - 1
[-,3,2,-,+,+] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => [2,4,3,6,5,1] => ? = 2 - 1
[-,3,2,+,-,+] => [3,4,6,1,2,5] => [3,4,6,1,2,5] => [2,5,3,4,6,1] => ? = 2 - 1
[-,3,2,-,-,+] => [3,6,1,2,4,5] => [1,3,4,6,2,5] => [1,5,3,4,6,2] => ? = 2 - 1
[-,3,2,5,4,+] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => [2,4,3,6,5,1] => ? = 2 - 1
[-,3,4,2,+,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [2,3,6,4,5,1] => ? = 2 - 1
[-,3,4,2,-,+] => [4,6,1,2,3,5] => [1,2,4,6,3,5] => [1,2,5,4,6,3] => ? = 2 - 1
[-,3,4,5,2,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
[-,3,5,2,4,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [2,3,6,4,5,1] => ? = 2 - 1
[-,3,5,+,2,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [2,3,6,4,5,1] => ? = 2 - 1
[-,3,5,-,2,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
[-,3,6,-,2,5] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,4,6,5,3] => ? = 2 - 1
Description
The number of strict 3-descents. A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Matching statistic: St001960
(load all 3 compositions to match this statistic)
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[+] => [1] => [1] => [1] => ? = 0 - 1
[-] => [1] => [1] => [1] => ? = 0 - 1
[-,+] => [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[-,+,+] => [2,3,1] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[-,-,+] => [3,1,2] => [3,2,1] => [2,3,1] => 0 = 1 - 1
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => [4,1,3,2] => 1 = 2 - 1
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => [3,4,2,1] => 1 = 2 - 1
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => [3,4,1,2] => 0 = 1 - 1
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => [4,1,3,2] => 1 = 2 - 1
[4,-,+,1] => [3,4,1,2] => [3,1,4,2] => [4,1,3,2] => 1 = 2 - 1
[-,+,+,+,+] => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 0 = 1 - 1
[-,-,+,+,+] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[-,+,-,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => [5,1,2,4,3] => 1 = 2 - 1
[-,+,+,-,+] => [2,3,5,1,4] => [5,4,1,2,3] => [4,5,2,3,1] => 1 = 2 - 1
[-,-,-,+,+] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [4,1,3,5,2] => 1 = 2 - 1
[-,+,-,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => [3,4,5,2,1] => 1 = 2 - 1
[-,-,-,-,+] => [5,1,2,3,4] => [5,4,3,2,1] => [3,4,5,1,2] => 0 = 1 - 1
[-,+,4,3,+] => [2,4,5,1,3] => [4,1,2,5,3] => [5,1,2,4,3] => 1 = 2 - 1
[-,-,4,3,+] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[-,3,2,+,+] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[-,3,2,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => [4,1,3,5,2] => 1 = 2 - 1
[-,3,4,2,+] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[-,4,2,3,+] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[-,4,+,2,+] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[-,4,-,2,+] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[-,5,-,2,4] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[-,5,-,+,2] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[2,4,+,1,+] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[2,5,-,+,1] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[4,-,+,1,+] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[5,-,+,1,4] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[5,-,+,+,1] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[5,+,-,+,1] => [2,4,5,1,3] => [4,1,2,5,3] => [5,1,2,4,3] => 1 = 2 - 1
[5,-,-,+,1] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[5,-,+,-,1] => [3,5,1,2,4] => [3,1,5,4,2] => [4,1,3,5,2] => 1 = 2 - 1
[5,-,4,3,1] => [4,5,1,2,3] => [5,3,1,4,2] => [3,4,2,5,1] => 1 = 2 - 1
[5,3,2,+,1] => [3,4,5,1,2] => [5,2,4,1,3] => [2,4,5,3,1] => 1 = 2 - 1
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 1 - 1
[-,-,+,+,+,+] => [3,4,5,6,1,2] => [5,1,3,6,2,4] => [3,1,6,5,4,2] => ? = 2 - 1
[-,+,-,+,+,+] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => [3,6,5,2,4,1] => ? = 2 - 1
[-,+,+,-,+,+] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => [6,1,2,3,5,4] => ? = 2 - 1
[-,+,+,+,-,+] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => [5,6,2,3,4,1] => ? = 2 - 1
[-,-,-,+,+,+] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [6,1,4,3,5,2] => ? = 2 - 1
[-,-,+,-,+,+] => [3,5,6,1,2,4] => [5,2,6,4,1,3] => [2,5,4,6,3,1] => ? = 2 - 1
[-,-,+,+,-,+] => [3,4,6,1,2,5] => [6,5,2,4,1,3] => [4,5,1,6,3,2] => ? = 2 - 1
[-,+,-,-,+,+] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [4,5,2,3,6,1] => ? = 2 - 1
[-,+,-,+,-,+] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => [5,1,2,4,6,3] => ? = 2 - 1
[-,+,+,-,-,+] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => ? = 2 - 1
[-,-,-,-,+,+] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[-,-,-,+,-,+] => [4,6,1,2,3,5] => [6,5,3,1,4,2] => [3,5,6,2,1,4] => ? = 2 - 1
[-,-,+,-,-,+] => [3,6,1,2,4,5] => [3,1,6,5,4,2] => [5,1,3,6,2,4] => ? = 2 - 1
[-,+,-,-,-,+] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => [4,5,6,1,3,2] => ? = 2 - 1
[-,-,-,-,-,+] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => [4,5,6,1,2,3] => ? = 1 - 1
[-,+,+,5,4,+] => [2,3,5,6,1,4] => [5,1,2,3,6,4] => [6,1,2,3,5,4] => ? = 2 - 1
[-,-,+,5,4,+] => [3,5,6,1,2,4] => [5,2,6,4,1,3] => [2,5,4,6,3,1] => ? = 2 - 1
[-,+,-,5,4,+] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [4,5,2,3,6,1] => ? = 2 - 1
[-,-,-,5,4,+] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[-,+,4,3,+,+] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => [3,6,5,2,4,1] => ? = 2 - 1
[-,-,4,3,+,+] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [6,1,4,3,5,2] => ? = 2 - 1
[-,+,4,3,-,+] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => [5,1,2,4,6,3] => ? = 2 - 1
[-,-,4,3,-,+] => [4,6,1,2,3,5] => [6,5,3,1,4,2] => [3,5,6,2,1,4] => ? = 2 - 1
[-,+,4,5,3,+] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [4,5,2,3,6,1] => ? = 2 - 1
[-,-,4,5,3,+] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[-,+,5,3,4,+] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => [3,6,5,2,4,1] => ? = 2 - 1
[-,-,5,3,4,+] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [6,1,4,3,5,2] => ? = 2 - 1
[-,+,5,+,3,+] => [2,4,5,6,1,3] => [6,3,5,1,2,4] => [3,6,5,2,4,1] => ? = 2 - 1
[-,-,5,+,3,+] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [6,1,4,3,5,2] => ? = 2 - 1
[-,+,5,-,3,+] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [4,5,2,3,6,1] => ? = 2 - 1
[-,-,5,-,3,+] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[-,+,6,-,3,5] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [4,5,2,3,6,1] => ? = 2 - 1
[-,-,6,-,3,5] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[-,+,6,-,+,3] => [2,5,6,1,3,4] => [6,4,1,2,5,3] => [4,5,2,3,6,1] => ? = 2 - 1
[-,-,6,-,+,3] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[-,3,2,+,+,+] => [3,4,5,6,1,2] => [5,1,3,6,2,4] => [3,1,6,5,4,2] => ? = 2 - 1
[-,3,2,-,+,+] => [3,5,6,1,2,4] => [5,2,6,4,1,3] => [2,5,4,6,3,1] => ? = 2 - 1
[-,3,2,+,-,+] => [3,4,6,1,2,5] => [6,5,2,4,1,3] => [4,5,1,6,3,2] => ? = 2 - 1
[-,3,2,-,-,+] => [3,6,1,2,4,5] => [3,1,6,5,4,2] => [5,1,3,6,2,4] => ? = 2 - 1
[-,3,2,5,4,+] => [3,5,6,1,2,4] => [5,2,6,4,1,3] => [2,5,4,6,3,1] => ? = 2 - 1
[-,3,4,2,+,+] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [6,1,4,3,5,2] => ? = 2 - 1
[-,3,4,2,-,+] => [4,6,1,2,3,5] => [6,5,3,1,4,2] => [3,5,6,2,1,4] => ? = 2 - 1
[-,3,4,5,2,+] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[-,3,5,2,4,+] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [6,1,4,3,5,2] => ? = 2 - 1
[-,3,5,+,2,+] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [6,1,4,3,5,2] => ? = 2 - 1
[-,3,5,-,2,+] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[-,3,6,-,2,5] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
Description
The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001864
Mapping
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 75%
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[-,+] => [2,1] => [2,1] => [2,1] => 1
[-,+,+] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[-,-,+] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[-,+,+,+] => [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[-,-,+,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[-,+,-,+] => [2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[-,3,2,+] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[4,-,+,1] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 2
[-,+,+,+,+] => [2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[-,-,+,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[-,+,-,+,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[-,+,+,-,+] => [2,3,5,1,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[-,-,-,+,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[-,-,+,-,+] => [3,5,1,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[-,+,-,-,+] => [2,5,1,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[-,-,-,-,+] => [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[-,+,4,3,+] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[-,-,4,3,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[-,3,2,+,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[-,3,2,-,+] => [3,5,1,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[-,3,4,2,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[-,4,2,3,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[-,4,+,2,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[-,4,-,2,+] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[-,5,-,2,4] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[-,5,-,+,2] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[2,4,+,1,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[2,5,-,+,1] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[4,-,+,1,+] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[5,-,+,1,4] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[5,-,+,+,1] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[5,+,-,+,1] => [2,4,5,1,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[5,-,-,+,1] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[5,-,+,-,1] => [3,5,1,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 2
[5,-,4,3,1] => [4,5,1,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[5,3,2,+,1] => [3,4,5,1,2] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 1
[-,-,+,+,+,+] => [3,4,5,6,1,2] => [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[-,+,-,+,+,+] => [2,4,5,6,1,3] => [2,4,1,3,5,6] => [2,4,1,3,5,6] => ? = 2
[-,+,+,-,+,+] => [2,3,5,6,1,4] => [2,5,1,3,4,6] => [2,5,1,3,4,6] => ? = 2
[-,+,+,+,-,+] => [2,3,4,6,1,5] => [2,6,1,3,4,5] => [2,6,1,3,4,5] => ? = 2
[-,-,-,+,+,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2
[-,-,+,-,+,+] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => ? = 2
[-,-,+,+,-,+] => [3,4,6,1,2,5] => [3,4,6,1,2,5] => [3,4,6,1,2,5] => ? = 2
[-,+,-,-,+,+] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => ? = 2
[-,+,-,+,-,+] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => ? = 2
[-,+,+,-,-,+] => [2,3,6,1,4,5] => [2,3,6,1,4,5] => [2,3,6,1,4,5] => ? = 2
[-,-,-,-,+,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[-,-,-,+,-,+] => [4,6,1,2,3,5] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => ? = 2
[-,-,+,-,-,+] => [3,6,1,2,4,5] => [1,3,4,6,2,5] => [1,3,4,6,2,5] => ? = 2
[-,+,-,-,-,+] => [2,6,1,3,4,5] => [2,3,4,6,1,5] => [2,3,4,6,1,5] => ? = 2
[-,-,-,-,-,+] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1
[-,+,+,5,4,+] => [2,3,5,6,1,4] => [2,5,1,3,4,6] => [2,5,1,3,4,6] => ? = 2
[-,-,+,5,4,+] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => [3,5,6,1,2,4] => ? = 2
[-,+,-,5,4,+] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => ? = 2
[-,-,-,5,4,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[-,+,4,3,+,+] => [2,4,5,6,1,3] => [2,4,1,3,5,6] => [2,4,1,3,5,6] => ? = 2
[-,-,4,3,+,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2
[-,+,4,3,-,+] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => ? = 2
[-,-,4,3,-,+] => [4,6,1,2,3,5] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => ? = 2
[-,+,4,5,3,+] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => ? = 2
[-,-,4,5,3,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[-,+,5,3,4,+] => [2,4,5,6,1,3] => [2,4,1,3,5,6] => [2,4,1,3,5,6] => ? = 2
[-,-,5,3,4,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2
[-,+,5,+,3,+] => [2,4,5,6,1,3] => [2,4,1,3,5,6] => [2,4,1,3,5,6] => ? = 2
[-,-,5,+,3,+] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2
[-,+,5,-,3,+] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => ? = 2
[-,-,5,-,3,+] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[-,+,6,-,3,5] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => ? = 2
[-,-,6,-,3,5] => [5,6,1,2,3,4] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[-,+,6,-,+,3] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => [2,5,6,1,3,4] => ? = 2
Description
The number of excedances of a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.