Identifier
Identifier
Values
=>
[1]=>1 [1,1]=>0 [1,2]=>2 [2,1]=>1 [1,1,1]=>0 [1,1,2]=>0 [1,2,1]=>0 [2,1,1]=>0 [1,1,3]=>1 [1,3,1]=>1 [3,1,1]=>1 [1,2,2]=>1 [2,1,2]=>0 [2,2,1]=>0 [1,2,3]=>3 [1,3,2]=>2 [2,1,3]=>2 [2,3,1]=>2 [3,1,2]=>1 [3,2,1]=>1 [1,1,1,1]=>0 [1,1,1,2]=>0 [1,1,2,1]=>0 [1,2,1,1]=>0 [2,1,1,1]=>0 [1,1,1,3]=>0 [1,1,3,1]=>0 [1,3,1,1]=>0 [3,1,1,1]=>0 [1,1,1,4]=>1 [1,1,4,1]=>1 [1,4,1,1]=>1 [4,1,1,1]=>1 [1,1,2,2]=>0 [1,2,1,2]=>0 [1,2,2,1]=>0 [2,1,1,2]=>0 [2,1,2,1]=>0 [2,2,1,1]=>0 [1,1,2,3]=>0 [1,1,3,2]=>0 [1,2,1,3]=>0 [1,2,3,1]=>0 [1,3,1,2]=>0 [1,3,2,1]=>0 [2,1,1,3]=>0 [2,1,3,1]=>0 [2,3,1,1]=>0 [3,1,1,2]=>0 [3,1,2,1]=>0 [3,2,1,1]=>0 [1,1,2,4]=>1 [1,1,4,2]=>1 [1,2,1,4]=>1 [1,2,4,1]=>1 [1,4,1,2]=>1 [1,4,2,1]=>1 [2,1,1,4]=>1 [2,1,4,1]=>1 [2,4,1,1]=>1 [4,1,1,2]=>1 [4,1,2,1]=>1 [4,2,1,1]=>1 [1,1,3,3]=>0 [1,3,1,3]=>0 [1,3,3,1]=>0 [3,1,1,3]=>0 [3,1,3,1]=>0 [3,3,1,1]=>0 [1,1,3,4]=>2 [1,1,4,3]=>1 [1,3,1,4]=>2 [1,3,4,1]=>2 [1,4,1,3]=>1 [1,4,3,1]=>1 [3,1,1,4]=>2 [3,1,4,1]=>2 [3,4,1,1]=>2 [4,1,1,3]=>1 [4,1,3,1]=>1 [4,3,1,1]=>1 [1,2,2,2]=>1 [2,1,2,2]=>0 [2,2,1,2]=>0 [2,2,2,1]=>0 [1,2,2,3]=>1 [1,2,3,2]=>1 [1,3,2,2]=>1 [2,1,2,3]=>0 [2,1,3,2]=>0 [2,2,1,3]=>0 [2,2,3,1]=>0 [2,3,1,2]=>0 [2,3,2,1]=>0 [3,1,2,2]=>0 [3,2,1,2]=>0 [3,2,2,1]=>0 [1,2,2,4]=>2 [1,2,4,2]=>2 [1,4,2,2]=>2 [2,1,2,4]=>1 [2,1,4,2]=>1 [2,2,1,4]=>1 [2,2,4,1]=>1 [2,4,1,2]=>1 [2,4,2,1]=>1 [4,1,2,2]=>1 [4,2,1,2]=>1 [4,2,2,1]=>1 [1,2,3,3]=>2 [1,3,2,3]=>1 [1,3,3,2]=>1 [2,1,3,3]=>1 [2,3,1,3]=>1 [2,3,3,1]=>1 [3,1,2,3]=>0 [3,1,3,2]=>0 [3,2,1,3]=>0 [3,2,3,1]=>0 [3,3,1,2]=>0 [3,3,2,1]=>0 [1,2,3,4]=>4 [1,2,4,3]=>3 [1,3,2,4]=>3 [1,3,4,2]=>3 [1,4,2,3]=>2 [1,4,3,2]=>2 [2,1,3,4]=>3 [2,1,4,3]=>2 [2,3,1,4]=>3 [2,3,4,1]=>3 [2,4,1,3]=>2 [2,4,3,1]=>2 [3,1,2,4]=>2 [3,1,4,2]=>2 [3,2,1,4]=>2 [3,2,4,1]=>2 [3,4,1,2]=>2 [3,4,2,1]=>2 [4,1,2,3]=>1 [4,1,3,2]=>1 [4,2,1,3]=>1 [4,2,3,1]=>1 [4,3,1,2]=>1 [4,3,2,1]=>1 [1,1,1,1,1]=>0 [1,1,1,1,2]=>0 [1,1,1,2,1]=>0 [1,1,2,1,1]=>0 [1,2,1,1,1]=>0 [2,1,1,1,1]=>0 [1,1,1,1,3]=>0 [1,1,1,3,1]=>0 [1,1,3,1,1]=>0 [1,3,1,1,1]=>0 [3,1,1,1,1]=>0 [1,1,1,1,4]=>0 [1,1,1,4,1]=>0 [1,1,4,1,1]=>0 [1,4,1,1,1]=>0 [4,1,1,1,1]=>0 [1,1,1,1,5]=>1 [1,1,1,5,1]=>1 [1,1,5,1,1]=>1 [1,5,1,1,1]=>1 [5,1,1,1,1]=>1 [1,1,1,2,2]=>0 [1,1,2,1,2]=>0 [1,1,2,2,1]=>0 [1,2,1,1,2]=>0 [1,2,1,2,1]=>0 [1,2,2,1,1]=>0 [2,1,1,1,2]=>0 [2,1,1,2,1]=>0 [2,1,2,1,1]=>0 [2,2,1,1,1]=>0 [1,1,1,2,3]=>0 [1,1,1,3,2]=>0 [1,1,2,1,3]=>0 [1,1,2,3,1]=>0 [1,1,3,1,2]=>0 [1,1,3,2,1]=>0 [1,2,1,1,3]=>0 [1,2,1,3,1]=>0 [1,2,3,1,1]=>0 [1,3,1,1,2]=>0 [1,3,1,2,1]=>0 [1,3,2,1,1]=>0 [2,1,1,1,3]=>0 [2,1,1,3,1]=>0 [2,1,3,1,1]=>0 [2,3,1,1,1]=>0 [3,1,1,1,2]=>0 [3,1,1,2,1]=>0 [3,1,2,1,1]=>0 [3,2,1,1,1]=>0 [1,1,1,2,4]=>0 [1,1,1,4,2]=>0 [1,1,2,1,4]=>0 [1,1,2,4,1]=>0 [1,1,4,1,2]=>0 [1,1,4,2,1]=>0 [1,2,1,1,4]=>0 [1,2,1,4,1]=>0 [1,2,4,1,1]=>0 [1,4,1,1,2]=>0 [1,4,1,2,1]=>0 [1,4,2,1,1]=>0 [2,1,1,1,4]=>0 [2,1,1,4,1]=>0 [2,1,4,1,1]=>0 [2,4,1,1,1]=>0 [4,1,1,1,2]=>0 [4,1,1,2,1]=>0 [4,1,2,1,1]=>0 [4,2,1,1,1]=>0 [1,1,1,2,5]=>1 [1,1,1,5,2]=>1 [1,1,2,1,5]=>1 [1,1,2,5,1]=>1 [1,1,5,1,2]=>1 [1,1,5,2,1]=>1 [1,2,1,1,5]=>1 [1,2,1,5,1]=>1 [1,2,5,1,1]=>1 [1,5,1,1,2]=>1 [1,5,1,2,1]=>1 [1,5,2,1,1]=>1 [2,1,1,1,5]=>1 [2,1,1,5,1]=>1 [2,1,5,1,1]=>1 [2,5,1,1,1]=>1 [5,1,1,1,2]=>1 [5,1,1,2,1]=>1 [5,1,2,1,1]=>1 [5,2,1,1,1]=>1 [1,1,1,3,3]=>0 [1,1,3,1,3]=>0 [1,1,3,3,1]=>0 [1,3,1,1,3]=>0 [1,3,1,3,1]=>0 [1,3,3,1,1]=>0 [3,1,1,1,3]=>0 [3,1,1,3,1]=>0 [3,1,3,1,1]=>0 [3,3,1,1,1]=>0 [1,1,1,3,4]=>0 [1,1,1,4,3]=>0 [1,1,3,1,4]=>0 [1,1,3,4,1]=>0 [1,1,4,1,3]=>0 [1,1,4,3,1]=>0 [1,3,1,1,4]=>0 [1,3,1,4,1]=>0 [1,3,4,1,1]=>0 [1,4,1,1,3]=>0 [1,4,1,3,1]=>0 [1,4,3,1,1]=>0 [3,1,1,1,4]=>0 [3,1,1,4,1]=>0 [3,1,4,1,1]=>0 [3,4,1,1,1]=>0 [4,1,1,1,3]=>0 [4,1,1,3,1]=>0 [4,1,3,1,1]=>0 [4,3,1,1,1]=>0 [1,1,1,3,5]=>1 [1,1,1,5,3]=>1 [1,1,3,1,5]=>1 [1,1,3,5,1]=>1 [1,1,5,1,3]=>1 [1,1,5,3,1]=>1 [1,3,1,1,5]=>1 [1,3,1,5,1]=>1 [1,3,5,1,1]=>1 [1,5,1,1,3]=>1 [1,5,1,3,1]=>1 [1,5,3,1,1]=>1 [3,1,1,1,5]=>1 [3,1,1,5,1]=>1 [3,1,5,1,1]=>1 [3,5,1,1,1]=>1 [5,1,1,1,3]=>1 [5,1,1,3,1]=>1 [5,1,3,1,1]=>1 [5,3,1,1,1]=>1 [1,1,1,4,4]=>0 [1,1,4,1,4]=>0 [1,1,4,4,1]=>0 [1,4,1,1,4]=>0 [1,4,1,4,1]=>0 [1,4,4,1,1]=>0 [4,1,1,1,4]=>0 [4,1,1,4,1]=>0 [4,1,4,1,1]=>0 [4,4,1,1,1]=>0 [1,1,1,4,5]=>2 [1,1,1,5,4]=>1 [1,1,4,1,5]=>2 [1,1,4,5,1]=>2 [1,1,5,1,4]=>1 [1,1,5,4,1]=>1 [1,4,1,1,5]=>2 [1,4,1,5,1]=>2 [1,4,5,1,1]=>2 [1,5,1,1,4]=>1 [1,5,1,4,1]=>1 [1,5,4,1,1]=>1 [4,1,1,1,5]=>2 [4,1,1,5,1]=>2 [4,1,5,1,1]=>2 [4,5,1,1,1]=>2 [5,1,1,1,4]=>1 [5,1,1,4,1]=>1 [5,1,4,1,1]=>1 [5,4,1,1,1]=>1 [1,1,2,2,2]=>0 [1,2,1,2,2]=>0 [1,2,2,1,2]=>0 [1,2,2,2,1]=>0 [2,1,1,2,2]=>0 [2,1,2,1,2]=>0 [2,1,2,2,1]=>0 [2,2,1,1,2]=>0 [2,2,1,2,1]=>0 [2,2,2,1,1]=>0 [1,1,2,2,3]=>0 [1,1,2,3,2]=>0 [1,1,3,2,2]=>0 [1,2,1,2,3]=>0 [1,2,1,3,2]=>0 [1,2,2,1,3]=>0 [1,2,2,3,1]=>0 [1,2,3,1,2]=>0 [1,2,3,2,1]=>0 [1,3,1,2,2]=>0 [1,3,2,1,2]=>0 [1,3,2,2,1]=>0 [2,1,1,2,3]=>0 [2,1,1,3,2]=>0 [2,1,2,1,3]=>0 [2,1,2,3,1]=>0 [2,1,3,1,2]=>0 [2,1,3,2,1]=>0 [2,2,1,1,3]=>0 [2,2,1,3,1]=>0 [2,2,3,1,1]=>0 [2,3,1,1,2]=>0 [2,3,1,2,1]=>0 [2,3,2,1,1]=>0 [3,1,1,2,2]=>0 [3,1,2,1,2]=>0 [3,1,2,2,1]=>0 [3,2,1,1,2]=>0 [3,2,1,2,1]=>0 [3,2,2,1,1]=>0 [1,1,2,2,4]=>0 [1,1,2,4,2]=>0 [1,1,4,2,2]=>0 [1,2,1,2,4]=>0 [1,2,1,4,2]=>0 [1,2,2,1,4]=>0 [1,2,2,4,1]=>0 [1,2,4,1,2]=>0 [1,2,4,2,1]=>0 [1,4,1,2,2]=>0 [1,4,2,1,2]=>0 [1,4,2,2,1]=>0 [2,1,1,2,4]=>0 [2,1,1,4,2]=>0 [2,1,2,1,4]=>0 [2,1,2,4,1]=>0 [2,1,4,1,2]=>0 [2,1,4,2,1]=>0 [2,2,1,1,4]=>0 [2,2,1,4,1]=>0 [2,2,4,1,1]=>0 [2,4,1,1,2]=>0 [2,4,1,2,1]=>0 [2,4,2,1,1]=>0 [4,1,1,2,2]=>0 [4,1,2,1,2]=>0 [4,1,2,2,1]=>0 [4,2,1,1,2]=>0 [4,2,1,2,1]=>0 [4,2,2,1,1]=>0 [1,1,2,2,5]=>1 [1,1,2,5,2]=>1 [1,1,5,2,2]=>1 [1,2,1,2,5]=>1 [1,2,1,5,2]=>1 [1,2,2,1,5]=>1 [1,2,2,5,1]=>1 [1,2,5,1,2]=>1 [1,2,5,2,1]=>1 [1,5,1,2,2]=>1 [1,5,2,1,2]=>1 [1,5,2,2,1]=>1 [2,1,1,2,5]=>1 [2,1,1,5,2]=>1 [2,1,2,1,5]=>1 [2,1,2,5,1]=>1 [2,1,5,1,2]=>1 [2,1,5,2,1]=>1 [2,2,1,1,5]=>1 [2,2,1,5,1]=>1 [2,2,5,1,1]=>1 [2,5,1,1,2]=>1 [2,5,1,2,1]=>1 [2,5,2,1,1]=>1 [5,1,1,2,2]=>1 [5,1,2,1,2]=>1 [5,1,2,2,1]=>1 [5,2,1,1,2]=>1 [5,2,1,2,1]=>1 [5,2,2,1,1]=>1 [1,1,2,3,3]=>0 [1,1,3,2,3]=>0 [1,1,3,3,2]=>0 [1,2,1,3,3]=>0 [1,2,3,1,3]=>0 [1,2,3,3,1]=>0 [1,3,1,2,3]=>0 [1,3,1,3,2]=>0 [1,3,2,1,3]=>0 [1,3,2,3,1]=>0 [1,3,3,1,2]=>0 [1,3,3,2,1]=>0 [2,1,1,3,3]=>0 [2,1,3,1,3]=>0 [2,1,3,3,1]=>0 [2,3,1,1,3]=>0 [2,3,1,3,1]=>0 [2,3,3,1,1]=>0 [3,1,1,2,3]=>0 [3,1,1,3,2]=>0 [3,1,2,1,3]=>0 [3,1,2,3,1]=>0 [3,1,3,1,2]=>0 [3,1,3,2,1]=>0 [3,2,1,1,3]=>0 [3,2,1,3,1]=>0 [3,2,3,1,1]=>0 [3,3,1,1,2]=>0 [3,3,1,2,1]=>0 [3,3,2,1,1]=>0 [1,1,2,3,4]=>0 [1,1,2,4,3]=>0 [1,1,3,2,4]=>0 [1,1,3,4,2]=>0 [1,1,4,2,3]=>0 [1,1,4,3,2]=>0 [1,2,1,3,4]=>0 [1,2,1,4,3]=>0 [1,2,3,1,4]=>0 [1,2,3,4,1]=>0 [1,2,4,1,3]=>0 [1,2,4,3,1]=>0 [1,3,1,2,4]=>0 [1,3,1,4,2]=>0 [1,3,2,1,4]=>0 [1,3,2,4,1]=>0 [1,3,4,1,2]=>0 [1,3,4,2,1]=>0 [1,4,1,2,3]=>0 [1,4,1,3,2]=>0 [1,4,2,1,3]=>0 [1,4,2,3,1]=>0 [1,4,3,1,2]=>0 [1,4,3,2,1]=>0 [2,1,1,3,4]=>0 [2,1,1,4,3]=>0 [2,1,3,1,4]=>0 [2,1,3,4,1]=>0 [2,1,4,1,3]=>0 [2,1,4,3,1]=>0 [2,3,1,1,4]=>0 [2,3,1,4,1]=>0 [2,3,4,1,1]=>0 [2,4,1,1,3]=>0 [2,4,1,3,1]=>0 [2,4,3,1,1]=>0 [3,1,1,2,4]=>0 [3,1,1,4,2]=>0 [3,1,2,1,4]=>0 [3,1,2,4,1]=>0 [3,1,4,1,2]=>0 [3,1,4,2,1]=>0 [3,2,1,1,4]=>0 [3,2,1,4,1]=>0 [3,2,4,1,1]=>0 [3,4,1,1,2]=>0 [3,4,1,2,1]=>0 [3,4,2,1,1]=>0 [4,1,1,2,3]=>0 [4,1,1,3,2]=>0 [4,1,2,1,3]=>0 [4,1,2,3,1]=>0 [4,1,3,1,2]=>0 [4,1,3,2,1]=>0 [4,2,1,1,3]=>0 [4,2,1,3,1]=>0 [4,2,3,1,1]=>0 [4,3,1,1,2]=>0 [4,3,1,2,1]=>0 [4,3,2,1,1]=>0 [1,1,2,3,5]=>1 [1,1,2,5,3]=>1 [1,1,3,2,5]=>1 [1,1,3,5,2]=>1 [1,1,5,2,3]=>1 [1,1,5,3,2]=>1 [1,2,1,3,5]=>1 [1,2,1,5,3]=>1 [1,2,3,1,5]=>1 [1,2,3,5,1]=>1 [1,2,5,1,3]=>1 [1,2,5,3,1]=>1 [1,3,1,2,5]=>1 [1,3,1,5,2]=>1 [1,3,2,1,5]=>1 [1,3,2,5,1]=>1 [1,3,5,1,2]=>1 [1,3,5,2,1]=>1 [1,5,1,2,3]=>1 [1,5,1,3,2]=>1 [1,5,2,1,3]=>1 [1,5,2,3,1]=>1 [1,5,3,1,2]=>1 [1,5,3,2,1]=>1 [2,1,1,3,5]=>1 [2,1,1,5,3]=>1 [2,1,3,1,5]=>1 [2,1,3,5,1]=>1 [2,1,5,1,3]=>1 [2,1,5,3,1]=>1 [2,3,1,1,5]=>1 [2,3,1,5,1]=>1 [2,3,5,1,1]=>1 [2,5,1,1,3]=>1 [2,5,1,3,1]=>1 [2,5,3,1,1]=>1 [3,1,1,2,5]=>1 [3,1,1,5,2]=>1 [3,1,2,1,5]=>1 [3,1,2,5,1]=>1 [3,1,5,1,2]=>1 [3,1,5,2,1]=>1 [3,2,1,1,5]=>1 [3,2,1,5,1]=>1 [3,2,5,1,1]=>1 [3,5,1,1,2]=>1 [3,5,1,2,1]=>1 [3,5,2,1,1]=>1 [5,1,1,2,3]=>1 [5,1,1,3,2]=>1 [5,1,2,1,3]=>1 [5,1,2,3,1]=>1 [5,1,3,1,2]=>1 [5,1,3,2,1]=>1 [5,2,1,1,3]=>1 [5,2,1,3,1]=>1 [5,2,3,1,1]=>1 [5,3,1,1,2]=>1 [5,3,1,2,1]=>1 [5,3,2,1,1]=>1 [1,1,2,4,4]=>0 [1,1,4,2,4]=>0 [1,1,4,4,2]=>0 [1,2,1,4,4]=>0 [1,2,4,1,4]=>0 [1,2,4,4,1]=>0 [1,4,1,2,4]=>0 [1,4,1,4,2]=>0 [1,4,2,1,4]=>0 [1,4,2,4,1]=>0 [1,4,4,1,2]=>0 [1,4,4,2,1]=>0 [2,1,1,4,4]=>0 [2,1,4,1,4]=>0 [2,1,4,4,1]=>0 [2,4,1,1,4]=>0 [2,4,1,4,1]=>0 [2,4,4,1,1]=>0 [4,1,1,2,4]=>0 [4,1,1,4,2]=>0 [4,1,2,1,4]=>0 [4,1,2,4,1]=>0 [4,1,4,1,2]=>0 [4,1,4,2,1]=>0 [4,2,1,1,4]=>0 [4,2,1,4,1]=>0 [4,2,4,1,1]=>0 [4,4,1,1,2]=>0 [4,4,1,2,1]=>0 [4,4,2,1,1]=>0 [1,1,2,4,5]=>2 [1,1,2,5,4]=>1 [1,1,4,2,5]=>2 [1,1,4,5,2]=>2 [1,1,5,2,4]=>1 [1,1,5,4,2]=>1 [1,2,1,4,5]=>2 [1,2,1,5,4]=>1 [1,2,4,1,5]=>2 [1,2,4,5,1]=>2 [1,2,5,1,4]=>1 [1,2,5,4,1]=>1 [1,4,1,2,5]=>2 [1,4,1,5,2]=>2 [1,4,2,1,5]=>2 [1,4,2,5,1]=>2 [1,4,5,1,2]=>2 [1,4,5,2,1]=>2 [1,5,1,2,4]=>1 [1,5,1,4,2]=>1 [1,5,2,1,4]=>1 [1,5,2,4,1]=>1 [1,5,4,1,2]=>1 [1,5,4,2,1]=>1 [2,1,1,4,5]=>2 [2,1,1,5,4]=>1 [2,1,4,1,5]=>2 [2,1,4,5,1]=>2 [2,1,5,1,4]=>1 [2,1,5,4,1]=>1 [2,4,1,1,5]=>2 [2,4,1,5,1]=>2 [2,4,5,1,1]=>2 [2,5,1,1,4]=>1 [2,5,1,4,1]=>1 [2,5,4,1,1]=>1 [4,1,1,2,5]=>2 [4,1,1,5,2]=>2 [4,1,2,1,5]=>2 [4,1,2,5,1]=>2 [4,1,5,1,2]=>2 [4,1,5,2,1]=>2 [4,2,1,1,5]=>2 [4,2,1,5,1]=>2 [4,2,5,1,1]=>2 [4,5,1,1,2]=>2 [4,5,1,2,1]=>2 [4,5,2,1,1]=>2 [5,1,1,2,4]=>1 [5,1,1,4,2]=>1 [5,1,2,1,4]=>1 [5,1,2,4,1]=>1 [5,1,4,1,2]=>1 [5,1,4,2,1]=>1 [5,2,1,1,4]=>1 [5,2,1,4,1]=>1 [5,2,4,1,1]=>1 [5,4,1,1,2]=>1 [5,4,1,2,1]=>1 [5,4,2,1,1]=>1 [1,1,3,3,3]=>0 [1,3,1,3,3]=>0 [1,3,3,1,3]=>0 [1,3,3,3,1]=>0 [3,1,1,3,3]=>0 [3,1,3,1,3]=>0 [3,1,3,3,1]=>0 [3,3,1,1,3]=>0 [3,3,1,3,1]=>0 [3,3,3,1,1]=>0 [1,1,3,3,4]=>0 [1,1,3,4,3]=>0 [1,1,4,3,3]=>0 [1,3,1,3,4]=>0 [1,3,1,4,3]=>0 [1,3,3,1,4]=>0 [1,3,3,4,1]=>0 [1,3,4,1,3]=>0 [1,3,4,3,1]=>0 [1,4,1,3,3]=>0 [1,4,3,1,3]=>0 [1,4,3,3,1]=>0 [3,1,1,3,4]=>0 [3,1,1,4,3]=>0 [3,1,3,1,4]=>0 [3,1,3,4,1]=>0 [3,1,4,1,3]=>0 [3,1,4,3,1]=>0 [3,3,1,1,4]=>0 [3,3,1,4,1]=>0 [3,3,4,1,1]=>0 [3,4,1,1,3]=>0 [3,4,1,3,1]=>0 [3,4,3,1,1]=>0 [4,1,1,3,3]=>0 [4,1,3,1,3]=>0 [4,1,3,3,1]=>0 [4,3,1,1,3]=>0 [4,3,1,3,1]=>0 [4,3,3,1,1]=>0 [1,1,3,3,5]=>1 [1,1,3,5,3]=>1 [1,1,5,3,3]=>1 [1,3,1,3,5]=>1 [1,3,1,5,3]=>1 [1,3,3,1,5]=>1 [1,3,3,5,1]=>1 [1,3,5,1,3]=>1 [1,3,5,3,1]=>1 [1,5,1,3,3]=>1 [1,5,3,1,3]=>1 [1,5,3,3,1]=>1 [3,1,1,3,5]=>1 [3,1,1,5,3]=>1 [3,1,3,1,5]=>1 [3,1,3,5,1]=>1 [3,1,5,1,3]=>1 [3,1,5,3,1]=>1 [3,3,1,1,5]=>1 [3,3,1,5,1]=>1 [3,3,5,1,1]=>1 [3,5,1,1,3]=>1 [3,5,1,3,1]=>1 [3,5,3,1,1]=>1 [5,1,1,3,3]=>1 [5,1,3,1,3]=>1 [5,1,3,3,1]=>1 [5,3,1,1,3]=>1 [5,3,1,3,1]=>1 [5,3,3,1,1]=>1 [1,1,3,4,4]=>1 [1,1,4,3,4]=>0 [1,1,4,4,3]=>0 [1,3,1,4,4]=>1 [1,3,4,1,4]=>1 [1,3,4,4,1]=>1 [1,4,1,3,4]=>0 [1,4,1,4,3]=>0 [1,4,3,1,4]=>0 [1,4,3,4,1]=>0 [1,4,4,1,3]=>0 [1,4,4,3,1]=>0 [3,1,1,4,4]=>1 [3,1,4,1,4]=>1 [3,1,4,4,1]=>1 [3,4,1,1,4]=>1 [3,4,1,4,1]=>1 [3,4,4,1,1]=>1 [4,1,1,3,4]=>0 [4,1,1,4,3]=>0 [4,1,3,1,4]=>0 [4,1,3,4,1]=>0 [4,1,4,1,3]=>0 [4,1,4,3,1]=>0 [4,3,1,1,4]=>0 [4,3,1,4,1]=>0 [4,3,4,1,1]=>0 [4,4,1,1,3]=>0 [4,4,1,3,1]=>0 [4,4,3,1,1]=>0 [1,1,3,4,5]=>3 [1,1,3,5,4]=>2 [1,1,4,3,5]=>2 [1,1,4,5,3]=>2 [1,1,5,3,4]=>1 [1,1,5,4,3]=>1 [1,3,1,4,5]=>3 [1,3,1,5,4]=>2 [1,3,4,1,5]=>3 [1,3,4,5,1]=>3 [1,3,5,1,4]=>2 [1,3,5,4,1]=>2 [1,4,1,3,5]=>2 [1,4,1,5,3]=>2 [1,4,3,1,5]=>2 [1,4,3,5,1]=>2 [1,4,5,1,3]=>2 [1,4,5,3,1]=>2 [1,5,1,3,4]=>1 [1,5,1,4,3]=>1 [1,5,3,1,4]=>1 [1,5,3,4,1]=>1 [1,5,4,1,3]=>1 [1,5,4,3,1]=>1 [3,1,1,4,5]=>3 [3,1,1,5,4]=>2 [3,1,4,1,5]=>3 [3,1,4,5,1]=>3 [3,1,5,1,4]=>2 [3,1,5,4,1]=>2 [3,4,1,1,5]=>3 [3,4,1,5,1]=>3 [3,4,5,1,1]=>3 [3,5,1,1,4]=>2 [3,5,1,4,1]=>2 [3,5,4,1,1]=>2 [4,1,1,3,5]=>2 [4,1,1,5,3]=>2 [4,1,3,1,5]=>2 [4,1,3,5,1]=>2 [4,1,5,1,3]=>2 [4,1,5,3,1]=>2 [4,3,1,1,5]=>2 [4,3,1,5,1]=>2 [4,3,5,1,1]=>2 [4,5,1,1,3]=>2 [4,5,1,3,1]=>2 [4,5,3,1,1]=>2 [5,1,1,3,4]=>1 [5,1,1,4,3]=>1 [5,1,3,1,4]=>1 [5,1,3,4,1]=>1 [5,1,4,1,3]=>1 [5,1,4,3,1]=>1 [5,3,1,1,4]=>1 [5,3,1,4,1]=>1 [5,3,4,1,1]=>1 [5,4,1,1,3]=>1 [5,4,1,3,1]=>1 [5,4,3,1,1]=>1 [1,2,2,2,2]=>1 [2,1,2,2,2]=>0 [2,2,1,2,2]=>0 [2,2,2,1,2]=>0 [2,2,2,2,1]=>0 [1,2,2,2,3]=>1 [1,2,2,3,2]=>1 [1,2,3,2,2]=>1 [1,3,2,2,2]=>1 [2,1,2,2,3]=>0 [2,1,2,3,2]=>0 [2,1,3,2,2]=>0 [2,2,1,2,3]=>0 [2,2,1,3,2]=>0 [2,2,2,1,3]=>0 [2,2,2,3,1]=>0 [2,2,3,1,2]=>0 [2,2,3,2,1]=>0 [2,3,1,2,2]=>0 [2,3,2,1,2]=>0 [2,3,2,2,1]=>0 [3,1,2,2,2]=>0 [3,2,1,2,2]=>0 [3,2,2,1,2]=>0 [3,2,2,2,1]=>0 [1,2,2,2,4]=>1 [1,2,2,4,2]=>1 [1,2,4,2,2]=>1 [1,4,2,2,2]=>1 [2,1,2,2,4]=>0 [2,1,2,4,2]=>0 [2,1,4,2,2]=>0 [2,2,1,2,4]=>0 [2,2,1,4,2]=>0 [2,2,2,1,4]=>0 [2,2,2,4,1]=>0 [2,2,4,1,2]=>0 [2,2,4,2,1]=>0 [2,4,1,2,2]=>0 [2,4,2,1,2]=>0 [2,4,2,2,1]=>0 [4,1,2,2,2]=>0 [4,2,1,2,2]=>0 [4,2,2,1,2]=>0 [4,2,2,2,1]=>0 [1,2,2,2,5]=>2 [1,2,2,5,2]=>2 [1,2,5,2,2]=>2 [1,5,2,2,2]=>2 [2,1,2,2,5]=>1 [2,1,2,5,2]=>1 [2,1,5,2,2]=>1 [2,2,1,2,5]=>1 [2,2,1,5,2]=>1 [2,2,2,1,5]=>1 [2,2,2,5,1]=>1 [2,2,5,1,2]=>1 [2,2,5,2,1]=>1 [2,5,1,2,2]=>1 [2,5,2,1,2]=>1 [2,5,2,2,1]=>1 [5,1,2,2,2]=>1 [5,2,1,2,2]=>1 [5,2,2,1,2]=>1 [5,2,2,2,1]=>1 [1,2,2,3,3]=>1 [1,2,3,2,3]=>1 [1,2,3,3,2]=>1 [1,3,2,2,3]=>1 [1,3,2,3,2]=>1 [1,3,3,2,2]=>1 [2,1,2,3,3]=>0 [2,1,3,2,3]=>0 [2,1,3,3,2]=>0 [2,2,1,3,3]=>0 [2,2,3,1,3]=>0 [2,2,3,3,1]=>0 [2,3,1,2,3]=>0 [2,3,1,3,2]=>0 [2,3,2,1,3]=>0 [2,3,2,3,1]=>0 [2,3,3,1,2]=>0 [2,3,3,2,1]=>0 [3,1,2,2,3]=>0 [3,1,2,3,2]=>0 [3,1,3,2,2]=>0 [3,2,1,2,3]=>0 [3,2,1,3,2]=>0 [3,2,2,1,3]=>0 [3,2,2,3,1]=>0 [3,2,3,1,2]=>0 [3,2,3,2,1]=>0 [3,3,1,2,2]=>0 [3,3,2,1,2]=>0 [3,3,2,2,1]=>0 [1,2,2,3,4]=>1 [1,2,2,4,3]=>1 [1,2,3,2,4]=>1 [1,2,3,4,2]=>1 [1,2,4,2,3]=>1 [1,2,4,3,2]=>1 [1,3,2,2,4]=>1 [1,3,2,4,2]=>1 [1,3,4,2,2]=>1 [1,4,2,2,3]=>1 [1,4,2,3,2]=>1 [1,4,3,2,2]=>1 [2,1,2,3,4]=>0 [2,1,2,4,3]=>0 [2,1,3,2,4]=>0 [2,1,3,4,2]=>0 [2,1,4,2,3]=>0 [2,1,4,3,2]=>0 [2,2,1,3,4]=>0 [2,2,1,4,3]=>0 [2,2,3,1,4]=>0 [2,2,3,4,1]=>0 [2,2,4,1,3]=>0 [2,2,4,3,1]=>0 [2,3,1,2,4]=>0 [2,3,1,4,2]=>0 [2,3,2,1,4]=>0 [2,3,2,4,1]=>0 [2,3,4,1,2]=>0 [2,3,4,2,1]=>0 [2,4,1,2,3]=>0 [2,4,1,3,2]=>0 [2,4,2,1,3]=>0 [2,4,2,3,1]=>0 [2,4,3,1,2]=>0 [2,4,3,2,1]=>0 [3,1,2,2,4]=>0 [3,1,2,4,2]=>0 [3,1,4,2,2]=>0 [3,2,1,2,4]=>0 [3,2,1,4,2]=>0 [3,2,2,1,4]=>0 [3,2,2,4,1]=>0 [3,2,4,1,2]=>0 [3,2,4,2,1]=>0 [3,4,1,2,2]=>0 [3,4,2,1,2]=>0 [3,4,2,2,1]=>0 [4,1,2,2,3]=>0 [4,1,2,3,2]=>0 [4,1,3,2,2]=>0 [4,2,1,2,3]=>0 [4,2,1,3,2]=>0 [4,2,2,1,3]=>0 [4,2,2,3,1]=>0 [4,2,3,1,2]=>0 [4,2,3,2,1]=>0 [4,3,1,2,2]=>0 [4,3,2,1,2]=>0 [4,3,2,2,1]=>0 [1,2,2,3,5]=>2 [1,2,2,5,3]=>2 [1,2,3,2,5]=>2 [1,2,3,5,2]=>2 [1,2,5,2,3]=>2 [1,2,5,3,2]=>2 [1,3,2,2,5]=>2 [1,3,2,5,2]=>2 [1,3,5,2,2]=>2 [1,5,2,2,3]=>2 [1,5,2,3,2]=>2 [1,5,3,2,2]=>2 [2,1,2,3,5]=>1 [2,1,2,5,3]=>1 [2,1,3,2,5]=>1 [2,1,3,5,2]=>1 [2,1,5,2,3]=>1 [2,1,5,3,2]=>1 [2,2,1,3,5]=>1 [2,2,1,5,3]=>1 [2,2,3,1,5]=>1 [2,2,3,5,1]=>1 [2,2,5,1,3]=>1 [2,2,5,3,1]=>1 [2,3,1,2,5]=>1 [2,3,1,5,2]=>1 [2,3,2,1,5]=>1 [2,3,2,5,1]=>1 [2,3,5,1,2]=>1 [2,3,5,2,1]=>1 [2,5,1,2,3]=>1 [2,5,1,3,2]=>1 [2,5,2,1,3]=>1 [2,5,2,3,1]=>1 [2,5,3,1,2]=>1 [2,5,3,2,1]=>1 [3,1,2,2,5]=>1 [3,1,2,5,2]=>1 [3,1,5,2,2]=>1 [3,2,1,2,5]=>1 [3,2,1,5,2]=>1 [3,2,2,1,5]=>1 [3,2,2,5,1]=>1 [3,2,5,1,2]=>1 [3,2,5,2,1]=>1 [3,5,1,2,2]=>1 [3,5,2,1,2]=>1 [3,5,2,2,1]=>1 [5,1,2,2,3]=>1 [5,1,2,3,2]=>1 [5,1,3,2,2]=>1 [5,2,1,2,3]=>1 [5,2,1,3,2]=>1 [5,2,2,1,3]=>1 [5,2,2,3,1]=>1 [5,2,3,1,2]=>1 [5,2,3,2,1]=>1 [5,3,1,2,2]=>1 [5,3,2,1,2]=>1 [5,3,2,2,1]=>1 [1,2,2,4,4]=>1 [1,2,4,2,4]=>1 [1,2,4,4,2]=>1 [1,4,2,2,4]=>1 [1,4,2,4,2]=>1 [1,4,4,2,2]=>1 [2,1,2,4,4]=>0 [2,1,4,2,4]=>0 [2,1,4,4,2]=>0 [2,2,1,4,4]=>0 [2,2,4,1,4]=>0 [2,2,4,4,1]=>0 [2,4,1,2,4]=>0 [2,4,1,4,2]=>0 [2,4,2,1,4]=>0 [2,4,2,4,1]=>0 [2,4,4,1,2]=>0 [2,4,4,2,1]=>0 [4,1,2,2,4]=>0 [4,1,2,4,2]=>0 [4,1,4,2,2]=>0 [4,2,1,2,4]=>0 [4,2,1,4,2]=>0 [4,2,2,1,4]=>0 [4,2,2,4,1]=>0 [4,2,4,1,2]=>0 [4,2,4,2,1]=>0 [4,4,1,2,2]=>0 [4,4,2,1,2]=>0 [4,4,2,2,1]=>0 [1,2,2,4,5]=>3 [1,2,2,5,4]=>2 [1,2,4,2,5]=>3 [1,2,4,5,2]=>3 [1,2,5,2,4]=>2 [1,2,5,4,2]=>2 [1,4,2,2,5]=>3 [1,4,2,5,2]=>3 [1,4,5,2,2]=>3 [1,5,2,2,4]=>2 [1,5,2,4,2]=>2 [1,5,4,2,2]=>2 [2,1,2,4,5]=>2 [2,1,2,5,4]=>1 [2,1,4,2,5]=>2 [2,1,4,5,2]=>2 [2,1,5,2,4]=>1 [2,1,5,4,2]=>1 [2,2,1,4,5]=>2 [2,2,1,5,4]=>1 [2,2,4,1,5]=>2 [2,2,4,5,1]=>2 [2,2,5,1,4]=>1 [2,2,5,4,1]=>1 [2,4,1,2,5]=>2 [2,4,1,5,2]=>2 [2,4,2,1,5]=>2 [2,4,2,5,1]=>2 [2,4,5,1,2]=>2 [2,4,5,2,1]=>2 [2,5,1,2,4]=>1 [2,5,1,4,2]=>1 [2,5,2,1,4]=>1 [2,5,2,4,1]=>1 [2,5,4,1,2]=>1 [2,5,4,2,1]=>1 [4,1,2,2,5]=>2 [4,1,2,5,2]=>2 [4,1,5,2,2]=>2 [4,2,1,2,5]=>2 [4,2,1,5,2]=>2 [4,2,2,1,5]=>2 [4,2,2,5,1]=>2 [4,2,5,1,2]=>2 [4,2,5,2,1]=>2 [4,5,1,2,2]=>2 [4,5,2,1,2]=>2 [4,5,2,2,1]=>2 [5,1,2,2,4]=>1 [5,1,2,4,2]=>1 [5,1,4,2,2]=>1 [5,2,1,2,4]=>1 [5,2,1,4,2]=>1 [5,2,2,1,4]=>1 [5,2,2,4,1]=>1 [5,2,4,1,2]=>1 [5,2,4,2,1]=>1 [5,4,1,2,2]=>1 [5,4,2,1,2]=>1 [5,4,2,2,1]=>1 [1,2,3,3,3]=>2 [1,3,2,3,3]=>1 [1,3,3,2,3]=>1 [1,3,3,3,2]=>1 [2,1,3,3,3]=>1 [2,3,1,3,3]=>1 [2,3,3,1,3]=>1 [2,3,3,3,1]=>1 [3,1,2,3,3]=>0 [3,1,3,2,3]=>0 [3,1,3,3,2]=>0 [3,2,1,3,3]=>0 [3,2,3,1,3]=>0 [3,2,3,3,1]=>0 [3,3,1,2,3]=>0 [3,3,1,3,2]=>0 [3,3,2,1,3]=>0 [3,3,2,3,1]=>0 [3,3,3,1,2]=>0 [3,3,3,2,1]=>0 [1,2,3,3,4]=>2 [1,2,3,4,3]=>2 [1,2,4,3,3]=>2 [1,3,2,3,4]=>1 [1,3,2,4,3]=>1 [1,3,3,2,4]=>1 [1,3,3,4,2]=>1 [1,3,4,2,3]=>1 [1,3,4,3,2]=>1 [1,4,2,3,3]=>1 [1,4,3,2,3]=>1 [1,4,3,3,2]=>1 [2,1,3,3,4]=>1 [2,1,3,4,3]=>1 [2,1,4,3,3]=>1 [2,3,1,3,4]=>1 [2,3,1,4,3]=>1 [2,3,3,1,4]=>1 [2,3,3,4,1]=>1 [2,3,4,1,3]=>1 [2,3,4,3,1]=>1 [2,4,1,3,3]=>1 [2,4,3,1,3]=>1 [2,4,3,3,1]=>1 [3,1,2,3,4]=>0 [3,1,2,4,3]=>0 [3,1,3,2,4]=>0 [3,1,3,4,2]=>0 [3,1,4,2,3]=>0 [3,1,4,3,2]=>0 [3,2,1,3,4]=>0 [3,2,1,4,3]=>0 [3,2,3,1,4]=>0 [3,2,3,4,1]=>0 [3,2,4,1,3]=>0 [3,2,4,3,1]=>0 [3,3,1,2,4]=>0 [3,3,1,4,2]=>0 [3,3,2,1,4]=>0 [3,3,2,4,1]=>0 [3,3,4,1,2]=>0 [3,3,4,2,1]=>0 [3,4,1,2,3]=>0 [3,4,1,3,2]=>0 [3,4,2,1,3]=>0 [3,4,2,3,1]=>0 [3,4,3,1,2]=>0 [3,4,3,2,1]=>0 [4,1,2,3,3]=>0 [4,1,3,2,3]=>0 [4,1,3,3,2]=>0 [4,2,1,3,3]=>0 [4,2,3,1,3]=>0 [4,2,3,3,1]=>0 [4,3,1,2,3]=>0 [4,3,1,3,2]=>0 [4,3,2,1,3]=>0 [4,3,2,3,1]=>0 [4,3,3,1,2]=>0
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Description
The number of critical left to right maxima of the parking functions.
An entry $p$ in a parking function is critical, if there are exactly $p-1$ entries smaller than $p$ and $n-p$ entries larger than $p$. It is a left to right maximum, if there are no larger entries before it.
This statistic allows the computation of the Tutte polynomial of the complete graph $K_{n+1}$, via
$$ \sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P}, $$
where the sum is over all parking functions of length $n$, see [1, thm.13.5.16].
References
[1] Handbook of enumerative combinatorics MathSciNet:3408702
Code
def statistic(P):
    """
    sage: R. = PolynomialRing(ZZ)
    sage: n=3; sum(x^statistic(P)*y^(binomial(len(P)+1,2)-sum(P)) for P in ParkingFunctions(n))
    x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y

    sage: G = graphs.CompleteGraph(n+1)
    sage: G.tutte_polynomial()
    x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y
    
    """    
    return len([1 for i, j in enumerate(P)
                if ((i == 0 or max(P[:i]) < j)
                    and len([1 for k in P if k < j]) == j-1
                    and len([1 for k in P if k > j]) == len(P)-j)])

Created
Aug 12, 2017 at 23:31 by Martin Rubey
Updated
Aug 12, 2017 at 23:31 by Martin Rubey