edit this statistic or download as text // json
Identifier
Values
=>
[]=>0 [1]=>1 [1,2]=>4 [2,1]=>0 [1,2,3]=>9 [1,3,2]=>4 [2,1,3]=>0 [2,3,1]=>4 [3,1,2]=>0 [3,2,1]=>1 [1,2,3,4]=>16 [1,2,4,3]=>10 [1,3,2,4]=>6 [1,3,4,2]=>10 [1,4,2,3]=>6 [1,4,3,2]=>6 [2,1,3,4]=>0 [2,1,4,3]=>0 [2,3,1,4]=>6 [2,3,4,1]=>10 [2,4,1,3]=>4 [2,4,3,1]=>6 [3,1,2,4]=>0 [3,1,4,2]=>0 [3,2,1,4]=>2 [3,2,4,1]=>0 [3,4,1,2]=>4 [3,4,2,1]=>6 [4,1,2,3]=>0 [4,1,3,2]=>0 [4,2,1,3]=>2 [4,2,3,1]=>0 [4,3,1,2]=>2 [4,3,2,1]=>0 [1,2,3,4,5]=>25 [1,2,3,5,4]=>18 [1,2,4,3,5]=>14 [1,2,4,5,3]=>18 [1,2,5,3,4]=>14 [1,2,5,4,3]=>13 [1,3,2,4,5]=>8 [1,3,2,5,4]=>7 [1,3,4,2,5]=>14 [1,3,4,5,2]=>18 [1,3,5,2,4]=>11 [1,3,5,4,2]=>13 [1,4,2,3,5]=>8 [1,4,2,5,3]=>7 [1,4,3,2,5]=>9 [1,4,3,5,2]=>7 [1,4,5,2,3]=>11 [1,4,5,3,2]=>13 [1,5,2,3,4]=>8 [1,5,2,4,3]=>7 [1,5,3,2,4]=>9 [1,5,3,4,2]=>7 [1,5,4,2,3]=>9 [1,5,4,3,2]=>6 [2,1,3,4,5]=>0 [2,1,3,5,4]=>0 [2,1,4,3,5]=>0 [2,1,4,5,3]=>0 [2,1,5,3,4]=>0 [2,1,5,4,3]=>0 [2,3,1,4,5]=>8 [2,3,1,5,4]=>7 [2,3,4,1,5]=>14 [2,3,4,5,1]=>18 [2,3,5,1,4]=>11 [2,3,5,4,1]=>13 [2,4,1,3,5]=>5 [2,4,1,5,3]=>7 [2,4,3,1,5]=>9 [2,4,3,5,1]=>7 [2,4,5,1,3]=>11 [2,4,5,3,1]=>13 [2,5,1,3,4]=>5 [2,5,1,4,3]=>5 [2,5,3,1,4]=>9 [2,5,3,4,1]=>7 [2,5,4,1,3]=>7 [2,5,4,3,1]=>6 [3,1,2,4,5]=>0 [3,1,2,5,4]=>0 [3,1,4,2,5]=>0 [3,1,4,5,2]=>0 [3,1,5,2,4]=>0 [3,1,5,4,2]=>0 [3,2,1,4,5]=>3 [3,2,1,5,4]=>3 [3,2,4,1,5]=>0 [3,2,4,5,1]=>0 [3,2,5,1,4]=>0 [3,2,5,4,1]=>0 [3,4,1,2,5]=>5 [3,4,1,5,2]=>7 [3,4,2,1,5]=>9 [3,4,2,5,1]=>7 [3,4,5,1,2]=>11 [3,4,5,2,1]=>13 [3,5,1,2,4]=>5 [3,5,1,4,2]=>5 [3,5,2,1,4]=>7 [3,5,2,4,1]=>5 [3,5,4,1,2]=>7 [3,5,4,2,1]=>6 [4,1,2,3,5]=>0 [4,1,2,5,3]=>0 [4,1,3,2,5]=>0 [4,1,3,5,2]=>0 [4,1,5,2,3]=>0 [4,1,5,3,2]=>0 [4,2,1,3,5]=>3 [4,2,1,5,3]=>3 [4,2,3,1,5]=>0 [4,2,3,5,1]=>0 [4,2,5,1,3]=>0 [4,2,5,3,1]=>0 [4,3,1,2,5]=>3 [4,3,1,5,2]=>3 [4,3,2,1,5]=>0 [4,3,2,5,1]=>2 [4,3,5,1,2]=>0 [4,3,5,2,1]=>0 [4,5,1,2,3]=>5 [4,5,1,3,2]=>5 [4,5,2,1,3]=>7 [4,5,2,3,1]=>5 [4,5,3,1,2]=>7 [4,5,3,2,1]=>6 [5,1,2,3,4]=>0 [5,1,2,4,3]=>0 [5,1,3,2,4]=>0 [5,1,3,4,2]=>0 [5,1,4,2,3]=>0 [5,1,4,3,2]=>0 [5,2,1,3,4]=>3 [5,2,1,4,3]=>3 [5,2,3,1,4]=>0 [5,2,3,4,1]=>0 [5,2,4,1,3]=>0 [5,2,4,3,1]=>0 [5,3,1,2,4]=>3 [5,3,1,4,2]=>3 [5,3,2,1,4]=>0 [5,3,2,4,1]=>2 [5,3,4,1,2]=>0 [5,3,4,2,1]=>0 [5,4,1,2,3]=>3 [5,4,1,3,2]=>2 [5,4,2,1,3]=>0 [5,4,2,3,1]=>2 [5,4,3,1,2]=>0 [5,4,3,2,1]=>1 [1,2,3,4,5,6]=>36 [1,2,3,4,6,5]=>28 [1,2,3,5,4,6]=>24 [1,2,3,5,6,4]=>28 [1,2,3,6,4,5]=>24 [1,2,3,6,5,4]=>22 [1,2,4,3,5,6]=>18 [1,2,4,3,6,5]=>16 [1,2,4,5,3,6]=>24 [1,2,4,5,6,3]=>28 [1,2,4,6,3,5]=>20 [1,2,4,6,5,3]=>22 [1,2,5,3,4,6]=>18 [1,2,5,3,6,4]=>16 [1,2,5,4,3,6]=>18 [1,2,5,4,6,3]=>16 [1,2,5,6,3,4]=>20 [1,2,5,6,4,3]=>22 [1,2,6,3,4,5]=>18 [1,2,6,3,5,4]=>16 [1,2,6,4,3,5]=>18 [1,2,6,4,5,3]=>16 [1,2,6,5,3,4]=>18 [1,2,6,5,4,3]=>14 [1,3,2,4,5,6]=>10 [1,3,2,4,6,5]=>9 [1,3,2,5,4,6]=>9 [1,3,2,5,6,4]=>9 [1,3,2,6,4,5]=>9 [1,3,2,6,5,4]=>8 [1,3,4,2,5,6]=>18 [1,3,4,2,6,5]=>16 [1,3,4,5,2,6]=>24 [1,3,4,5,6,2]=>28 [1,3,4,6,2,5]=>20 [1,3,4,6,5,2]=>22 [1,3,5,2,4,6]=>14 [1,3,5,2,6,4]=>16 [1,3,5,4,2,6]=>18 [1,3,5,4,6,2]=>16 [1,3,5,6,2,4]=>20 [1,3,5,6,4,2]=>22 [1,3,6,2,4,5]=>14 [1,3,6,2,5,4]=>13 [1,3,6,4,2,5]=>18 [1,3,6,4,5,2]=>16 [1,3,6,5,2,4]=>15 [1,3,6,5,4,2]=>14 [1,4,2,3,5,6]=>10 [1,4,2,3,6,5]=>9 [1,4,2,5,3,6]=>9 [1,4,2,5,6,3]=>9 [1,4,2,6,3,5]=>9 [1,4,2,6,5,3]=>8 [1,4,3,2,5,6]=>12 [1,4,3,2,6,5]=>11 [1,4,3,5,2,6]=>9 [1,4,3,5,6,2]=>9 [1,4,3,6,2,5]=>8 [1,4,3,6,5,2]=>8 [1,4,5,2,3,6]=>14 [1,4,5,2,6,3]=>16 [1,4,5,3,2,6]=>18 [1,4,5,3,6,2]=>16 [1,4,5,6,2,3]=>20 [1,4,5,6,3,2]=>22 [1,4,6,2,3,5]=>14 [1,4,6,2,5,3]=>13 [1,4,6,3,2,5]=>15 [1,4,6,3,5,2]=>13 [1,4,6,5,2,3]=>15 [1,4,6,5,3,2]=>14 [1,5,2,3,4,6]=>10 [1,5,2,3,6,4]=>9 [1,5,2,4,3,6]=>9 [1,5,2,4,6,3]=>9 [1,5,2,6,3,4]=>9 [1,5,2,6,4,3]=>8 [1,5,3,2,4,6]=>12 [1,5,3,2,6,4]=>11 [1,5,3,4,2,6]=>9 [1,5,3,4,6,2]=>9 [1,5,3,6,2,4]=>8 [1,5,3,6,4,2]=>8 [1,5,4,2,3,6]=>12 [1,5,4,2,6,3]=>11 [1,5,4,3,2,6]=>8 [1,5,4,3,6,2]=>10 [1,5,4,6,2,3]=>8 [1,5,4,6,3,2]=>8 [1,5,6,2,3,4]=>14 [1,5,6,2,4,3]=>13 [1,5,6,3,2,4]=>15 [1,5,6,3,4,2]=>13 [1,5,6,4,2,3]=>15 [1,5,6,4,3,2]=>14 [1,6,2,3,4,5]=>10 [1,6,2,3,5,4]=>9 [1,6,2,4,3,5]=>9 [1,6,2,4,5,3]=>9 [1,6,2,5,3,4]=>9 [1,6,2,5,4,3]=>8 [1,6,3,2,4,5]=>12 [1,6,3,2,5,4]=>11 [1,6,3,4,2,5]=>9 [1,6,3,4,5,2]=>9 [1,6,3,5,2,4]=>8 [1,6,3,5,4,2]=>8 [1,6,4,2,3,5]=>12 [1,6,4,2,5,3]=>11 [1,6,4,3,2,5]=>8 [1,6,4,3,5,2]=>10 [1,6,4,5,2,3]=>8 [1,6,4,5,3,2]=>8 [1,6,5,2,3,4]=>12 [1,6,5,2,4,3]=>10 [1,6,5,3,2,4]=>8 [1,6,5,3,4,2]=>10 [1,6,5,4,2,3]=>8 [1,6,5,4,3,2]=>8 [2,1,3,4,5,6]=>0 [2,1,3,4,6,5]=>0 [2,1,3,5,4,6]=>0 [2,1,3,5,6,4]=>0 [2,1,3,6,4,5]=>0 [2,1,3,6,5,4]=>0 [2,1,4,3,5,6]=>0 [2,1,4,3,6,5]=>0 [2,1,4,5,3,6]=>0 [2,1,4,5,6,3]=>0 [2,1,4,6,3,5]=>0 [2,1,4,6,5,3]=>0 [2,1,5,3,4,6]=>0 [2,1,5,3,6,4]=>0 [2,1,5,4,3,6]=>0 [2,1,5,4,6,3]=>0 [2,1,5,6,3,4]=>0 [2,1,5,6,4,3]=>0 [2,1,6,3,4,5]=>0 [2,1,6,3,5,4]=>0 [2,1,6,4,3,5]=>0 [2,1,6,4,5,3]=>0 [2,1,6,5,3,4]=>0 [2,1,6,5,4,3]=>0 [2,3,1,4,5,6]=>10 [2,3,1,4,6,5]=>9 [2,3,1,5,4,6]=>9 [2,3,1,5,6,4]=>9 [2,3,1,6,4,5]=>9 [2,3,1,6,5,4]=>8 [2,3,4,1,5,6]=>18 [2,3,4,1,6,5]=>16 [2,3,4,5,1,6]=>24 [2,3,4,5,6,1]=>28 [2,3,4,6,1,5]=>20 [2,3,4,6,5,1]=>22 [2,3,5,1,4,6]=>14 [2,3,5,1,6,4]=>16 [2,3,5,4,1,6]=>18 [2,3,5,4,6,1]=>16 [2,3,5,6,1,4]=>20 [2,3,5,6,4,1]=>22 [2,3,6,1,4,5]=>14 [2,3,6,1,5,4]=>13 [2,3,6,4,1,5]=>18 [2,3,6,4,5,1]=>16 [2,3,6,5,1,4]=>15 [2,3,6,5,4,1]=>14 [2,4,1,3,5,6]=>6 [2,4,1,3,6,5]=>7 [2,4,1,5,3,6]=>9 [2,4,1,5,6,3]=>9 [2,4,1,6,3,5]=>7 [2,4,1,6,5,3]=>8 [2,4,3,1,5,6]=>12 [2,4,3,1,6,5]=>11 [2,4,3,5,1,6]=>9 [2,4,3,5,6,1]=>9 [2,4,3,6,1,5]=>8 [2,4,3,6,5,1]=>8 [2,4,5,1,3,6]=>14 [2,4,5,1,6,3]=>16 [2,4,5,3,1,6]=>18 [2,4,5,3,6,1]=>16 [2,4,5,6,1,3]=>20 [2,4,5,6,3,1]=>22 [2,4,6,1,3,5]=>12 [2,4,6,1,5,3]=>13 [2,4,6,3,1,5]=>15 [2,4,6,3,5,1]=>13 [2,4,6,5,1,3]=>15 [2,4,6,5,3,1]=>14 [2,5,1,3,4,6]=>6 [2,5,1,3,6,4]=>7 [2,5,1,4,3,6]=>6 [2,5,1,4,6,3]=>6 [2,5,1,6,3,4]=>7 [2,5,1,6,4,3]=>8 [2,5,3,1,4,6]=>12 [2,5,3,1,6,4]=>11 [2,5,3,4,1,6]=>9 [2,5,3,4,6,1]=>9 [2,5,3,6,1,4]=>8 [2,5,3,6,4,1]=>8 [2,5,4,1,3,6]=>9 [2,5,4,1,6,3]=>11 [2,5,4,3,1,6]=>8 [2,5,4,3,6,1]=>10 [2,5,4,6,1,3]=>8 [2,5,4,6,3,1]=>8 [2,5,6,1,3,4]=>12 [2,5,6,1,4,3]=>13 [2,5,6,3,1,4]=>15 [2,5,6,3,4,1]=>13 [2,5,6,4,1,3]=>15 [2,5,6,4,3,1]=>14 [2,6,1,3,4,5]=>6 [2,6,1,3,5,4]=>6 [2,6,1,4,3,5]=>6 [2,6,1,4,5,3]=>6 [2,6,1,5,3,4]=>6 [2,6,1,5,4,3]=>6 [2,6,3,1,4,5]=>12 [2,6,3,1,5,4]=>11 [2,6,3,4,1,5]=>9 [2,6,3,4,5,1]=>9 [2,6,3,5,1,4]=>8 [2,6,3,5,4,1]=>8 [2,6,4,1,3,5]=>9 [2,6,4,1,5,3]=>11 [2,6,4,3,1,5]=>8 [2,6,4,3,5,1]=>10 [2,6,4,5,1,3]=>8 [2,6,4,5,3,1]=>8 [2,6,5,1,3,4]=>9 [2,6,5,1,4,3]=>8 [2,6,5,3,1,4]=>8 [2,6,5,3,4,1]=>10 [2,6,5,4,1,3]=>6 [2,6,5,4,3,1]=>8 [3,1,2,4,5,6]=>0 [3,1,2,4,6,5]=>0 [3,1,2,5,4,6]=>0 [3,1,2,5,6,4]=>0 [3,1,2,6,4,5]=>0 [3,1,2,6,5,4]=>0 [3,1,4,2,5,6]=>0 [3,1,4,2,6,5]=>0 [3,1,4,5,2,6]=>0 [3,1,4,5,6,2]=>0 [3,1,4,6,2,5]=>0 [3,1,4,6,5,2]=>0 [3,1,5,2,4,6]=>0 [3,1,5,2,6,4]=>0 [3,1,5,4,2,6]=>0 [3,1,5,4,6,2]=>0 [3,1,5,6,2,4]=>0 [3,1,5,6,4,2]=>0 [3,1,6,2,4,5]=>0 [3,1,6,2,5,4]=>0 [3,1,6,4,2,5]=>0 [3,1,6,4,5,2]=>0 [3,1,6,5,2,4]=>0 [3,1,6,5,4,2]=>0 [3,2,1,4,5,6]=>4 [3,2,1,4,6,5]=>4 [3,2,1,5,4,6]=>4 [3,2,1,5,6,4]=>4 [3,2,1,6,4,5]=>4 [3,2,1,6,5,4]=>5 [3,2,4,1,5,6]=>0 [3,2,4,1,6,5]=>0 [3,2,4,5,1,6]=>0 [3,2,4,5,6,1]=>0 [3,2,4,6,1,5]=>0 [3,2,4,6,5,1]=>0 [3,2,5,1,4,6]=>0 [3,2,5,1,6,4]=>0 [3,2,5,4,1,6]=>0 [3,2,5,4,6,1]=>0 [3,2,5,6,1,4]=>0 [3,2,5,6,4,1]=>0 [3,2,6,1,4,5]=>0 [3,2,6,1,5,4]=>0 [3,2,6,4,1,5]=>0 [3,2,6,4,5,1]=>0 [3,2,6,5,1,4]=>0 [3,2,6,5,4,1]=>0 [3,4,1,2,5,6]=>6 [3,4,1,2,6,5]=>7 [3,4,1,5,2,6]=>9 [3,4,1,5,6,2]=>9 [3,4,1,6,2,5]=>7 [3,4,1,6,5,2]=>8 [3,4,2,1,5,6]=>12 [3,4,2,1,6,5]=>11 [3,4,2,5,1,6]=>9 [3,4,2,5,6,1]=>9 [3,4,2,6,1,5]=>8 [3,4,2,6,5,1]=>8 [3,4,5,1,2,6]=>14 [3,4,5,1,6,2]=>16 [3,4,5,2,1,6]=>18 [3,4,5,2,6,1]=>16 [3,4,5,6,1,2]=>20 [3,4,5,6,2,1]=>22 [3,4,6,1,2,5]=>12 [3,4,6,1,5,2]=>13 [3,4,6,2,1,5]=>15 [3,4,6,2,5,1]=>13 [3,4,6,5,1,2]=>15 [3,4,6,5,2,1]=>14 [3,5,1,2,4,6]=>6 [3,5,1,2,6,4]=>7 [3,5,1,4,2,6]=>6 [3,5,1,4,6,2]=>6 [3,5,1,6,2,4]=>7 [3,5,1,6,4,2]=>8 [3,5,2,1,4,6]=>9 [3,5,2,1,6,4]=>11 [3,5,2,4,1,6]=>6 [3,5,2,4,6,1]=>6 [3,5,2,6,1,4]=>8 [3,5,2,6,4,1]=>8 [3,5,4,1,2,6]=>9 [3,5,4,1,6,2]=>11 [3,5,4,2,1,6]=>8 [3,5,4,2,6,1]=>10 [3,5,4,6,1,2]=>8 [3,5,4,6,2,1]=>8 [3,5,6,1,2,4]=>12 [3,5,6,1,4,2]=>13 [3,5,6,2,1,4]=>15 [3,5,6,2,4,1]=>13 [3,5,6,4,1,2]=>15 [3,5,6,4,2,1]=>14 [3,6,1,2,4,5]=>6 [3,6,1,2,5,4]=>6 [3,6,1,4,2,5]=>6 [3,6,1,4,5,2]=>6 [3,6,1,5,2,4]=>6 [3,6,1,5,4,2]=>6 [3,6,2,1,4,5]=>9 [3,6,2,1,5,4]=>8 [3,6,2,4,1,5]=>6 [3,6,2,4,5,1]=>6 [3,6,2,5,1,4]=>5 [3,6,2,5,4,1]=>6 [3,6,4,1,2,5]=>9 [3,6,4,1,5,2]=>11 [3,6,4,2,1,5]=>8 [3,6,4,2,5,1]=>10 [3,6,4,5,1,2]=>8 [3,6,4,5,2,1]=>8 [3,6,5,1,2,4]=>9 [3,6,5,1,4,2]=>8 [3,6,5,2,1,4]=>6 [3,6,5,2,4,1]=>8 [3,6,5,4,1,2]=>6 [3,6,5,4,2,1]=>8 [4,1,2,3,5,6]=>0 [4,1,2,3,6,5]=>0 [4,1,2,5,3,6]=>0 [4,1,2,5,6,3]=>0 [4,1,2,6,3,5]=>0 [4,1,2,6,5,3]=>0 [4,1,3,2,5,6]=>0 [4,1,3,2,6,5]=>0 [4,1,3,5,2,6]=>0 [4,1,3,5,6,2]=>0 [4,1,3,6,2,5]=>0 [4,1,3,6,5,2]=>0 [4,1,5,2,3,6]=>0 [4,1,5,2,6,3]=>0 [4,1,5,3,2,6]=>0 [4,1,5,3,6,2]=>0 [4,1,5,6,2,3]=>0 [4,1,5,6,3,2]=>0 [4,1,6,2,3,5]=>0 [4,1,6,2,5,3]=>0 [4,1,6,3,2,5]=>0 [4,1,6,3,5,2]=>0 [4,1,6,5,2,3]=>0 [4,1,6,5,3,2]=>0 [4,2,1,3,5,6]=>4 [4,2,1,3,6,5]=>4 [4,2,1,5,3,6]=>4 [4,2,1,5,6,3]=>4 [4,2,1,6,3,5]=>4 [4,2,1,6,5,3]=>5 [4,2,3,1,5,6]=>0 [4,2,3,1,6,5]=>0 [4,2,3,5,1,6]=>0 [4,2,3,5,6,1]=>0 [4,2,3,6,1,5]=>0 [4,2,3,6,5,1]=>0 [4,2,5,1,3,6]=>0 [4,2,5,1,6,3]=>0 [4,2,5,3,1,6]=>0 [4,2,5,3,6,1]=>0 [4,2,5,6,1,3]=>0 [4,2,5,6,3,1]=>0 [4,2,6,1,3,5]=>0 [4,2,6,1,5,3]=>0 [4,2,6,3,1,5]=>0 [4,2,6,3,5,1]=>0 [4,2,6,5,1,3]=>0 [4,2,6,5,3,1]=>0 [4,3,1,2,5,6]=>4 [4,3,1,2,6,5]=>4 [4,3,1,5,2,6]=>4 [4,3,1,5,6,2]=>4 [4,3,1,6,2,5]=>4 [4,3,1,6,5,2]=>5 [4,3,2,1,5,6]=>0 [4,3,2,1,6,5]=>0 [4,3,2,5,1,6]=>3 [4,3,2,5,6,1]=>3 [4,3,2,6,1,5]=>3 [4,3,2,6,5,1]=>3 [4,3,5,1,2,6]=>0 [4,3,5,1,6,2]=>0 [4,3,5,2,1,6]=>0 [4,3,5,2,6,1]=>0 [4,3,5,6,1,2]=>0 [4,3,5,6,2,1]=>0 [4,3,6,1,2,5]=>0 [4,3,6,1,5,2]=>0 [4,3,6,2,1,5]=>0 [4,3,6,2,5,1]=>0 [4,3,6,5,1,2]=>0 [4,3,6,5,2,1]=>0 [4,5,1,2,3,6]=>6 [4,5,1,2,6,3]=>7 [4,5,1,3,2,6]=>6 [4,5,1,3,6,2]=>6 [4,5,1,6,2,3]=>7 [4,5,1,6,3,2]=>8 [4,5,2,1,3,6]=>9 [4,5,2,1,6,3]=>11 [4,5,2,3,1,6]=>6 [4,5,2,3,6,1]=>6 [4,5,2,6,1,3]=>8 [4,5,2,6,3,1]=>8 [4,5,3,1,2,6]=>9 [4,5,3,1,6,2]=>11 [4,5,3,2,1,6]=>8 [4,5,3,2,6,1]=>10 [4,5,3,6,1,2]=>8 [4,5,3,6,2,1]=>8 [4,5,6,1,2,3]=>12 [4,5,6,1,3,2]=>13 [4,5,6,2,1,3]=>15 [4,5,6,2,3,1]=>13 [4,5,6,3,1,2]=>15 [4,5,6,3,2,1]=>14 [4,6,1,2,3,5]=>6 [4,6,1,2,5,3]=>6 [4,6,1,3,2,5]=>6 [4,6,1,3,5,2]=>6 [4,6,1,5,2,3]=>6 [4,6,1,5,3,2]=>6 [4,6,2,1,3,5]=>9 [4,6,2,1,5,3]=>8 [4,6,2,3,1,5]=>6 [4,6,2,3,5,1]=>6 [4,6,2,5,1,3]=>5 [4,6,2,5,3,1]=>6 [4,6,3,1,2,5]=>9 [4,6,3,1,5,2]=>8 [4,6,3,2,1,5]=>6 [4,6,3,2,5,1]=>8 [4,6,3,5,1,2]=>5 [4,6,3,5,2,1]=>6 [4,6,5,1,2,3]=>9 [4,6,5,1,3,2]=>8 [4,6,5,2,1,3]=>6 [4,6,5,2,3,1]=>8 [4,6,5,3,1,2]=>6 [4,6,5,3,2,1]=>8 [5,1,2,3,4,6]=>0 [5,1,2,3,6,4]=>0 [5,1,2,4,3,6]=>0 [5,1,2,4,6,3]=>0 [5,1,2,6,3,4]=>0 [5,1,2,6,4,3]=>0 [5,1,3,2,4,6]=>0 [5,1,3,2,6,4]=>0 [5,1,3,4,2,6]=>0 [5,1,3,4,6,2]=>0 [5,1,3,6,2,4]=>0 [5,1,3,6,4,2]=>0 [5,1,4,2,3,6]=>0 [5,1,4,2,6,3]=>0 [5,1,4,3,2,6]=>0 [5,1,4,3,6,2]=>0 [5,1,4,6,2,3]=>0 [5,1,4,6,3,2]=>0 [5,1,6,2,3,4]=>0 [5,1,6,2,4,3]=>0 [5,1,6,3,2,4]=>0 [5,1,6,3,4,2]=>0 [5,1,6,4,2,3]=>0 [5,1,6,4,3,2]=>0 [5,2,1,3,4,6]=>4 [5,2,1,3,6,4]=>4 [5,2,1,4,3,6]=>4 [5,2,1,4,6,3]=>4 [5,2,1,6,3,4]=>4 [5,2,1,6,4,3]=>5 [5,2,3,1,4,6]=>0 [5,2,3,1,6,4]=>0 [5,2,3,4,1,6]=>0 [5,2,3,4,6,1]=>0 [5,2,3,6,1,4]=>0 [5,2,3,6,4,1]=>0 [5,2,4,1,3,6]=>0 [5,2,4,1,6,3]=>0 [5,2,4,3,1,6]=>0 [5,2,4,3,6,1]=>0 [5,2,4,6,1,3]=>0 [5,2,4,6,3,1]=>0 [5,2,6,1,3,4]=>0 [5,2,6,1,4,3]=>0 [5,2,6,3,1,4]=>0 [5,2,6,3,4,1]=>0 [5,2,6,4,1,3]=>0 [5,2,6,4,3,1]=>0 [5,3,1,2,4,6]=>4 [5,3,1,2,6,4]=>4 [5,3,1,4,2,6]=>4 [5,3,1,4,6,2]=>4 [5,3,1,6,2,4]=>4 [5,3,1,6,4,2]=>5 [5,3,2,1,4,6]=>0 [5,3,2,1,6,4]=>0 [5,3,2,4,1,6]=>3 [5,3,2,4,6,1]=>3 [5,3,2,6,1,4]=>3 [5,3,2,6,4,1]=>3 [5,3,4,1,2,6]=>0 [5,3,4,1,6,2]=>0 [5,3,4,2,1,6]=>0 [5,3,4,2,6,1]=>0 [5,3,4,6,1,2]=>0 [5,3,4,6,2,1]=>0 [5,3,6,1,2,4]=>0 [5,3,6,1,4,2]=>0 [5,3,6,2,1,4]=>0 [5,3,6,2,4,1]=>0 [5,3,6,4,1,2]=>0 [5,3,6,4,2,1]=>0 [5,4,1,2,3,6]=>4 [5,4,1,2,6,3]=>4 [5,4,1,3,2,6]=>3 [5,4,1,3,6,2]=>3 [5,4,1,6,2,3]=>4 [5,4,1,6,3,2]=>3 [5,4,2,1,3,6]=>0 [5,4,2,1,6,3]=>0 [5,4,2,3,1,6]=>3 [5,4,2,3,6,1]=>3 [5,4,2,6,1,3]=>3 [5,4,2,6,3,1]=>3 [5,4,3,1,2,6]=>0 [5,4,3,1,6,2]=>0 [5,4,3,2,1,6]=>2 [5,4,3,2,6,1]=>0 [5,4,3,6,1,2]=>3 [5,4,3,6,2,1]=>2 [5,4,6,1,2,3]=>0 [5,4,6,1,3,2]=>0 [5,4,6,2,1,3]=>0 [5,4,6,2,3,1]=>0 [5,4,6,3,1,2]=>0 [5,4,6,3,2,1]=>0 [5,6,1,2,3,4]=>6 [5,6,1,2,4,3]=>6 [5,6,1,3,2,4]=>6 [5,6,1,3,4,2]=>6 [5,6,1,4,2,3]=>6 [5,6,1,4,3,2]=>6 [5,6,2,1,3,4]=>9 [5,6,2,1,4,3]=>8 [5,6,2,3,1,4]=>6 [5,6,2,3,4,1]=>6 [5,6,2,4,1,3]=>5 [5,6,2,4,3,1]=>6 [5,6,3,1,2,4]=>9 [5,6,3,1,4,2]=>8 [5,6,3,2,1,4]=>6 [5,6,3,2,4,1]=>8 [5,6,3,4,1,2]=>5 [5,6,3,4,2,1]=>6 [5,6,4,1,2,3]=>9 [5,6,4,1,3,2]=>8 [5,6,4,2,1,3]=>6 [5,6,4,2,3,1]=>8 [5,6,4,3,1,2]=>6 [5,6,4,3,2,1]=>8 [6,1,2,3,4,5]=>0 [6,1,2,3,5,4]=>0 [6,1,2,4,3,5]=>0 [6,1,2,4,5,3]=>0 [6,1,2,5,3,4]=>0 [6,1,2,5,4,3]=>0 [6,1,3,2,4,5]=>0 [6,1,3,2,5,4]=>0 [6,1,3,4,2,5]=>0 [6,1,3,4,5,2]=>0 [6,1,3,5,2,4]=>0 [6,1,3,5,4,2]=>0 [6,1,4,2,3,5]=>0 [6,1,4,2,5,3]=>0 [6,1,4,3,2,5]=>0 [6,1,4,3,5,2]=>0 [6,1,4,5,2,3]=>0 [6,1,4,5,3,2]=>0 [6,1,5,2,3,4]=>0 [6,1,5,2,4,3]=>0 [6,1,5,3,2,4]=>0 [6,1,5,3,4,2]=>0 [6,1,5,4,2,3]=>0 [6,1,5,4,3,2]=>0 [6,2,1,3,4,5]=>4 [6,2,1,3,5,4]=>4 [6,2,1,4,3,5]=>4 [6,2,1,4,5,3]=>4 [6,2,1,5,3,4]=>4 [6,2,1,5,4,3]=>3 [6,2,3,1,4,5]=>0 [6,2,3,1,5,4]=>0 [6,2,3,4,1,5]=>0 [6,2,3,4,5,1]=>0 [6,2,3,5,1,4]=>0 [6,2,3,5,4,1]=>0 [6,2,4,1,3,5]=>0 [6,2,4,1,5,3]=>0 [6,2,4,3,1,5]=>0 [6,2,4,3,5,1]=>0 [6,2,4,5,1,3]=>0 [6,2,4,5,3,1]=>0 [6,2,5,1,3,4]=>0 [6,2,5,1,4,3]=>0 [6,2,5,3,1,4]=>0 [6,2,5,3,4,1]=>0 [6,2,5,4,1,3]=>0 [6,2,5,4,3,1]=>0 [6,3,1,2,4,5]=>4 [6,3,1,2,5,4]=>4 [6,3,1,4,2,5]=>4 [6,3,1,4,5,2]=>4 [6,3,1,5,2,4]=>4 [6,3,1,5,4,2]=>3 [6,3,2,1,4,5]=>0 [6,3,2,1,5,4]=>0 [6,3,2,4,1,5]=>3 [6,3,2,4,5,1]=>3 [6,3,2,5,1,4]=>3 [6,3,2,5,4,1]=>3 [6,3,4,1,2,5]=>0 [6,3,4,1,5,2]=>0 [6,3,4,2,1,5]=>0 [6,3,4,2,5,1]=>0 [6,3,4,5,1,2]=>0 [6,3,4,5,2,1]=>0 [6,3,5,1,2,4]=>0 [6,3,5,1,4,2]=>0 [6,3,5,2,1,4]=>0 [6,3,5,2,4,1]=>0 [6,3,5,4,1,2]=>0 [6,3,5,4,2,1]=>0 [6,4,1,2,3,5]=>4 [6,4,1,2,5,3]=>4 [6,4,1,3,2,5]=>3 [6,4,1,3,5,2]=>3 [6,4,1,5,2,3]=>4 [6,4,1,5,3,2]=>3 [6,4,2,1,3,5]=>0 [6,4,2,1,5,3]=>0 [6,4,2,3,1,5]=>3 [6,4,2,3,5,1]=>3 [6,4,2,5,1,3]=>3 [6,4,2,5,3,1]=>3 [6,4,3,1,2,5]=>0 [6,4,3,1,5,2]=>0 [6,4,3,2,1,5]=>2 [6,4,3,2,5,1]=>0 [6,4,3,5,1,2]=>3 [6,4,3,5,2,1]=>2 [6,4,5,1,2,3]=>0 [6,4,5,1,3,2]=>0 [6,4,5,2,1,3]=>0 [6,4,5,2,3,1]=>0 [6,4,5,3,1,2]=>0 [6,4,5,3,2,1]=>0 [6,5,1,2,3,4]=>4 [6,5,1,2,4,3]=>3 [6,5,1,3,2,4]=>3 [6,5,1,3,4,2]=>3 [6,5,1,4,2,3]=>3 [6,5,1,4,3,2]=>2 [6,5,2,1,3,4]=>0 [6,5,2,1,4,3]=>0 [6,5,2,3,1,4]=>3 [6,5,2,3,4,1]=>3 [6,5,2,4,1,3]=>3 [6,5,2,4,3,1]=>2 [6,5,3,1,2,4]=>0 [6,5,3,1,4,2]=>0 [6,5,3,2,1,4]=>2 [6,5,3,2,4,1]=>0 [6,5,3,4,1,2]=>3 [6,5,3,4,2,1]=>2 [6,5,4,1,2,3]=>0 [6,5,4,1,3,2]=>0 [6,5,4,2,1,3]=>2 [6,5,4,2,3,1]=>0 [6,5,4,3,1,2]=>2 [6,5,4,3,2,1]=>0
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Description
Eigenvalues of the random-to-random operator acting on the regular representation.
This statistic is defined for a permutation $w$ as:
$$ \left[\binom{\ell(w) + 1}{2} + \operatorname{diag}\left(Q(w)\right)\right] - \left[\binom{\ell(u) + 1}{2} + \operatorname{diag}\left(Q(u)\right)\right] $$
where:
  • $u$ is the longest suffix of $w$ (viewed as a word) whose first ascent is even;
  • $\ell(w)$ is the size of the permutation $w$ (equivalently, the length of the word $w$);
  • $Q(w), Q(u)$ denote the recording tableaux of $w, u$ under the RSK correspondence;
  • $\operatorname{diag}(\lambda)$ denotes the diagonal index (or content) of an integer partition $\lambda$;
  • and $\operatorname{diag}(T)$ of a tableau $T$ denotes the diagonal index of the partition given by the shape of $T$.
The regular representation of the symmetric group of degree n has dimension n!, so any linear operator acting on this vector space has n! eigenvalues (counting multiplicities). Hence, the eigenvalues of the random-to-random operator can be indexed by permutations; and the values of this statistic give all the eigenvalues of the operator (Theorem 12 of [1]).
References
[1] Dieker, A. B., Saliola, F. Spectral analysis of random-to-random Markov chains arXiv:1509.08580
Code
def is_desarrangement_word(w):
    r"""
    EXAMPLES::

        sage: for n in range(4):
        ....:     for perm in Permutations(n):
        ....:         print "%s => %s" % (perm, is_desarrangement_word(perm))
        [] => True
        [1] => False
        [1, 2] => False
        [2, 1] => True
        [1, 2, 3] => False
        [1, 3, 2] => False
        [2, 1, 3] => True
        [2, 3, 1] => False
        [3, 1, 2] => True
        [3, 2, 1] => False
    """
    X = [k for k in range(len(w)-1) if w[k] <= w[k+1]]
    if X:
        return X[0] % 2 == 1
    else:
        return len(w) % 2 == 0

def desarrangement_factorization(w):
    r"""
    EXAMPLES::

        sage: for n in range(4):
        ....:     for perm in Permutations(n):
        ....:         print "%s => %s" % (perm, desarrangement_factorization(perm))
        [] => ([], [])
        [1] => ([1], [])
        [1, 2] => ([1, 2], [])
        [2, 1] => ([], [2, 1])
        [1, 2, 3] => ([1, 2, 3], [])
        [1, 3, 2] => ([1], [3, 2])
        [2, 1, 3] => ([], [2, 1, 3])
        [2, 3, 1] => ([2], [3, 1])
        [3, 1, 2] => ([], [3, 1, 2])
        [3, 2, 1] => ([3], [2, 1])
    """
    for i in range(len(w) + 1):
        u = Word(w[i:]).standard_permutation()
        if is_desarrangement_word(u):
            return w[:i], w[i:]

def diagonal_index_of_partition(la):
    return sum((j - i) for (i, j) in la.cells())

def binomial_shifted_diagonal_index_of_partition(partition):
    return binomial(partition.size() + 1, 2) + diagonal_index_of_partition(partition)

def statistic(w):
    r"""
    EXAMPLES::

        sage: for n in range(5):
        ....:     for perm in Permutations(n):
        ....:         print "%s => %s" % (perm, statistic(perm))
        [] => 0
        [1] => 1
        [1, 2] => 4
        [2, 1] => 0
        [1, 2, 3] => 9
        [1, 3, 2] => 4
        [2, 1, 3] => 0
        [2, 3, 1] => 4
        [3, 1, 2] => 0
        [3, 2, 1] => 1
    """
    Qw = RSK(w)[1]
    _, u = desarrangement_factorization(w)
    Qu = RSK(u)[1]
    Qw_index = binomial_shifted_diagonal_index_of_partition(Qw.shape())
    Qu_index = binomial_shifted_diagonal_index_of_partition(Qu.shape())
    return Qw_index - Qu_index

Created
May 23, 2016 at 21:58 by Franco Saliola
Updated
Jan 13, 2018 at 15:45 by Martin Rubey