Identifier
Values
=>
[+]=>0
[-]=>0
[+,+]=>0
[-,+]=>0
[+,-]=>0
[-,-]=>0
[2,1]=>0
[+,+,+]=>0
[-,+,+]=>0
[+,-,+]=>0
[+,+,-]=>0
[-,-,+]=>0
[-,+,-]=>0
[+,-,-]=>0
[-,-,-]=>0
[+,3,2]=>0
[-,3,2]=>0
[2,1,+]=>0
[2,1,-]=>0
[2,3,1]=>0
[3,1,2]=>0
[3,+,1]=>0
[3,-,1]=>0
[+,+,+,+]=>0
[-,+,+,+]=>0
[+,-,+,+]=>0
[+,+,-,+]=>0
[+,+,+,-]=>0
[-,-,+,+]=>0
[-,+,-,+]=>0
[-,+,+,-]=>0
[+,-,-,+]=>0
[+,-,+,-]=>0
[+,+,-,-]=>0
[-,-,-,+]=>0
[-,-,+,-]=>0
[-,+,-,-]=>0
[+,-,-,-]=>0
[-,-,-,-]=>0
[+,+,4,3]=>0
[-,+,4,3]=>0
[+,-,4,3]=>0
[-,-,4,3]=>0
[+,3,2,+]=>0
[-,3,2,+]=>0
[+,3,2,-]=>0
[-,3,2,-]=>0
[+,3,4,2]=>0
[-,3,4,2]=>0
[+,4,2,3]=>0
[-,4,2,3]=>0
[+,4,+,2]=>0
[-,4,+,2]=>0
[+,4,-,2]=>0
[-,4,-,2]=>0
[2,1,+,+]=>0
[2,1,-,+]=>0
[2,1,+,-]=>0
[2,1,-,-]=>0
[2,1,4,3]=>0
[2,3,1,+]=>0
[2,3,1,-]=>0
[2,3,4,1]=>0
[2,4,1,3]=>1
[2,4,+,1]=>0
[2,4,-,1]=>0
[3,1,2,+]=>0
[3,1,2,-]=>0
[3,1,4,2]=>1
[3,+,1,+]=>0
[3,-,1,+]=>0
[3,+,1,-]=>0
[3,-,1,-]=>0
[3,+,4,1]=>0
[3,-,4,1]=>0
[3,4,1,2]=>0
[3,4,2,1]=>1
[4,1,2,3]=>0
[4,1,+,2]=>0
[4,1,-,2]=>0
[4,+,1,3]=>0
[4,-,1,3]=>0
[4,+,+,1]=>0
[4,-,+,1]=>0
[4,+,-,1]=>0
[4,-,-,1]=>0
[4,3,1,2]=>1
[4,3,2,1]=>0
[+,+,+,+,+]=>0
[-,+,+,+,+]=>0
[+,-,+,+,+]=>0
[+,+,-,+,+]=>0
[+,+,+,-,+]=>0
[+,+,+,+,-]=>0
[-,-,+,+,+]=>0
[-,+,-,+,+]=>0
[-,+,+,-,+]=>0
[-,+,+,+,-]=>0
[+,-,-,+,+]=>0
[+,-,+,-,+]=>0
[+,-,+,+,-]=>0
[+,+,-,-,+]=>0
[+,+,-,+,-]=>0
[+,+,+,-,-]=>0
[-,-,-,+,+]=>0
[-,-,+,-,+]=>0
[-,-,+,+,-]=>0
[-,+,-,-,+]=>0
[-,+,-,+,-]=>0
[-,+,+,-,-]=>0
[+,-,-,-,+]=>0
[+,-,-,+,-]=>0
[+,-,+,-,-]=>0
[+,+,-,-,-]=>0
[-,-,-,-,+]=>0
[-,-,-,+,-]=>0
[-,-,+,-,-]=>0
[-,+,-,-,-]=>0
[+,-,-,-,-]=>0
[-,-,-,-,-]=>0
[+,+,+,5,4]=>0
[-,+,+,5,4]=>0
[+,-,+,5,4]=>0
[+,+,-,5,4]=>0
[-,-,+,5,4]=>0
[-,+,-,5,4]=>0
[+,-,-,5,4]=>0
[-,-,-,5,4]=>0
[+,+,4,3,+]=>0
[-,+,4,3,+]=>0
[+,-,4,3,+]=>0
[+,+,4,3,-]=>0
[-,-,4,3,+]=>0
[-,+,4,3,-]=>0
[+,-,4,3,-]=>0
[-,-,4,3,-]=>0
[+,+,4,5,3]=>0
[-,+,4,5,3]=>0
[+,-,4,5,3]=>0
[-,-,4,5,3]=>0
[+,+,5,3,4]=>0
[-,+,5,3,4]=>0
[+,-,5,3,4]=>0
[-,-,5,3,4]=>0
[+,+,5,+,3]=>0
[-,+,5,+,3]=>0
[+,-,5,+,3]=>0
[+,+,5,-,3]=>0
[-,-,5,+,3]=>0
[-,+,5,-,3]=>0
[+,-,5,-,3]=>0
[-,-,5,-,3]=>0
[+,3,2,+,+]=>0
[-,3,2,+,+]=>0
[+,3,2,-,+]=>0
[+,3,2,+,-]=>0
[-,3,2,-,+]=>0
[-,3,2,+,-]=>0
[+,3,2,-,-]=>0
[-,3,2,-,-]=>0
[+,3,2,5,4]=>0
[-,3,2,5,4]=>0
[+,3,4,2,+]=>0
[-,3,4,2,+]=>0
[+,3,4,2,-]=>0
[-,3,4,2,-]=>0
[+,3,4,5,2]=>0
[-,3,4,5,2]=>0
[+,3,5,2,4]=>1
[-,3,5,2,4]=>1
[+,3,5,+,2]=>0
[-,3,5,+,2]=>0
[+,3,5,-,2]=>0
[-,3,5,-,2]=>0
[+,4,2,3,+]=>0
[-,4,2,3,+]=>0
[+,4,2,3,-]=>0
[-,4,2,3,-]=>0
[+,4,2,5,3]=>1
[-,4,2,5,3]=>1
[+,4,+,2,+]=>0
[-,4,+,2,+]=>0
[+,4,-,2,+]=>0
[+,4,+,2,-]=>0
[-,4,-,2,+]=>0
[-,4,+,2,-]=>0
[+,4,-,2,-]=>0
[-,4,-,2,-]=>0
[+,4,+,5,2]=>0
[-,4,+,5,2]=>0
[+,4,-,5,2]=>0
[-,4,-,5,2]=>0
[+,4,5,2,3]=>0
[-,4,5,2,3]=>0
[+,4,5,3,2]=>1
[-,4,5,3,2]=>1
[+,5,2,3,4]=>0
[-,5,2,3,4]=>0
[+,5,2,+,3]=>0
[-,5,2,+,3]=>0
[+,5,2,-,3]=>0
[-,5,2,-,3]=>0
[+,5,+,2,4]=>0
[-,5,+,2,4]=>0
[+,5,-,2,4]=>0
[-,5,-,2,4]=>0
[+,5,+,+,2]=>0
[-,5,+,+,2]=>0
[+,5,-,+,2]=>0
[+,5,+,-,2]=>0
[-,5,-,+,2]=>0
[-,5,+,-,2]=>0
[+,5,-,-,2]=>0
[-,5,-,-,2]=>0
[+,5,4,2,3]=>1
[-,5,4,2,3]=>1
[+,5,4,3,2]=>0
[-,5,4,3,2]=>0
[2,1,+,+,+]=>0
[2,1,-,+,+]=>0
[2,1,+,-,+]=>0
[2,1,+,+,-]=>0
[2,1,-,-,+]=>0
[2,1,-,+,-]=>0
[2,1,+,-,-]=>0
[2,1,-,-,-]=>0
[2,1,+,5,4]=>0
[2,1,-,5,4]=>0
[2,1,4,3,+]=>0
[2,1,4,3,-]=>0
[2,1,4,5,3]=>0
[2,1,5,3,4]=>0
[2,1,5,+,3]=>0
[2,1,5,-,3]=>0
[2,3,1,+,+]=>0
[2,3,1,-,+]=>0
[2,3,1,+,-]=>0
[2,3,1,-,-]=>0
[2,3,1,5,4]=>0
[2,3,4,1,+]=>0
[2,3,4,1,-]=>0
[2,3,4,5,1]=>0
[2,3,5,1,4]=>2
[2,3,5,+,1]=>0
[2,3,5,-,1]=>0
[2,4,1,3,+]=>1
[2,4,1,3,-]=>1
[2,4,1,5,3]=>2
[2,4,+,1,+]=>0
[2,4,-,1,+]=>0
[2,4,+,1,-]=>0
[2,4,-,1,-]=>0
[2,4,+,5,1]=>0
[2,4,-,5,1]=>0
[2,4,5,1,3]=>1
[2,4,5,3,1]=>2
[2,5,1,3,4]=>2
[2,5,1,+,3]=>1
[2,5,1,-,3]=>1
[2,5,+,1,4]=>1
[2,5,-,1,4]=>1
[2,5,+,+,1]=>0
[2,5,-,+,1]=>0
[2,5,+,-,1]=>0
[2,5,-,-,1]=>0
[2,5,4,1,3]=>2
[2,5,4,3,1]=>0
[3,1,2,+,+]=>0
[3,1,2,-,+]=>0
[3,1,2,+,-]=>0
[3,1,2,-,-]=>0
[3,1,2,5,4]=>0
[3,1,4,2,+]=>1
[3,1,4,2,-]=>1
[3,1,4,5,2]=>2
[3,1,5,2,4]=>2
[3,1,5,+,2]=>1
[3,1,5,-,2]=>1
[3,+,1,+,+]=>0
[3,-,1,+,+]=>0
[3,+,1,-,+]=>0
[3,+,1,+,-]=>0
[3,-,1,-,+]=>0
[3,-,1,+,-]=>0
[3,+,1,-,-]=>0
[3,-,1,-,-]=>0
[3,+,1,5,4]=>0
[3,-,1,5,4]=>0
[3,+,4,1,+]=>0
[3,-,4,1,+]=>0
[3,+,4,1,-]=>0
[3,-,4,1,-]=>0
[3,+,4,5,1]=>0
[3,-,4,5,1]=>0
[3,+,5,1,4]=>1
[3,-,5,1,4]=>1
[3,+,5,+,1]=>0
[3,-,5,+,1]=>0
[3,+,5,-,1]=>0
[3,-,5,-,1]=>0
[3,4,1,2,+]=>0
[3,4,1,2,-]=>0
[3,4,1,5,2]=>1
[3,4,2,1,+]=>1
[3,4,2,1,-]=>1
[3,4,2,5,1]=>2
[3,4,5,1,2]=>0
[3,4,5,2,1]=>1
[3,5,1,2,4]=>1
[3,5,1,+,2]=>0
[3,5,1,-,2]=>0
[3,5,2,1,4]=>2
[3,5,2,+,1]=>1
[3,5,2,-,1]=>1
[3,5,4,1,2]=>1
[3,5,4,2,1]=>2
[4,1,2,3,+]=>0
[4,1,2,3,-]=>0
[4,1,2,5,3]=>2
[4,1,+,2,+]=>0
[4,1,-,2,+]=>0
[4,1,+,2,-]=>0
[4,1,-,2,-]=>0
[4,1,+,5,2]=>1
[4,1,-,5,2]=>1
[4,1,5,2,3]=>1
[4,1,5,3,2]=>2
[4,+,1,3,+]=>0
[4,-,1,3,+]=>0
[4,+,1,3,-]=>0
[4,-,1,3,-]=>0
[4,+,1,5,3]=>1
[4,-,1,5,3]=>1
[4,+,+,1,+]=>0
[4,-,+,1,+]=>0
[4,+,-,1,+]=>0
[4,+,+,1,-]=>0
[4,-,-,1,+]=>0
[4,-,+,1,-]=>0
[4,+,-,1,-]=>0
[4,-,-,1,-]=>0
[4,+,+,5,1]=>0
[4,-,+,5,1]=>0
[4,+,-,5,1]=>0
[4,-,-,5,1]=>0
[4,+,5,1,3]=>0
[4,-,5,1,3]=>0
[4,+,5,3,1]=>1
[4,-,5,3,1]=>1
[4,3,1,2,+]=>1
[4,3,1,2,-]=>1
[4,3,1,5,2]=>2
[4,3,2,1,+]=>0
[4,3,2,1,-]=>0
[4,3,2,5,1]=>0
[4,3,5,1,2]=>1
[4,3,5,2,1]=>2
[4,5,1,2,3]=>0
[4,5,1,3,2]=>1
[4,5,2,1,3]=>1
[4,5,2,3,1]=>2
[4,5,+,1,2]=>0
[4,5,-,1,2]=>0
[4,5,+,2,1]=>1
[4,5,-,2,1]=>1
[5,1,2,3,4]=>0
[5,1,2,+,3]=>0
[5,1,2,-,3]=>0
[5,1,+,2,4]=>0
[5,1,-,2,4]=>0
[5,1,+,+,2]=>0
[5,1,-,+,2]=>0
[5,1,+,-,2]=>0
[5,1,-,-,2]=>0
[5,1,4,2,3]=>2
[5,1,4,3,2]=>0
[5,+,1,3,4]=>0
[5,-,1,3,4]=>0
[5,+,1,+,3]=>0
[5,-,1,+,3]=>0
[5,+,1,-,3]=>0
[5,-,1,-,3]=>0
[5,+,+,1,4]=>0
[5,-,+,1,4]=>0
[5,+,-,1,4]=>0
[5,-,-,1,4]=>0
[5,+,+,+,1]=>0
[5,-,+,+,1]=>0
[5,+,-,+,1]=>0
[5,+,+,-,1]=>0
[5,-,-,+,1]=>0
[5,-,+,-,1]=>0
[5,+,-,-,1]=>0
[5,-,-,-,1]=>0
[5,+,4,1,3]=>1
[5,-,4,1,3]=>1
[5,+,4,3,1]=>0
[5,-,4,3,1]=>0
[5,3,1,2,4]=>2
[5,3,1,+,2]=>1
[5,3,1,-,2]=>1
[5,3,2,1,4]=>0
[5,3,2,+,1]=>0
[5,3,2,-,1]=>0
[5,3,4,1,2]=>2
[5,3,4,2,1]=>0
[5,4,1,2,3]=>1
[5,4,1,3,2]=>2
[5,4,2,1,3]=>2
[5,4,2,3,1]=>0
[5,4,+,1,2]=>1
[5,4,-,1,2]=>1
[5,4,+,2,1]=>0
[5,4,-,2,1]=>0
[+,+,+,+,+,+]=>0
[-,+,+,+,+,+]=>0
[+,-,+,+,+,+]=>0
[+,+,-,+,+,+]=>0
[+,+,+,-,+,+]=>0
[+,+,+,+,-,+]=>0
[+,+,+,+,+,-]=>0
[-,-,+,+,+,+]=>0
[-,+,-,+,+,+]=>0
[-,+,+,-,+,+]=>0
[-,+,+,+,-,+]=>0
[-,+,+,+,+,-]=>0
[+,-,-,+,+,+]=>0
[+,-,+,-,+,+]=>0
[+,-,+,+,-,+]=>0
[+,-,+,+,+,-]=>0
[+,+,-,-,+,+]=>0
[+,+,-,+,-,+]=>0
[+,+,-,+,+,-]=>0
[+,+,+,-,-,+]=>0
[+,+,+,-,+,-]=>0
[+,+,+,+,-,-]=>0
[-,-,-,+,+,+]=>0
[-,-,+,-,+,+]=>0
[-,-,+,+,-,+]=>0
[-,-,+,+,+,-]=>0
[-,+,-,-,+,+]=>0
[-,+,-,+,-,+]=>0
[-,+,-,+,+,-]=>0
[-,+,+,-,-,+]=>0
[-,+,+,-,+,-]=>0
[-,+,+,+,-,-]=>0
[+,-,-,-,+,+]=>0
[+,-,-,+,-,+]=>0
[+,-,-,+,+,-]=>0
[+,-,+,-,-,+]=>0
[+,-,+,-,+,-]=>0
[+,-,+,+,-,-]=>0
[+,+,-,-,-,+]=>0
[+,+,-,-,+,-]=>0
[+,+,-,+,-,-]=>0
[+,+,+,-,-,-]=>0
[-,-,-,-,+,+]=>0
[-,-,-,+,-,+]=>0
[-,-,-,+,+,-]=>0
[-,-,+,-,-,+]=>0
[-,-,+,-,+,-]=>0
[-,-,+,+,-,-]=>0
[-,+,-,-,-,+]=>0
[-,+,-,-,+,-]=>0
[-,+,-,+,-,-]=>0
[-,+,+,-,-,-]=>0
[+,-,-,-,-,+]=>0
[+,-,-,-,+,-]=>0
[+,-,-,+,-,-]=>0
[+,-,+,-,-,-]=>0
[+,+,-,-,-,-]=>0
[-,-,-,-,-,+]=>0
[-,-,-,-,+,-]=>0
[-,-,-,+,-,-]=>0
[-,-,+,-,-,-]=>0
[-,+,-,-,-,-]=>0
[+,-,-,-,-,-]=>0
[-,-,-,-,-,-]=>0
[+,+,+,+,6,5]=>0
[-,+,+,+,6,5]=>0
[+,-,+,+,6,5]=>0
[+,+,-,+,6,5]=>0
[+,+,+,-,6,5]=>0
[-,-,+,+,6,5]=>0
[-,+,-,+,6,5]=>0
[-,+,+,-,6,5]=>0
[+,-,-,+,6,5]=>0
[+,-,+,-,6,5]=>0
[+,+,-,-,6,5]=>0
[-,-,-,+,6,5]=>0
[-,-,+,-,6,5]=>0
[-,+,-,-,6,5]=>0
[+,-,-,-,6,5]=>0
[-,-,-,-,6,5]=>0
[+,+,+,5,4,+]=>0
[-,+,+,5,4,+]=>0
[+,-,+,5,4,+]=>0
[+,+,-,5,4,+]=>0
[+,+,+,5,4,-]=>0
[-,-,+,5,4,+]=>0
[-,+,-,5,4,+]=>0
[-,+,+,5,4,-]=>0
[+,-,-,5,4,+]=>0
[+,-,+,5,4,-]=>0
[+,+,-,5,4,-]=>0
[-,-,-,5,4,+]=>0
[-,-,+,5,4,-]=>0
[-,+,-,5,4,-]=>0
[+,-,-,5,4,-]=>0
[-,-,-,5,4,-]=>0
[+,+,+,5,6,4]=>0
[-,+,+,5,6,4]=>0
[+,-,+,5,6,4]=>0
[+,+,-,5,6,4]=>0
[-,-,+,5,6,4]=>0
[-,+,-,5,6,4]=>0
[+,-,-,5,6,4]=>0
[-,-,-,5,6,4]=>0
[+,+,+,6,4,5]=>0
[-,+,+,6,4,5]=>0
[+,-,+,6,4,5]=>0
[+,+,-,6,4,5]=>0
[-,-,+,6,4,5]=>0
[-,+,-,6,4,5]=>0
[+,-,-,6,4,5]=>0
[-,-,-,6,4,5]=>0
[+,+,+,6,+,4]=>0
[-,+,+,6,+,4]=>0
[+,-,+,6,+,4]=>0
[+,+,-,6,+,4]=>0
[+,+,+,6,-,4]=>0
[-,-,+,6,+,4]=>0
[-,+,-,6,+,4]=>0
[-,+,+,6,-,4]=>0
[+,-,-,6,+,4]=>0
[+,-,+,6,-,4]=>0
[+,+,-,6,-,4]=>0
[-,-,-,6,+,4]=>0
[-,-,+,6,-,4]=>0
[-,+,-,6,-,4]=>0
[+,-,-,6,-,4]=>0
[-,-,-,6,-,4]=>0
[+,+,4,3,+,+]=>0
[-,+,4,3,+,+]=>0
[+,-,4,3,+,+]=>0
[+,+,4,3,-,+]=>0
[+,+,4,3,+,-]=>0
[-,-,4,3,+,+]=>0
[-,+,4,3,-,+]=>0
[-,+,4,3,+,-]=>0
[+,-,4,3,-,+]=>0
[+,-,4,3,+,-]=>0
[+,+,4,3,-,-]=>0
[-,-,4,3,-,+]=>0
[-,-,4,3,+,-]=>0
[-,+,4,3,-,-]=>0
[+,-,4,3,-,-]=>0
[-,-,4,3,-,-]=>0
[+,+,4,3,6,5]=>0
[-,+,4,3,6,5]=>0
[+,-,4,3,6,5]=>0
[-,-,4,3,6,5]=>0
[+,+,4,5,3,+]=>0
[-,+,4,5,3,+]=>0
[+,-,4,5,3,+]=>0
[+,+,4,5,3,-]=>0
[-,-,4,5,3,+]=>0
[-,+,4,5,3,-]=>0
[+,-,4,5,3,-]=>0
[-,-,4,5,3,-]=>0
[+,+,4,5,6,3]=>0
[-,+,4,5,6,3]=>0
[+,-,4,5,6,3]=>0
[-,-,4,5,6,3]=>0
[+,+,4,6,3,5]=>1
[-,+,4,6,3,5]=>1
[+,-,4,6,3,5]=>1
[-,-,4,6,3,5]=>1
[+,+,4,6,+,3]=>0
[-,+,4,6,+,3]=>0
[+,-,4,6,+,3]=>0
[+,+,4,6,-,3]=>0
[-,-,4,6,+,3]=>0
[-,+,4,6,-,3]=>0
[+,-,4,6,-,3]=>0
[-,-,4,6,-,3]=>0
[+,+,5,3,4,+]=>0
[-,+,5,3,4,+]=>0
[+,-,5,3,4,+]=>0
[+,+,5,3,4,-]=>0
[-,-,5,3,4,+]=>0
[-,+,5,3,4,-]=>0
[+,-,5,3,4,-]=>0
[-,-,5,3,4,-]=>0
[+,+,5,3,6,4]=>1
[-,+,5,3,6,4]=>1
[+,-,5,3,6,4]=>1
[-,-,5,3,6,4]=>1
[+,+,5,+,3,+]=>0
[-,+,5,+,3,+]=>0
[+,-,5,+,3,+]=>0
[+,+,5,-,3,+]=>0
[+,+,5,+,3,-]=>0
[-,-,5,+,3,+]=>0
[-,+,5,-,3,+]=>0
[-,+,5,+,3,-]=>0
[+,-,5,-,3,+]=>0
[+,-,5,+,3,-]=>0
[+,+,5,-,3,-]=>0
[-,-,5,-,3,+]=>0
[-,-,5,+,3,-]=>0
[-,+,5,-,3,-]=>0
[+,-,5,-,3,-]=>0
[-,-,5,-,3,-]=>0
[+,+,5,+,6,3]=>0
[-,+,5,+,6,3]=>0
[+,-,5,+,6,3]=>0
[+,+,5,-,6,3]=>0
[-,-,5,+,6,3]=>0
[-,+,5,-,6,3]=>0
[+,-,5,-,6,3]=>0
[-,-,5,-,6,3]=>0
[+,+,5,6,3,4]=>0
[-,+,5,6,3,4]=>0
[+,-,5,6,3,4]=>0
[-,-,5,6,3,4]=>0
[+,+,5,6,4,3]=>1
[-,+,5,6,4,3]=>1
[+,-,5,6,4,3]=>1
[-,-,5,6,4,3]=>1
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[2,4,1,3,-,+]=>1
[2,4,1,3,+,-]=>1
[2,4,1,3,-,-]=>1
[2,4,1,3,6,5]=>1
[2,4,1,5,3,+]=>2
[2,4,1,5,3,-]=>2
[2,4,1,5,6,3]=>3
[2,4,1,6,3,5]=>3
[2,4,1,6,+,3]=>2
[2,4,1,6,-,3]=>2
[2,4,+,1,+,+]=>0
[2,4,-,1,+,+]=>0
[2,4,+,1,-,+]=>0
[2,4,+,1,+,-]=>0
[2,4,-,1,-,+]=>0
[2,4,-,1,+,-]=>0
[2,4,+,1,-,-]=>0
[2,4,-,1,-,-]=>0
[2,4,+,1,6,5]=>0
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Description
The number of crossed alignments in a decorated permutation.
Let $w$ be a decorated permutation.
An alignment $(i \mapsto w(i), j \mapsto w(j))$ is crossed if there exists a third arc $(h \mapsto w(h))$ which forms a crossing with both arcs in the alignment.
Let $w$ be a decorated permutation.
An alignment $(i \mapsto w(i), j \mapsto w(j))$ is crossed if there exists a third arc $(h \mapsto w(h))$ which forms a crossing with both arcs in the alignment.
References
[1] Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs arXiv:2207.06508
Code
def cyclic_interval(a, b, n):
if a <= b:
return list(range(a, b+1))
return list(range(a, n+1)) + list(range(1, b+1))
def alignments(x):
"""
sage: x = DecoratedPermutation([8,9,5,-4,7,6,1,3,2])
sage: (3, 5, 6, 6) in alignments(x)
True
sage: len(alignments(x))
13
sage: x = DecoratedPermutation([8,9,5,4,7,-6,1,3,2])
sage: (3, 5, 6, 6) in alignments(x)
False
"""
pi = x.to_signed_permutation().permutation()
n = len(pi)
al = []
for i, e in enumerate(pi, 1):
for j, f in enumerate(pi, 1):
if i == j:
continue
a, b = sorted([i, e])
c, d = sorted([j, f])
if a <= c <= b <= d or c <= a <= d <= b:
continue
if (e not in cyclic_interval(i, f-1, n)
or f not in cyclic_interval(e+1, j, n)):
continue
if j == f and x[j-1] < 0:
continue
if i == e and x[i-1] > 0:
continue
al.append((i, e, j, f))
return al
def statistic(x):
A = alignments(x)
pi = x.to_signed_permutation().permutation()
n = len(pi)
cal = []
for i, e, j, f in A:
a, b = sorted([i, e])
c, d = sorted([j, f])
for k, g in enumerate(pi, 1):
r, s = sorted([k, g])
if ((a <= r <= b <= s or r <= a <= s <= b) and
(c <= r <= d <= s or r <= c <= s <= d)):
cal.append((i, e, j, f, r, k))
break
return len(cal)
Created
Jul 22, 2022 at 12:07 by Martin Rubey
Updated
Jul 22, 2022 at 12:07 by Martin Rubey
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