Identifier
Values
[+] => [1] => [1] => [1] => 0
[-] => [1] => [1] => [1] => 0
[+,+] => [1,2] => [1,2] => [1,2] => 0
[-,+] => [1,2] => [1,2] => [1,2] => 0
[+,-] => [1,2] => [1,2] => [1,2] => 0
[-,-] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[+,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,-,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,+] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,+,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-,-,-] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[+,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[-,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,+] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,1,-] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,+,1] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[3,-,1] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[+,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[-,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[+,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[-,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[+,4,+,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0
[-,4,+,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0
[+,4,-,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0
[-,4,-,2] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,+] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,1,-] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[2,4,+,1] => [2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[2,4,-,1] => [2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[3,1,2,+] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,2,-] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 0
[3,+,1,+] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[3,-,1,+] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[3,+,1,-] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[3,-,1,-] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[3,+,4,1] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 2
[3,-,4,1] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 2
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,4,2,1] => [3,4,2,1] => [2,4,3,1] => [2,4,3,1] => 1
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[4,1,+,2] => [4,1,3,2] => [4,3,1,2] => [4,3,1,2] => 1
[4,1,-,2] => [4,1,3,2] => [4,3,1,2] => [4,3,1,2] => 1
[4,+,1,3] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[4,-,1,3] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[4,+,+,1] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,-,+,1] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,+,-,1] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,-,-,1] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[4,3,1,2] => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 0
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 0
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
>>> Load all 218 entries. <<<
[+,+,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,+,-,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,+,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,+,-,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,+,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,-,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,+,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,-,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,+,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,+,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,-,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[-,-,-,-,-] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[-,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[+,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[+,+,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[-,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[-,+,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[+,-,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[-,-,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[+,+,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[-,+,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[+,-,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[+,+,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[-,-,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[-,+,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[+,-,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[-,-,4,3,-] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[+,+,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[-,+,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[+,-,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[-,-,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[+,+,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[-,+,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[+,-,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[-,-,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[+,+,5,+,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[-,+,5,+,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[+,-,5,+,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[+,+,5,-,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[-,-,5,+,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[-,+,5,-,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[+,-,5,-,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[-,-,5,-,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[+,3,2,+,+] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[-,3,2,+,+] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[+,3,2,-,+] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[+,3,2,+,-] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[-,3,2,-,+] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[-,3,2,+,-] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[+,3,2,-,-] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[-,3,2,-,-] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[+,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[-,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[+,3,4,2,+] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[-,3,4,2,+] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[+,3,4,2,-] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[-,3,4,2,-] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[+,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[-,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[+,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[-,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[+,3,5,+,2] => [1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[-,3,5,+,2] => [1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[+,3,5,-,2] => [1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[-,3,5,-,2] => [1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[+,4,2,3,+] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[-,4,2,3,+] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[+,4,2,3,-] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[-,4,2,3,-] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[+,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[-,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[+,4,+,2,+] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[-,4,+,2,+] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[+,4,-,2,+] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[+,4,+,2,-] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[-,4,-,2,+] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[-,4,+,2,-] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[+,4,-,2,-] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[-,4,-,2,-] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[+,4,+,5,2] => [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[-,4,+,5,2] => [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[+,4,-,5,2] => [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[-,4,-,5,2] => [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[+,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[-,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[+,4,5,3,2] => [1,4,5,3,2] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[-,4,5,3,2] => [1,4,5,3,2] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[+,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[-,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[+,5,2,+,3] => [1,5,2,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[-,5,2,+,3] => [1,5,2,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[+,5,2,-,3] => [1,5,2,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[-,5,2,-,3] => [1,5,2,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[+,5,+,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[-,5,+,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[+,5,-,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[-,5,-,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[+,5,+,+,2] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[-,5,+,+,2] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[+,5,-,+,2] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[+,5,+,-,2] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[-,5,-,+,2] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[-,5,+,-,2] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[+,5,-,-,2] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[-,5,-,-,2] => [1,5,3,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[+,5,4,2,3] => [1,5,4,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[-,5,4,2,3] => [1,5,4,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[+,5,4,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[-,5,4,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
  • $-i < -j < -\pi(j) < -\pi(i)$, or
  • $-i < j \leq \pi(j) < -\pi(i)$, or
  • $i < j \leq \pi(j) < \pi(i)$.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation $\sigma$ is a pair $i < j$ such that $i < \sigma(j) < \sigma(i)$. The element $\sigma(j)$ is then an invisible inversion bottom.
A descent view in a permutation $\pi$ is an element $\pi(j)$ such that $\pi(i+1) < \pi(j) < \pi(i)$, and additionally the smallest element in the decreasing run containing $\pi(i)$ is smaller than the smallest element in the decreasing run containing $\pi(j)$.
This map is a bijection $\chi:\mathfrak S_n \to \mathfrak S_n$, such that
  • the multiset of descent views in $\pi$ is the multiset of invisible inversion bottoms in $\chi(\pi)$,
  • the set of left-to-right maxima of $\pi$ is the set of maximal elements in the cycles of $\chi(\pi)$,
  • the set of global ascent of $\pi$ is the set of global ascent of $\chi(\pi)$,
  • the set of maximal elements in the decreasing runs of $\pi$ is the set of weak deficiency positions of $\chi(\pi)$, and
  • the set of minimal elements in the decreasing runs of $\pi$ is the set of weak deficiency values of $\chi(\pi)$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
permutation
Description
The underlying permutation of the decorated permutation.