edit this statistic or download as text // json
Identifier
Values
=>
[1]=>1 [1,2]=>1 [2,1]=>1 [1,2,3]=>1 [1,3,2]=>2 [2,1,3]=>1 [2,3,1]=>1 [3,1,2]=>1 [3,2,1]=>1 [1,2,3,4]=>1 [1,2,4,3]=>3 [1,3,2,4]=>2 [1,3,4,2]=>3 [1,4,2,3]=>3 [1,4,3,2]=>5 [2,1,3,4]=>1 [2,1,4,3]=>3 [2,3,1,4]=>1 [2,3,4,1]=>1 [2,4,1,3]=>2 [2,4,3,1]=>2 [3,1,2,4]=>1 [3,1,4,2]=>2 [3,2,1,4]=>1 [3,2,4,1]=>1 [3,4,1,2]=>1 [3,4,2,1]=>1 [4,1,2,3]=>1 [4,1,3,2]=>2 [4,2,1,3]=>1 [4,2,3,1]=>1 [4,3,1,2]=>1 [4,3,2,1]=>1 [1,2,3,4,5]=>1 [1,2,3,5,4]=>4 [1,2,4,3,5]=>3 [1,2,4,5,3]=>6 [1,2,5,3,4]=>6 [1,2,5,4,3]=>14 [1,3,2,4,5]=>2 [1,3,2,5,4]=>8 [1,3,4,2,5]=>3 [1,3,4,5,2]=>4 [1,3,5,2,4]=>8 [1,3,5,4,2]=>11 [1,4,2,3,5]=>3 [1,4,2,5,3]=>8 [1,4,3,2,5]=>5 [1,4,3,5,2]=>7 [1,4,5,2,3]=>6 [1,4,5,3,2]=>9 [1,5,2,3,4]=>4 [1,5,2,4,3]=>11 [1,5,3,2,4]=>7 [1,5,3,4,2]=>10 [1,5,4,2,3]=>9 [1,5,4,3,2]=>14 [2,1,3,4,5]=>1 [2,1,3,5,4]=>4 [2,1,4,3,5]=>3 [2,1,4,5,3]=>6 [2,1,5,3,4]=>6 [2,1,5,4,3]=>14 [2,3,1,4,5]=>1 [2,3,1,5,4]=>4 [2,3,4,1,5]=>1 [2,3,4,5,1]=>1 [2,3,5,1,4]=>3 [2,3,5,4,1]=>3 [2,4,1,3,5]=>2 [2,4,1,5,3]=>5 [2,4,3,1,5]=>2 [2,4,3,5,1]=>2 [2,4,5,1,3]=>3 [2,4,5,3,1]=>3 [2,5,1,3,4]=>3 [2,5,1,4,3]=>8 [2,5,3,1,4]=>3 [2,5,3,4,1]=>3 [2,5,4,1,3]=>5 [2,5,4,3,1]=>5 [3,1,2,4,5]=>1 [3,1,2,5,4]=>4 [3,1,4,2,5]=>2 [3,1,4,5,2]=>3 [3,1,5,2,4]=>5 [3,1,5,4,2]=>8 [3,2,1,4,5]=>1 [3,2,1,5,4]=>4 [3,2,4,1,5]=>1 [3,2,4,5,1]=>1 [3,2,5,1,4]=>3 [3,2,5,4,1]=>3 [3,4,1,2,5]=>1 [3,4,1,5,2]=>2 [3,4,2,1,5]=>1 [3,4,2,5,1]=>1 [3,4,5,1,2]=>1 [3,4,5,2,1]=>1 [3,5,1,2,4]=>2 [3,5,1,4,2]=>4 [3,5,2,1,4]=>2 [3,5,2,4,1]=>2 [3,5,4,1,2]=>2 [3,5,4,2,1]=>2 [4,1,2,3,5]=>1 [4,1,2,5,3]=>3 [4,1,3,2,5]=>2 [4,1,3,5,2]=>3 [4,1,5,2,3]=>3 [4,1,5,3,2]=>5 [4,2,1,3,5]=>1 [4,2,1,5,3]=>3 [4,2,3,1,5]=>1 [4,2,3,5,1]=>1 [4,2,5,1,3]=>2 [4,2,5,3,1]=>2 [4,3,1,2,5]=>1 [4,3,1,5,2]=>2 [4,3,2,1,5]=>1 [4,3,2,5,1]=>1 [4,3,5,1,2]=>1 [4,3,5,2,1]=>1 [4,5,1,2,3]=>1 [4,5,1,3,2]=>2 [4,5,2,1,3]=>1 [4,5,2,3,1]=>1 [4,5,3,1,2]=>1 [4,5,3,2,1]=>1 [5,1,2,3,4]=>1 [5,1,2,4,3]=>3 [5,1,3,2,4]=>2 [5,1,3,4,2]=>3 [5,1,4,2,3]=>3 [5,1,4,3,2]=>5 [5,2,1,3,4]=>1 [5,2,1,4,3]=>3 [5,2,3,1,4]=>1 [5,2,3,4,1]=>1 [5,2,4,1,3]=>2 [5,2,4,3,1]=>2 [5,3,1,2,4]=>1 [5,3,1,4,2]=>2 [5,3,2,1,4]=>1 [5,3,2,4,1]=>1 [5,3,4,1,2]=>1 [5,3,4,2,1]=>1 [5,4,1,2,3]=>1 [5,4,1,3,2]=>2 [5,4,2,1,3]=>1 [5,4,2,3,1]=>1 [5,4,3,1,2]=>1 [5,4,3,2,1]=>1 [1,2,3,4,5,6]=>1 [1,2,3,4,6,5]=>5 [1,2,3,5,4,6]=>4 [1,2,3,5,6,4]=>10 [1,2,3,6,4,5]=>10 [1,2,3,6,5,4]=>30 [1,2,4,3,5,6]=>3 [1,2,4,3,6,5]=>15 [1,2,4,5,3,6]=>6 [1,2,4,5,6,3]=>10 [1,2,4,6,3,5]=>20 [1,2,4,6,5,3]=>35 [1,2,5,3,4,6]=>6 [1,2,5,3,6,4]=>20 [1,2,5,4,3,6]=>14 [1,2,5,4,6,3]=>25 [1,2,5,6,3,4]=>20 [1,2,5,6,4,3]=>40 [1,2,6,3,4,5]=>10 [1,2,6,3,5,4]=>35 [1,2,6,4,3,5]=>25 [1,2,6,4,5,3]=>46 [1,2,6,5,3,4]=>40 [1,2,6,5,4,3]=>84 [1,3,2,4,5,6]=>2 [1,3,2,4,6,5]=>10 [1,3,2,5,4,6]=>8 [1,3,2,5,6,4]=>20 [1,3,2,6,4,5]=>20 [1,3,2,6,5,4]=>60 [1,3,4,2,5,6]=>3 [1,3,4,2,6,5]=>15 [1,3,4,5,2,6]=>4 [1,3,4,5,6,2]=>5 [1,3,4,6,2,5]=>15 [1,3,4,6,5,2]=>19 [1,3,5,2,4,6]=>8 [1,3,5,2,6,4]=>25 [1,3,5,4,2,6]=>11 [1,3,5,4,6,2]=>14 [1,3,5,6,2,4]=>20 [1,3,5,6,4,2]=>26 [1,3,6,2,4,5]=>15 [1,3,6,2,5,4]=>51 [1,3,6,4,2,5]=>21 [1,3,6,4,5,2]=>27 [1,3,6,5,2,4]=>44 [1,3,6,5,4,2]=>58 [1,4,2,3,5,6]=>3 [1,4,2,3,6,5]=>15 [1,4,2,5,3,6]=>8 [1,4,2,5,6,3]=>15 [1,4,2,6,3,5]=>25 [1,4,2,6,5,3]=>51 [1,4,3,2,5,6]=>5 [1,4,3,2,6,5]=>25 [1,4,3,5,2,6]=>7 [1,4,3,5,6,2]=>9 [1,4,3,6,2,5]=>26 [1,4,3,6,5,2]=>34 [1,4,5,2,3,6]=>6 [1,4,5,2,6,3]=>15 [1,4,5,3,2,6]=>9 [1,4,5,3,6,2]=>12 [1,4,5,6,2,3]=>10 [1,4,5,6,3,2]=>14 [1,4,6,2,3,5]=>15 [1,4,6,2,5,3]=>38 [1,4,6,3,2,5]=>23 [1,4,6,3,5,2]=>31 [1,4,6,5,2,3]=>26 [1,4,6,5,3,2]=>37 [1,5,2,3,4,6]=>4 [1,5,2,3,6,4]=>15 [1,5,2,4,3,6]=>11 [1,5,2,4,6,3]=>21 [1,5,2,6,3,4]=>20 [1,5,2,6,4,3]=>44 [1,5,3,2,4,6]=>7 [1,5,3,2,6,4]=>26 [1,5,3,4,2,6]=>10 [1,5,3,4,6,2]=>13 [1,5,3,6,2,4]=>24 [1,5,3,6,4,2]=>32 [1,5,4,2,3,6]=>9 [1,5,4,2,6,3]=>23 [1,5,4,3,2,6]=>14 [1,5,4,3,6,2]=>19 [1,5,4,6,2,3]=>16 [1,5,4,6,3,2]=>23 [1,5,6,2,3,4]=>10 [1,5,6,2,4,3]=>26 [1,5,6,3,2,4]=>16 [1,5,6,3,4,2]=>22 [1,5,6,4,2,3]=>19 [1,5,6,4,3,2]=>28 [1,6,2,3,4,5]=>5 [1,6,2,3,5,4]=>19 [1,6,2,4,3,5]=>14 [1,6,2,4,5,3]=>27 [1,6,2,5,3,4]=>26 [1,6,2,5,4,3]=>58 [1,6,3,2,4,5]=>9 [1,6,3,2,5,4]=>34 [1,6,3,4,2,5]=>13 [1,6,3,4,5,2]=>17 [1,6,3,5,2,4]=>32 [1,6,3,5,4,2]=>43 [1,6,4,2,3,5]=>12 [1,6,4,2,5,3]=>31 [1,6,4,3,2,5]=>19 [1,6,4,3,5,2]=>26 [1,6,4,5,2,3]=>22 [1,6,4,5,3,2]=>32 [1,6,5,2,3,4]=>14 [1,6,5,2,4,3]=>37 [1,6,5,3,2,4]=>23 [1,6,5,3,4,2]=>32 [1,6,5,4,2,3]=>28 [1,6,5,4,3,2]=>42 [2,1,3,4,5,6]=>1 [2,1,3,4,6,5]=>5 [2,1,3,5,4,6]=>4 [2,1,3,5,6,4]=>10 [2,1,3,6,4,5]=>10 [2,1,3,6,5,4]=>30 [2,1,4,3,5,6]=>3 [2,1,4,3,6,5]=>15 [2,1,4,5,3,6]=>6 [2,1,4,5,6,3]=>10 [2,1,4,6,3,5]=>20 [2,1,4,6,5,3]=>35 [2,1,5,3,4,6]=>6 [2,1,5,3,6,4]=>20 [2,1,5,4,3,6]=>14 [2,1,5,4,6,3]=>25 [2,1,5,6,3,4]=>20 [2,1,5,6,4,3]=>40 [2,1,6,3,4,5]=>10 [2,1,6,3,5,4]=>35 [2,1,6,4,3,5]=>25 [2,1,6,4,5,3]=>46 [2,1,6,5,3,4]=>40 [2,1,6,5,4,3]=>84 [2,3,1,4,5,6]=>1 [2,3,1,4,6,5]=>5 [2,3,1,5,4,6]=>4 [2,3,1,5,6,4]=>10 [2,3,1,6,4,5]=>10 [2,3,1,6,5,4]=>30 [2,3,4,1,5,6]=>1 [2,3,4,1,6,5]=>5 [2,3,4,5,1,6]=>1 [2,3,4,5,6,1]=>1 [2,3,4,6,1,5]=>4 [2,3,4,6,5,1]=>4 [2,3,5,1,4,6]=>3 [2,3,5,1,6,4]=>9 [2,3,5,4,1,6]=>3 [2,3,5,4,6,1]=>3 [2,3,5,6,1,4]=>6 [2,3,5,6,4,1]=>6 [2,3,6,1,4,5]=>6 [2,3,6,1,5,4]=>20 [2,3,6,4,1,5]=>6 [2,3,6,4,5,1]=>6 [2,3,6,5,1,4]=>14 [2,3,6,5,4,1]=>14 [2,4,1,3,5,6]=>2 [2,4,1,3,6,5]=>10 [2,4,1,5,3,6]=>5 [2,4,1,5,6,3]=>9 [2,4,1,6,3,5]=>16 [2,4,1,6,5,3]=>31 [2,4,3,1,5,6]=>2 [2,4,3,1,6,5]=>10 [2,4,3,5,1,6]=>2 [2,4,3,5,6,1]=>2 [2,4,3,6,1,5]=>8 [2,4,3,6,5,1]=>8 [2,4,5,1,3,6]=>3 [2,4,5,1,6,3]=>7 [2,4,5,3,1,6]=>3 [2,4,5,3,6,1]=>3 [2,4,5,6,1,3]=>4 [2,4,5,6,3,1]=>4 [2,4,6,1,3,5]=>8 [2,4,6,1,5,3]=>19 [2,4,6,3,1,5]=>8 [2,4,6,3,5,1]=>8 [2,4,6,5,1,3]=>11 [2,4,6,5,3,1]=>11 [2,5,1,3,4,6]=>3 [2,5,1,3,6,4]=>11 [2,5,1,4,3,6]=>8 [2,5,1,4,6,3]=>15 [2,5,1,6,3,4]=>14 [2,5,1,6,4,3]=>30 [2,5,3,1,4,6]=>3 [2,5,3,1,6,4]=>11 [2,5,3,4,1,6]=>3 [2,5,3,4,6,1]=>3 [2,5,3,6,1,4]=>8 [2,5,3,6,4,1]=>8 [2,5,4,1,3,6]=>5 [2,5,4,1,6,3]=>12 [2,5,4,3,1,6]=>5 [2,5,4,3,6,1]=>5 [2,5,4,6,1,3]=>7 [2,5,4,6,3,1]=>7 [2,5,6,1,3,4]=>6 [2,5,6,1,4,3]=>15 [2,5,6,3,1,4]=>6 [2,5,6,3,4,1]=>6 [2,5,6,4,1,3]=>9 [2,5,6,4,3,1]=>9 [2,6,1,3,4,5]=>4 [2,6,1,3,5,4]=>15 [2,6,1,4,3,5]=>11 [2,6,1,4,5,3]=>21 [2,6,1,5,3,4]=>20 [2,6,1,5,4,3]=>44 [2,6,3,1,4,5]=>4 [2,6,3,1,5,4]=>15 [2,6,3,4,1,5]=>4 [2,6,3,4,5,1]=>4 [2,6,3,5,1,4]=>11 [2,6,3,5,4,1]=>11 [2,6,4,1,3,5]=>7 [2,6,4,1,5,3]=>17 [2,6,4,3,1,5]=>7 [2,6,4,3,5,1]=>7 [2,6,4,5,1,3]=>10 [2,6,4,5,3,1]=>10 [2,6,5,1,3,4]=>9 [2,6,5,1,4,3]=>23 [2,6,5,3,1,4]=>9 [2,6,5,3,4,1]=>9 [2,6,5,4,1,3]=>14 [2,6,5,4,3,1]=>14 [3,1,2,4,5,6]=>1 [3,1,2,4,6,5]=>5 [3,1,2,5,4,6]=>4 [3,1,2,5,6,4]=>10 [3,1,2,6,4,5]=>10 [3,1,2,6,5,4]=>30 [3,1,4,2,5,6]=>2 [3,1,4,2,6,5]=>10 [3,1,4,5,2,6]=>3 [3,1,4,5,6,2]=>4 [3,1,4,6,2,5]=>11 [3,1,4,6,5,2]=>15 [3,1,5,2,4,6]=>5 [3,1,5,2,6,4]=>16 [3,1,5,4,2,6]=>8 [3,1,5,4,6,2]=>11 [3,1,5,6,2,4]=>14 [3,1,5,6,4,2]=>20 [3,1,6,2,4,5]=>9 [3,1,6,2,5,4]=>31 [3,1,6,4,2,5]=>15 [3,1,6,4,5,2]=>21 [3,1,6,5,2,4]=>30 [3,1,6,5,4,2]=>44 [3,2,1,4,5,6]=>1 [3,2,1,4,6,5]=>5 [3,2,1,5,4,6]=>4 [3,2,1,5,6,4]=>10 [3,2,1,6,4,5]=>10 [3,2,1,6,5,4]=>30 [3,2,4,1,5,6]=>1 [3,2,4,1,6,5]=>5 [3,2,4,5,1,6]=>1 [3,2,4,5,6,1]=>1 [3,2,4,6,1,5]=>4 [3,2,4,6,5,1]=>4 [3,2,5,1,4,6]=>3 [3,2,5,1,6,4]=>9 [3,2,5,4,1,6]=>3 [3,2,5,4,6,1]=>3 [3,2,5,6,1,4]=>6 [3,2,5,6,4,1]=>6 [3,2,6,1,4,5]=>6 [3,2,6,1,5,4]=>20 [3,2,6,4,1,5]=>6 [3,2,6,4,5,1]=>6 [3,2,6,5,1,4]=>14 [3,2,6,5,4,1]=>14 [3,4,1,2,5,6]=>1 [3,4,1,2,6,5]=>5 [3,4,1,5,2,6]=>2 [3,4,1,5,6,2]=>3 [3,4,1,6,2,5]=>7 [3,4,1,6,5,2]=>11 [3,4,2,1,5,6]=>1 [3,4,2,1,6,5]=>5 [3,4,2,5,1,6]=>1 [3,4,2,5,6,1]=>1 [3,4,2,6,1,5]=>4 [3,4,2,6,5,1]=>4 [3,4,5,1,2,6]=>1 [3,4,5,1,6,2]=>2 [3,4,5,2,1,6]=>1 [3,4,5,2,6,1]=>1 [3,4,5,6,1,2]=>1 [3,4,5,6,2,1]=>1 [3,4,6,1,2,5]=>3 [3,4,6,1,5,2]=>6 [3,4,6,2,1,5]=>3 [3,4,6,2,5,1]=>3 [3,4,6,5,1,2]=>3 [3,4,6,5,2,1]=>3 [3,5,1,2,4,6]=>2 [3,5,1,2,6,4]=>7 [3,5,1,4,2,6]=>4 [3,5,1,4,6,2]=>6 [3,5,1,6,2,4]=>8 [3,5,1,6,4,2]=>13 [3,5,2,1,4,6]=>2 [3,5,2,1,6,4]=>7 [3,5,2,4,1,6]=>2 [3,5,2,4,6,1]=>2 [3,5,2,6,1,4]=>5 [3,5,2,6,4,1]=>5 [3,5,4,1,2,6]=>2 [3,5,4,1,6,2]=>4 [3,5,4,2,1,6]=>2 [3,5,4,2,6,1]=>2 [3,5,4,6,1,2]=>2 [3,5,4,6,2,1]=>2 [3,5,6,1,2,4]=>3 [3,5,6,1,4,2]=>6 [3,5,6,2,1,4]=>3 [3,5,6,2,4,1]=>3 [3,5,6,4,1,2]=>3 [3,5,6,4,2,1]=>3 [3,6,1,2,4,5]=>3 [3,6,1,2,5,4]=>11 [3,6,1,4,2,5]=>6 [3,6,1,4,5,2]=>9 [3,6,1,5,2,4]=>13 [3,6,1,5,4,2]=>21 [3,6,2,1,4,5]=>3 [3,6,2,1,5,4]=>11 [3,6,2,4,1,5]=>3 [3,6,2,4,5,1]=>3 [3,6,2,5,1,4]=>8 [3,6,2,5,4,1]=>8 [3,6,4,1,2,5]=>3 [3,6,4,1,5,2]=>6 [3,6,4,2,1,5]=>3 [3,6,4,2,5,1]=>3 [3,6,4,5,1,2]=>3 [3,6,4,5,2,1]=>3 [3,6,5,1,2,4]=>5 [3,6,5,1,4,2]=>10 [3,6,5,2,1,4]=>5 [3,6,5,2,4,1]=>5 [3,6,5,4,1,2]=>5 [3,6,5,4,2,1]=>5 [4,1,2,3,5,6]=>1 [4,1,2,3,6,5]=>5 [4,1,2,5,3,6]=>3 [4,1,2,5,6,3]=>6 [4,1,2,6,3,5]=>9 [4,1,2,6,5,3]=>20 [4,1,3,2,5,6]=>2 [4,1,3,2,6,5]=>10 [4,1,3,5,2,6]=>3 [4,1,3,5,6,2]=>4 [4,1,3,6,2,5]=>11 [4,1,3,6,5,2]=>15 [4,1,5,2,3,6]=>3 [4,1,5,2,6,3]=>8 [4,1,5,3,2,6]=>5 [4,1,5,3,6,2]=>7 [4,1,5,6,2,3]=>6 [4,1,5,6,3,2]=>9 [4,1,6,2,3,5]=>7 [4,1,6,2,5,3]=>19 [4,1,6,3,2,5]=>12 [4,1,6,3,5,2]=>17 [4,1,6,5,2,3]=>15 [4,1,6,5,3,2]=>23 [4,2,1,3,5,6]=>1 [4,2,1,3,6,5]=>5 [4,2,1,5,3,6]=>3 [4,2,1,5,6,3]=>6 [4,2,1,6,3,5]=>9 [4,2,1,6,5,3]=>20 [4,2,3,1,5,6]=>1 [4,2,3,1,6,5]=>5 [4,2,3,5,1,6]=>1 [4,2,3,5,6,1]=>1 [4,2,3,6,1,5]=>4 [4,2,3,6,5,1]=>4 [4,2,5,1,3,6]=>2 [4,2,5,1,6,3]=>5 [4,2,5,3,1,6]=>2 [4,2,5,3,6,1]=>2 [4,2,5,6,1,3]=>3 [4,2,5,6,3,1]=>3 [4,2,6,1,3,5]=>5 [4,2,6,1,5,3]=>13 [4,2,6,3,1,5]=>5 [4,2,6,3,5,1]=>5 [4,2,6,5,1,3]=>8 [4,2,6,5,3,1]=>8 [4,3,1,2,5,6]=>1 [4,3,1,2,6,5]=>5 [4,3,1,5,2,6]=>2 [4,3,1,5,6,2]=>3 [4,3,1,6,2,5]=>7 [4,3,1,6,5,2]=>11 [4,3,2,1,5,6]=>1 [4,3,2,1,6,5]=>5 [4,3,2,5,1,6]=>1 [4,3,2,5,6,1]=>1 [4,3,2,6,1,5]=>4 [4,3,2,6,5,1]=>4 [4,3,5,1,2,6]=>1 [4,3,5,1,6,2]=>2 [4,3,5,2,1,6]=>1 [4,3,5,2,6,1]=>1 [4,3,5,6,1,2]=>1 [4,3,5,6,2,1]=>1 [4,3,6,1,2,5]=>3 [4,3,6,1,5,2]=>6 [4,3,6,2,1,5]=>3 [4,3,6,2,5,1]=>3 [4,3,6,5,1,2]=>3 [4,3,6,5,2,1]=>3 [4,5,1,2,3,6]=>1 [4,5,1,2,6,3]=>3 [4,5,1,3,2,6]=>2 [4,5,1,3,6,2]=>3 [4,5,1,6,2,3]=>3 [4,5,1,6,3,2]=>5 [4,5,2,1,3,6]=>1 [4,5,2,1,6,3]=>3 [4,5,2,3,1,6]=>1 [4,5,2,3,6,1]=>1 [4,5,2,6,1,3]=>2 [4,5,2,6,3,1]=>2 [4,5,3,1,2,6]=>1 [4,5,3,1,6,2]=>2 [4,5,3,2,1,6]=>1 [4,5,3,2,6,1]=>1 [4,5,3,6,1,2]=>1 [4,5,3,6,2,1]=>1 [4,5,6,1,2,3]=>1 [4,5,6,1,3,2]=>2 [4,5,6,2,1,3]=>1 [4,5,6,2,3,1]=>1 [4,5,6,3,1,2]=>1 [4,5,6,3,2,1]=>1 [4,6,1,2,3,5]=>2 [4,6,1,2,5,3]=>6 [4,6,1,3,2,5]=>4 [4,6,1,3,5,2]=>6 [4,6,1,5,2,3]=>6 [4,6,1,5,3,2]=>10 [4,6,2,1,3,5]=>2 [4,6,2,1,5,3]=>6 [4,6,2,3,1,5]=>2 [4,6,2,3,5,1]=>2 [4,6,2,5,1,3]=>4 [4,6,2,5,3,1]=>4 [4,6,3,1,2,5]=>2 [4,6,3,1,5,2]=>4 [4,6,3,2,1,5]=>2 [4,6,3,2,5,1]=>2 [4,6,3,5,1,2]=>2 [4,6,3,5,2,1]=>2 [4,6,5,1,2,3]=>2 [4,6,5,1,3,2]=>4 [4,6,5,2,1,3]=>2 [4,6,5,2,3,1]=>2 [4,6,5,3,1,2]=>2 [4,6,5,3,2,1]=>2 [5,1,2,3,4,6]=>1 [5,1,2,3,6,4]=>4 [5,1,2,4,3,6]=>3 [5,1,2,4,6,3]=>6 [5,1,2,6,3,4]=>6 [5,1,2,6,4,3]=>14 [5,1,3,2,4,6]=>2 [5,1,3,2,6,4]=>8 [5,1,3,4,2,6]=>3 [5,1,3,4,6,2]=>4 [5,1,3,6,2,4]=>8 [5,1,3,6,4,2]=>11 [5,1,4,2,3,6]=>3 [5,1,4,2,6,3]=>8 [5,1,4,3,2,6]=>5 [5,1,4,3,6,2]=>7 [5,1,4,6,2,3]=>6 [5,1,4,6,3,2]=>9 [5,1,6,2,3,4]=>4 [5,1,6,2,4,3]=>11 [5,1,6,3,2,4]=>7 [5,1,6,3,4,2]=>10 [5,1,6,4,2,3]=>9 [5,1,6,4,3,2]=>14 [5,2,1,3,4,6]=>1 [5,2,1,3,6,4]=>4 [5,2,1,4,3,6]=>3 [5,2,1,4,6,3]=>6 [5,2,1,6,3,4]=>6 [5,2,1,6,4,3]=>14 [5,2,3,1,4,6]=>1 [5,2,3,1,6,4]=>4 [5,2,3,4,1,6]=>1 [5,2,3,4,6,1]=>1 [5,2,3,6,1,4]=>3 [5,2,3,6,4,1]=>3 [5,2,4,1,3,6]=>2 [5,2,4,1,6,3]=>5 [5,2,4,3,1,6]=>2 [5,2,4,3,6,1]=>2 [5,2,4,6,1,3]=>3 [5,2,4,6,3,1]=>3 [5,2,6,1,3,4]=>3 [5,2,6,1,4,3]=>8 [5,2,6,3,1,4]=>3 [5,2,6,3,4,1]=>3 [5,2,6,4,1,3]=>5 [5,2,6,4,3,1]=>5 [5,3,1,2,4,6]=>1 [5,3,1,2,6,4]=>4 [5,3,1,4,2,6]=>2 [5,3,1,4,6,2]=>3 [5,3,1,6,2,4]=>5 [5,3,1,6,4,2]=>8 [5,3,2,1,4,6]=>1 [5,3,2,1,6,4]=>4 [5,3,2,4,1,6]=>1 [5,3,2,4,6,1]=>1 [5,3,2,6,1,4]=>3 [5,3,2,6,4,1]=>3 [5,3,4,1,2,6]=>1 [5,3,4,1,6,2]=>2 [5,3,4,2,1,6]=>1 [5,3,4,2,6,1]=>1 [5,3,4,6,1,2]=>1 [5,3,4,6,2,1]=>1 [5,3,6,1,2,4]=>2 [5,3,6,1,4,2]=>4 [5,3,6,2,1,4]=>2 [5,3,6,2,4,1]=>2 [5,3,6,4,1,2]=>2 [5,3,6,4,2,1]=>2 [5,4,1,2,3,6]=>1 [5,4,1,2,6,3]=>3 [5,4,1,3,2,6]=>2 [5,4,1,3,6,2]=>3 [5,4,1,6,2,3]=>3 [5,4,1,6,3,2]=>5 [5,4,2,1,3,6]=>1 [5,4,2,1,6,3]=>3 [5,4,2,3,1,6]=>1 [5,4,2,3,6,1]=>1 [5,4,2,6,1,3]=>2 [5,4,2,6,3,1]=>2 [5,4,3,1,2,6]=>1 [5,4,3,1,6,2]=>2 [5,4,3,2,1,6]=>1 [5,4,3,2,6,1]=>1 [5,4,3,6,1,2]=>1 [5,4,3,6,2,1]=>1 [5,4,6,1,2,3]=>1 [5,4,6,1,3,2]=>2 [5,4,6,2,1,3]=>1 [5,4,6,2,3,1]=>1 [5,4,6,3,1,2]=>1 [5,4,6,3,2,1]=>1 [5,6,1,2,3,4]=>1 [5,6,1,2,4,3]=>3 [5,6,1,3,2,4]=>2 [5,6,1,3,4,2]=>3 [5,6,1,4,2,3]=>3 [5,6,1,4,3,2]=>5 [5,6,2,1,3,4]=>1 [5,6,2,1,4,3]=>3 [5,6,2,3,1,4]=>1 [5,6,2,3,4,1]=>1 [5,6,2,4,1,3]=>2 [5,6,2,4,3,1]=>2 [5,6,3,1,2,4]=>1 [5,6,3,1,4,2]=>2 [5,6,3,2,1,4]=>1 [5,6,3,2,4,1]=>1 [5,6,3,4,1,2]=>1 [5,6,3,4,2,1]=>1 [5,6,4,1,2,3]=>1 [5,6,4,1,3,2]=>2 [5,6,4,2,1,3]=>1 [5,6,4,2,3,1]=>1 [5,6,4,3,1,2]=>1 [5,6,4,3,2,1]=>1 [6,1,2,3,4,5]=>1 [6,1,2,3,5,4]=>4 [6,1,2,4,3,5]=>3 [6,1,2,4,5,3]=>6 [6,1,2,5,3,4]=>6 [6,1,2,5,4,3]=>14 [6,1,3,2,4,5]=>2 [6,1,3,2,5,4]=>8 [6,1,3,4,2,5]=>3 [6,1,3,4,5,2]=>4 [6,1,3,5,2,4]=>8 [6,1,3,5,4,2]=>11 [6,1,4,2,3,5]=>3 [6,1,4,2,5,3]=>8 [6,1,4,3,2,5]=>5 [6,1,4,3,5,2]=>7 [6,1,4,5,2,3]=>6 [6,1,4,5,3,2]=>9 [6,1,5,2,3,4]=>4 [6,1,5,2,4,3]=>11 [6,1,5,3,2,4]=>7 [6,1,5,3,4,2]=>10 [6,1,5,4,2,3]=>9 [6,1,5,4,3,2]=>14 [6,2,1,3,4,5]=>1 [6,2,1,3,5,4]=>4 [6,2,1,4,3,5]=>3 [6,2,1,4,5,3]=>6 [6,2,1,5,3,4]=>6 [6,2,1,5,4,3]=>14 [6,2,3,1,4,5]=>1 [6,2,3,1,5,4]=>4 [6,2,3,4,1,5]=>1 [6,2,3,4,5,1]=>1 [6,2,3,5,1,4]=>3 [6,2,3,5,4,1]=>3 [6,2,4,1,3,5]=>2 [6,2,4,1,5,3]=>5 [6,2,4,3,1,5]=>2 [6,2,4,3,5,1]=>2 [6,2,4,5,1,3]=>3 [6,2,4,5,3,1]=>3 [6,2,5,1,3,4]=>3 [6,2,5,1,4,3]=>8 [6,2,5,3,1,4]=>3 [6,2,5,3,4,1]=>3 [6,2,5,4,1,3]=>5 [6,2,5,4,3,1]=>5 [6,3,1,2,4,5]=>1 [6,3,1,2,5,4]=>4 [6,3,1,4,2,5]=>2 [6,3,1,4,5,2]=>3 [6,3,1,5,2,4]=>5 [6,3,1,5,4,2]=>8 [6,3,2,1,4,5]=>1 [6,3,2,1,5,4]=>4 [6,3,2,4,1,5]=>1 [6,3,2,4,5,1]=>1 [6,3,2,5,1,4]=>3 [6,3,2,5,4,1]=>3 [6,3,4,1,2,5]=>1 [6,3,4,1,5,2]=>2 [6,3,4,2,1,5]=>1 [6,3,4,2,5,1]=>1 [6,3,4,5,1,2]=>1 [6,3,4,5,2,1]=>1 [6,3,5,1,2,4]=>2 [6,3,5,1,4,2]=>4 [6,3,5,2,1,4]=>2 [6,3,5,2,4,1]=>2 [6,3,5,4,1,2]=>2 [6,3,5,4,2,1]=>2 [6,4,1,2,3,5]=>1 [6,4,1,2,5,3]=>3 [6,4,1,3,2,5]=>2 [6,4,1,3,5,2]=>3 [6,4,1,5,2,3]=>3 [6,4,1,5,3,2]=>5 [6,4,2,1,3,5]=>1 [6,4,2,1,5,3]=>3 [6,4,2,3,1,5]=>1 [6,4,2,3,5,1]=>1 [6,4,2,5,1,3]=>2 [6,4,2,5,3,1]=>2 [6,4,3,1,2,5]=>1 [6,4,3,1,5,2]=>2 [6,4,3,2,1,5]=>1 [6,4,3,2,5,1]=>1 [6,4,3,5,1,2]=>1 [6,4,3,5,2,1]=>1 [6,4,5,1,2,3]=>1 [6,4,5,1,3,2]=>2 [6,4,5,2,1,3]=>1 [6,4,5,2,3,1]=>1 [6,4,5,3,1,2]=>1 [6,4,5,3,2,1]=>1 [6,5,1,2,3,4]=>1 [6,5,1,2,4,3]=>3 [6,5,1,3,2,4]=>2 [6,5,1,3,4,2]=>3 [6,5,1,4,2,3]=>3 [6,5,1,4,3,2]=>5 [6,5,2,1,3,4]=>1 [6,5,2,1,4,3]=>3 [6,5,2,3,1,4]=>1 [6,5,2,3,4,1]=>1 [6,5,2,4,1,3]=>2 [6,5,2,4,3,1]=>2 [6,5,3,1,2,4]=>1 [6,5,3,1,4,2]=>2 [6,5,3,2,1,4]=>1 [6,5,3,2,4,1]=>1 [6,5,3,4,1,2]=>1 [6,5,3,4,2,1]=>1 [6,5,4,1,2,3]=>1 [6,5,4,1,3,2]=>2 [6,5,4,2,1,3]=>1 [6,5,4,2,3,1]=>1 [6,5,4,3,1,2]=>1 [6,5,4,3,2,1]=>1 [1,2,3,4,5,6,7]=>1 [1,2,3,4,5,7,6]=>6 [1,2,3,4,6,5,7]=>5 [1,2,3,6,4,7,5]=>40 [1,2,6,3,4,7,5]=>45 [1,3,2,4,5,6,7]=>2 [1,3,7,2,4,5,6]=>24 [1,3,7,6,5,4,2]=>378 [1,4,6,7,2,3,5]=>45 [1,4,6,7,5,3,2]=>97 [1,4,7,2,3,5,6]=>27 [1,4,7,6,5,3,2]=>211 [1,5,2,3,4,7,6]=>24 [1,5,2,7,3,4,6]=>56 [1,5,4,7,6,3,2]=>125 [1,5,6,2,7,3,4]=>45 [1,5,6,4,7,3,2]=>43 [1,5,6,7,2,3,4]=>20 [1,5,6,7,4,3,2]=>48 [1,5,7,2,3,4,6]=>24 [1,5,7,6,4,3,2]=>124 [1,6,2,3,4,7,5]=>24 [1,6,2,3,7,4,5]=>45 [1,6,2,7,3,4,5]=>40 [1,6,5,4,3,7,2]=>56 [1,6,5,4,7,3,2]=>66 [1,6,5,7,4,3,2]=>76 [1,6,7,2,3,4,5]=>15 [1,6,7,5,4,3,2]=>90 [1,7,2,3,4,5,6]=>6 [1,7,3,5,6,4,2]=>122 [1,7,3,6,5,4,2]=>251 [1,7,4,3,6,5,2]=>150 [1,7,4,5,3,6,2]=>54 [1,7,4,5,6,3,2]=>62 [1,7,4,6,5,3,2]=>151 [1,7,5,4,3,6,2]=>75 [1,7,5,4,6,3,2]=>89 [1,7,6,4,5,3,2]=>107 [1,7,6,5,4,3,2]=>132 [2,1,3,4,5,6,7]=>1 [2,1,3,4,7,5,6]=>15 [2,1,3,7,4,5,6]=>20 [2,1,6,3,4,5,7]=>10 [2,1,7,3,4,5,6]=>15 [2,1,7,6,5,4,3]=>594 [2,3,1,4,5,6,7]=>1 [2,3,4,1,5,6,7]=>1 [2,3,4,1,5,7,6]=>6 [2,3,4,5,1,6,7]=>1 [2,3,4,5,1,7,6]=>6 [2,3,4,5,6,1,7]=>1 [2,3,4,5,6,7,1]=>1 [2,3,4,5,7,1,6]=>5 [2,3,4,6,1,7,5]=>14 [2,3,4,6,7,1,5]=>10 [2,3,4,7,1,5,6]=>10 [2,3,5,6,7,1,4]=>10 [2,4,1,3,5,6,7]=>2 [2,5,1,3,4,6,7]=>3 [2,5,7,1,3,4,6]=>15 [2,6,1,3,4,5,7]=>4 [2,6,1,7,3,4,5]=>30 [2,6,5,4,3,1,7]=>14 [2,6,7,1,3,4,5]=>10 [2,7,1,3,4,5,6]=>5 [3,1,2,4,5,6,7]=>1 [3,1,2,4,5,7,6]=>6 [3,2,1,4,5,6,7]=>1 [3,2,4,5,6,7,1]=>1 [3,4,1,2,5,6,7]=>1 [3,4,2,5,6,7,1]=>1 [3,4,5,2,6,7,1]=>1 [3,4,5,6,2,7,1]=>1 [3,4,5,6,7,1,2]=>1 [3,5,6,1,2,4,7]=>3 [3,5,6,4,2,1,7]=>3 [3,6,1,2,4,5,7]=>3 [3,6,5,4,2,1,7]=>5 [4,1,2,3,5,6,7]=>1 [4,1,2,3,5,7,6]=>6 [4,1,6,2,3,5,7]=>7 [4,2,3,5,6,7,1]=>1 [4,2,5,1,3,6,7]=>2 [4,3,2,1,5,6,7]=>1 [4,3,2,5,6,7,1]=>1 [4,3,5,2,6,7,1]=>1 [4,3,6,5,2,1,7]=>3 [4,5,1,6,2,3,7]=>3 [4,5,2,3,6,7,1]=>1 [4,5,3,6,2,1,7]=>1 [4,5,6,1,2,3,7]=>1 [4,5,6,3,2,1,7]=>1 [4,6,1,2,3,5,7]=>2 [4,6,1,2,3,7,5]=>9 [4,6,5,3,2,1,7]=>2 [5,1,2,3,4,6,7]=>1 [5,1,2,3,4,7,6]=>6 [5,1,2,3,6,4,7]=>4 [5,1,2,3,6,7,4]=>10 [5,1,2,3,7,4,6]=>14 [5,1,2,6,3,4,7]=>6 [5,1,6,2,3,4,7]=>4 [5,1,6,2,3,7,4]=>15 [5,2,3,4,6,7,1]=>1 [5,2,6,7,1,3,4]=>6 [5,3,2,4,6,7,1]=>1 [5,3,2,6,1,4,7]=>3 [5,4,3,2,1,6,7]=>1 [5,4,3,2,1,7,6]=>6 [5,4,3,2,6,1,7]=>1 [5,4,3,6,2,1,7]=>1 [5,4,6,3,2,1,7]=>1 [5,6,1,2,3,4,7]=>1 [5,6,1,2,3,7,4]=>4 [5,6,3,4,1,2,7]=>1 [5,6,4,3,2,1,7]=>1 [5,6,7,1,2,3,4]=>1 [6,1,2,3,4,5,7]=>1 [6,1,2,3,4,7,5]=>5 [6,1,2,3,7,4,5]=>10 [6,1,2,7,3,4,5]=>10 [6,2,3,4,5,7,1]=>1 [6,2,4,5,3,1,7]=>3 [6,2,5,4,3,1,7]=>5 [6,3,2,5,4,1,7]=>3 [6,3,4,2,5,1,7]=>1 [6,3,4,5,2,1,7]=>1 [6,3,5,4,2,1,7]=>2 [6,4,3,2,5,1,7]=>1
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Description
The number of reduced Kogan faces with the permutation as type.
This is equivalent to finding the number of ways to represent the permutation $\pi \in S_{n+1}$ as a reduced subword of $s_n (s_{n-1} s_n) (s_{n-2} s_{n-1} s_n) \dotsm (s_1 \dotsm s_n)$, or the number of reduced pipe dreams for $\pi$.
References
[1] Kirichenko, V. A., Smirnov, E. Y., Timorin, V. A. Schubert calculus and Gelfand-Zetlin polytopes DOI:10.1070/rm2012v067n04abeh004804 arXiv:1101.0278
Code
def statistic(pi):
    n = len(pi)
    if pi == Permutations(n).one():
        return 1
    else:
        S = Word([j for i in range(n, 0, -1) for j in range(i, n+1)])
        return sum(S.number_of_subword_occurrences(Word(w)) for w in pi.reduced_words())
Created
Jun 15, 2015 at 18:00 by Per Alexandersson
Updated
May 19, 2023 at 14:57 by Martin Rubey