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]=>1 [2,1,3]=>1 [2,3,1]=>2 [3,1,2]=>2 [3,2,1]=>5 [1,2,3,4]=>1 [1,2,4,3]=>1 [1,3,2,4]=>1 [1,3,4,2]=>2 [1,4,2,3]=>2 [1,4,3,2]=>5 [2,1,3,4]=>1 [2,1,4,3]=>2 [2,3,1,4]=>2 [2,3,4,1]=>6 [2,4,1,3]=>6 [2,4,3,1]=>18 [3,1,2,4]=>2 [3,1,4,2]=>6 [3,2,1,4]=>5 [3,2,4,1]=>18 [3,4,1,2]=>22 [3,4,2,1]=>62 [4,1,2,3]=>6 [4,1,3,2]=>18 [4,2,1,3]=>18 [4,2,3,1]=>73 [4,3,1,2]=>62 [4,3,2,1]=>210 [1,2,3,4,5]=>1 [1,2,3,5,4]=>1 [1,2,4,3,5]=>1 [1,2,4,5,3]=>2 [1,2,5,3,4]=>2 [1,2,5,4,3]=>5 [1,3,2,4,5]=>1 [1,3,2,5,4]=>2 [1,3,4,2,5]=>2 [1,3,4,5,2]=>6 [1,3,5,2,4]=>6 [1,3,5,4,2]=>18 [1,4,2,3,5]=>2 [1,4,2,5,3]=>6 [1,4,3,2,5]=>5 [1,4,3,5,2]=>18 [1,4,5,2,3]=>22 [1,4,5,3,2]=>62 [1,5,2,3,4]=>6 [1,5,2,4,3]=>18 [1,5,3,2,4]=>18 [1,5,3,4,2]=>73 [1,5,4,2,3]=>62 [1,5,4,3,2]=>210 [2,1,3,4,5]=>1 [2,1,3,5,4]=>2 [2,1,4,3,5]=>2 [2,1,4,5,3]=>6 [2,1,5,3,4]=>6 [2,1,5,4,3]=>18 [2,3,1,4,5]=>2 [2,3,1,5,4]=>6 [2,3,4,1,5]=>6 [2,3,4,5,1]=>24 [2,3,5,1,4]=>24 [2,3,5,4,1]=>86 [2,4,1,3,5]=>6 [2,4,1,5,3]=>24 [2,4,3,1,5]=>18 [2,4,3,5,1]=>86 [2,4,5,1,3]=>106 [2,4,5,3,1]=>352 [2,5,1,3,4]=>24 [2,5,1,4,3]=>86 [2,5,3,1,4]=>86 [2,5,3,4,1]=>418 [2,5,4,1,3]=>352 [2,5,4,3,1]=>1382 [3,1,2,4,5]=>2 [3,1,2,5,4]=>6 [3,1,4,2,5]=>6 [3,1,4,5,2]=>24 [3,1,5,2,4]=>24 [3,1,5,4,2]=>86 [3,2,1,4,5]=>5 [3,2,1,5,4]=>18 [3,2,4,1,5]=>18 [3,2,4,5,1]=>86 [3,2,5,1,4]=>86 [3,2,5,4,1]=>354 [3,4,1,2,5]=>22 [3,4,1,5,2]=>106 [3,4,2,1,5]=>62 [3,4,2,5,1]=>352 [3,4,5,1,2]=>508 [3,4,5,2,1]=>1634 [3,5,1,2,4]=>106 [3,5,1,4,2]=>462 [3,5,2,1,4]=>352 [3,5,2,4,1]=>1958 [3,5,4,1,2]=>1926 [3,5,4,2,1]=>7160 [4,1,2,3,5]=>6 [4,1,2,5,3]=>24 [4,1,3,2,5]=>18 [4,1,3,5,2]=>86 [4,1,5,2,3]=>106 [4,1,5,3,2]=>352 [4,2,1,3,5]=>18 [4,2,1,5,3]=>86 [4,2,3,1,5]=>73 [4,2,3,5,1]=>418 [4,2,5,1,3]=>462 [4,2,5,3,1]=>1958 [4,3,1,2,5]=>62 [4,3,1,5,2]=>352 [4,3,2,1,5]=>210 [4,3,2,5,1]=>1382 [4,3,5,1,2]=>1926 [4,3,5,2,1]=>7160 [4,5,1,2,3]=>508 [4,5,1,3,2]=>1926 [4,5,2,1,3]=>1926 [4,5,2,3,1]=>10544 [4,5,3,1,2]=>8022 [4,5,3,2,1]=>34266 [5,1,2,3,4]=>24 [5,1,2,4,3]=>86 [5,1,3,2,4]=>86 [5,1,3,4,2]=>418 [5,1,4,2,3]=>352 [5,1,4,3,2]=>1382 [5,2,1,3,4]=>86 [5,2,1,4,3]=>354 [5,2,3,1,4]=>418 [5,2,3,4,1]=>2381 [5,2,4,1,3]=>1958 [5,2,4,3,1]=>9234 [5,3,1,2,4]=>352 [5,3,1,4,2]=>1958 [5,3,2,1,4]=>1382 [5,3,2,4,1]=>9234 [5,3,4,1,2]=>10544 [5,3,4,2,1]=>44062 [5,4,1,2,3]=>1634 [5,4,1,3,2]=>7160 [5,4,2,1,3]=>7160 [5,4,2,3,1]=>44062 [5,4,3,1,2]=>34266 [5,4,3,2,1]=>162482 [1,2,3,4,5,6]=>1 [1,2,3,4,6,5]=>1 [1,2,3,5,4,6]=>1 [1,2,3,5,6,4]=>2 [1,2,3,6,4,5]=>2 [1,2,3,6,5,4]=>5 [1,2,4,3,5,6]=>1 [1,2,4,3,6,5]=>2 [1,2,4,5,3,6]=>2 [1,2,4,5,6,3]=>6 [1,2,4,6,3,5]=>6 [1,2,4,6,5,3]=>18 [1,2,5,3,4,6]=>2 [1,2,5,3,6,4]=>6 [1,2,5,4,3,6]=>5 [1,2,5,4,6,3]=>18 [1,2,5,6,3,4]=>22 [1,2,5,6,4,3]=>62 [1,2,6,3,4,5]=>6 [1,2,6,3,5,4]=>18 [1,2,6,4,3,5]=>18 [1,2,6,4,5,3]=>73 [1,2,6,5,3,4]=>62 [1,2,6,5,4,3]=>210 [1,3,2,4,5,6]=>1 [1,3,2,4,6,5]=>2 [1,3,2,5,4,6]=>2 [1,3,2,5,6,4]=>6 [1,3,2,6,4,5]=>6 [1,3,2,6,5,4]=>18 [1,3,4,2,5,6]=>2 [1,3,4,2,6,5]=>6 [1,3,4,5,2,6]=>6 [1,3,4,5,6,2]=>24 [1,3,4,6,2,5]=>24 [1,3,4,6,5,2]=>86 [1,3,5,2,4,6]=>6 [1,3,5,2,6,4]=>24 [1,3,5,4,2,6]=>18 [1,3,5,4,6,2]=>86 [1,3,5,6,2,4]=>106 [1,3,5,6,4,2]=>352 [1,3,6,2,4,5]=>24 [1,3,6,2,5,4]=>86 [1,3,6,4,2,5]=>86 [1,3,6,4,5,2]=>418 [1,3,6,5,2,4]=>352 [1,3,6,5,4,2]=>1382 [1,4,2,3,5,6]=>2 [1,4,2,3,6,5]=>6 [1,4,2,5,3,6]=>6 [1,4,2,5,6,3]=>24 [1,4,2,6,3,5]=>24 [1,4,2,6,5,3]=>86 [1,4,3,2,5,6]=>5 [1,4,3,2,6,5]=>18 [1,4,3,5,2,6]=>18 [1,4,3,5,6,2]=>86 [1,4,3,6,2,5]=>86 [1,4,3,6,5,2]=>354 [1,4,5,2,3,6]=>22 [1,4,5,2,6,3]=>106 [1,4,5,3,2,6]=>62 [1,4,5,3,6,2]=>352 [1,4,5,6,2,3]=>508 [1,4,5,6,3,2]=>1634 [1,4,6,2,3,5]=>106 [1,4,6,2,5,3]=>462 [1,4,6,3,2,5]=>352 [1,4,6,3,5,2]=>1958 [1,4,6,5,2,3]=>1926 [1,4,6,5,3,2]=>7160 [1,5,2,3,4,6]=>6 [1,5,2,3,6,4]=>24 [1,5,2,4,3,6]=>18 [1,5,2,4,6,3]=>86 [1,5,2,6,3,4]=>106 [1,5,2,6,4,3]=>352 [1,5,3,2,4,6]=>18 [1,5,3,2,6,4]=>86 [1,5,3,4,2,6]=>73 [1,5,3,4,6,2]=>418 [1,5,3,6,2,4]=>462 [1,5,3,6,4,2]=>1958 [1,5,4,2,3,6]=>62 [1,5,4,2,6,3]=>352 [1,5,4,3,2,6]=>210 [1,5,4,3,6,2]=>1382 [1,5,4,6,2,3]=>1926 [1,5,4,6,3,2]=>7160 [1,5,6,2,3,4]=>508 [1,5,6,2,4,3]=>1926 [1,5,6,3,2,4]=>1926 [1,5,6,3,4,2]=>10544 [1,5,6,4,2,3]=>8022 [1,5,6,4,3,2]=>34266 [1,6,2,3,4,5]=>24 [1,6,2,3,5,4]=>86 [1,6,2,4,3,5]=>86 [1,6,2,4,5,3]=>418 [1,6,2,5,3,4]=>352 [1,6,2,5,4,3]=>1382 [1,6,3,2,4,5]=>86 [1,6,3,2,5,4]=>354 [1,6,3,4,2,5]=>418 [1,6,3,4,5,2]=>2381 [1,6,3,5,2,4]=>1958 [1,6,3,5,4,2]=>9234 [1,6,4,2,3,5]=>352 [1,6,4,2,5,3]=>1958 [1,6,4,3,2,5]=>1382 [1,6,4,3,5,2]=>9234 [1,6,4,5,2,3]=>10544 [1,6,4,5,3,2]=>44062 [1,6,5,2,3,4]=>1634 [1,6,5,2,4,3]=>7160 [1,6,5,3,2,4]=>7160 [1,6,5,3,4,2]=>44062 [1,6,5,4,2,3]=>34266 [1,6,5,4,3,2]=>162482 [2,1,3,4,5,6]=>1 [2,1,3,4,6,5]=>2 [2,1,3,5,4,6]=>2 [2,1,3,5,6,4]=>6 [2,1,3,6,4,5]=>6 [2,1,3,6,5,4]=>18 [2,1,4,3,5,6]=>2 [2,1,4,3,6,5]=>6 [2,1,4,5,3,6]=>6 [2,1,4,5,6,3]=>24 [2,1,4,6,3,5]=>24 [2,1,4,6,5,3]=>86 [2,1,5,3,4,6]=>6 [2,1,5,3,6,4]=>24 [2,1,5,4,3,6]=>18 [2,1,5,4,6,3]=>86 [2,1,5,6,3,4]=>106 [2,1,5,6,4,3]=>352 [2,1,6,3,4,5]=>24 [2,1,6,3,5,4]=>86 [2,1,6,4,3,5]=>86 [2,1,6,4,5,3]=>418 [2,1,6,5,3,4]=>352 [2,1,6,5,4,3]=>1382 [2,3,1,4,5,6]=>2 [2,3,1,4,6,5]=>6 [2,3,1,5,4,6]=>6 [2,3,1,5,6,4]=>24 [2,3,1,6,4,5]=>24 [2,3,1,6,5,4]=>86 [2,3,4,1,5,6]=>6 [2,3,4,1,6,5]=>24 [2,3,4,5,1,6]=>24 [2,3,4,5,6,1]=>120 [2,3,4,6,1,5]=>120 [2,3,4,6,5,1]=>504 [2,3,5,1,4,6]=>24 [2,3,5,1,6,4]=>120 [2,3,5,4,1,6]=>86 [2,3,5,4,6,1]=>504 [2,3,5,6,1,4]=>624 [2,3,5,6,4,1]=>2384 [2,3,6,1,4,5]=>120 [2,3,6,1,5,4]=>504 [2,3,6,4,1,5]=>504 [2,3,6,4,5,1]=>2854 [2,3,6,5,1,4]=>2384 [2,3,6,5,4,1]=>10632 [2,4,1,3,5,6]=>6 [2,4,1,3,6,5]=>24 [2,4,1,5,3,6]=>24 [2,4,1,5,6,3]=>120 [2,4,1,6,3,5]=>120 [2,4,1,6,5,3]=>504 [2,4,3,1,5,6]=>18 [2,4,3,1,6,5]=>86 [2,4,3,5,1,6]=>86 [2,4,3,5,6,1]=>504 [2,4,3,6,1,5]=>504 [2,4,3,6,5,1]=>2406 [2,4,5,1,3,6]=>106 [2,4,5,1,6,3]=>624 [2,4,5,3,1,6]=>352 [2,4,5,3,6,1]=>2384 [2,4,5,6,1,3]=>3468 [2,4,5,6,3,1]=>12608 [2,4,6,1,3,5]=>624 [2,4,6,1,5,3]=>3154 [2,4,6,3,1,5]=>2384 [2,4,6,3,5,1]=>15232 [2,4,6,5,1,3]=>14904 [2,4,6,5,3,1]=>62040 [2,5,1,3,4,6]=>24 [2,5,1,3,6,4]=>120 [2,5,1,4,3,6]=>86 [2,5,1,4,6,3]=>504 [2,5,1,6,3,4]=>624 [2,5,1,6,4,3]=>2384 [2,5,3,1,4,6]=>86 [2,5,3,1,6,4]=>504 [2,5,3,4,1,6]=>418 [2,5,3,4,6,1]=>2854 [2,5,3,6,1,4]=>3154 [2,5,3,6,4,1]=>15232 [2,5,4,1,3,6]=>352 [2,5,4,1,6,3]=>2384 [2,5,4,3,1,6]=>1382 [2,5,4,3,6,1]=>10632 [2,5,4,6,1,3]=>14904 [2,5,4,6,3,1]=>62040 [2,5,6,1,3,4]=>3468 [2,5,6,1,4,3]=>14904 [2,5,6,3,1,4]=>14904 [2,5,6,3,4,1]=>92128 [2,5,6,4,1,3]=>69482 [2,5,6,4,3,1]=>329304 [2,6,1,3,4,5]=>120 [2,6,1,3,5,4]=>504 [2,6,1,4,3,5]=>504 [2,6,1,4,5,3]=>2854 [2,6,1,5,3,4]=>2384 [2,6,1,5,4,3]=>10632 [2,6,3,1,4,5]=>504 [2,6,3,1,5,4]=>2406 [2,6,3,4,1,5]=>2854 [2,6,3,4,5,1]=>18610 [2,6,3,5,1,4]=>15232 [2,6,3,5,4,1]=>80582 [2,6,4,1,3,5]=>2384 [2,6,4,1,5,3]=>15232 [2,6,4,3,1,5]=>10632 [2,6,4,3,5,1]=>80582 [2,6,4,5,1,3]=>92128 [2,6,4,5,3,1]=>426328 [2,6,5,1,3,4]=>12608 [2,6,5,1,4,3]=>62040 [2,6,5,3,1,4]=>62040 [2,6,5,3,4,1]=>426328 [2,6,5,4,1,3]=>329304 [2,6,5,4,3,1]=>1718598 [3,1,2,4,5,6]=>2 [3,1,2,4,6,5]=>6 [3,1,2,5,4,6]=>6 [3,1,2,5,6,4]=>24 [3,1,2,6,4,5]=>24 [3,1,2,6,5,4]=>86 [3,1,4,2,5,6]=>6 [3,1,4,2,6,5]=>24 [3,1,4,5,2,6]=>24 [3,1,4,5,6,2]=>120 [3,1,4,6,2,5]=>120 [3,1,4,6,5,2]=>504 [3,1,5,2,4,6]=>24 [3,1,5,2,6,4]=>120 [3,1,5,4,2,6]=>86 [3,1,5,4,6,2]=>504 [3,1,5,6,2,4]=>624 [3,1,5,6,4,2]=>2384 [3,1,6,2,4,5]=>120 [3,1,6,2,5,4]=>504 [3,1,6,4,2,5]=>504 [3,1,6,4,5,2]=>2854 [3,1,6,5,2,4]=>2384 [3,1,6,5,4,2]=>10632 [3,2,1,4,5,6]=>5 [3,2,1,4,6,5]=>18 [3,2,1,5,4,6]=>18 [3,2,1,5,6,4]=>86 [3,2,1,6,4,5]=>86 [3,2,1,6,5,4]=>354 [3,2,4,1,5,6]=>18 [3,2,4,1,6,5]=>86 [3,2,4,5,1,6]=>86 [3,2,4,5,6,1]=>504 [3,2,4,6,1,5]=>504 [3,2,4,6,5,1]=>2406 [3,2,5,1,4,6]=>86 [3,2,5,1,6,4]=>504 [3,2,5,4,1,6]=>354 [3,2,5,4,6,1]=>2406 [3,2,5,6,1,4]=>2986 [3,2,5,6,4,1]=>12800 [3,2,6,1,4,5]=>504 [3,2,6,1,5,4]=>2406 [3,2,6,4,1,5]=>2406 [3,2,6,4,5,1]=>15410 [3,2,6,5,1,4]=>12800 [3,2,6,5,4,1]=>63510 [3,4,1,2,5,6]=>22 [3,4,1,2,6,5]=>106 [3,4,1,5,2,6]=>106 [3,4,1,5,6,2]=>624 [3,4,1,6,2,5]=>624 [3,4,1,6,5,2]=>2986 [3,4,2,1,5,6]=>62 [3,4,2,1,6,5]=>352 [3,4,2,5,1,6]=>352 [3,4,2,5,6,1]=>2384 [3,4,2,6,1,5]=>2384 [3,4,2,6,5,1]=>12800 [3,4,5,1,2,6]=>508 [3,4,5,1,6,2]=>3468 [3,4,5,2,1,6]=>1634 [3,4,5,2,6,1]=>12608 [3,4,5,6,1,2]=>20944 [3,4,5,6,2,1]=>74024 [3,4,6,1,2,5]=>3468 [3,4,6,1,5,2]=>20356 [3,4,6,2,1,5]=>12608 [3,4,6,2,5,1]=>90002 [3,4,6,5,1,2]=>100380 [3,4,6,5,2,1]=>400224 [3,5,1,2,4,6]=>106 [3,5,1,2,6,4]=>624 [3,5,1,4,2,6]=>462 [3,5,1,4,6,2]=>3154 [3,5,1,6,2,4]=>3774 [3,5,1,6,4,2]=>16832 [3,5,2,1,4,6]=>352 [3,5,2,1,6,4]=>2384 [3,5,2,4,1,6]=>1958 [3,5,2,4,6,1]=>15232 [3,5,2,6,1,4]=>16832 [3,5,2,6,4,1]=>90456 [3,5,4,1,2,6]=>1926 [3,5,4,1,6,2]=>14904 [3,5,4,2,1,6]=>7160 [3,5,4,2,6,1]=>62040 [3,5,4,6,1,2]=>99744 [3,5,4,6,2,1]=>398456 [3,5,6,1,2,4]=>21736 [3,5,6,1,4,2]=>110764 [3,5,6,2,1,4]=>87676 [3,5,6,2,4,1]=>598728 [3,5,6,4,1,2]=>512684 [3,5,6,4,2,1]=>2300762 [3,6,1,2,4,5]=>624 [3,6,1,2,5,4]=>2986 [3,6,1,4,2,5]=>3154 [3,6,1,4,5,2]=>20374 [3,6,1,5,2,4]=>16832 [3,6,1,5,4,2]=>85802 [3,6,2,1,4,5]=>2384 [3,6,2,1,5,4]=>12800 [3,6,2,4,1,5]=>15232 [3,6,2,4,5,1]=>110878 [3,6,2,5,1,4]=>90456 [3,6,2,5,4,1]=>527840 [3,6,4,1,2,5]=>14904 [3,6,4,1,5,2]=>108326 [3,6,4,2,1,5]=>62040 [3,6,4,2,5,1]=>521224 [3,6,4,5,1,2]=>683144 [3,6,4,5,2,1]=>3000024 [3,6,5,1,2,4]=>87676 [3,6,5,1,4,2]=>506992 [3,6,5,2,1,4]=>399248 [3,6,5,2,4,1]=>3013576 [3,6,5,4,1,2]=>2651262 [3,6,5,4,2,1]=>12996712 [4,1,2,3,5,6]=>6 [4,1,2,3,6,5]=>24 [4,1,2,5,3,6]=>24 [4,1,2,5,6,3]=>120 [4,1,2,6,3,5]=>120 [4,1,2,6,5,3]=>504 [4,1,3,2,5,6]=>18 [4,1,3,2,6,5]=>86 [4,1,3,5,2,6]=>86 [4,1,3,5,6,2]=>504 [4,1,3,6,2,5]=>504 [4,1,3,6,5,2]=>2406 [4,1,5,2,3,6]=>106 [4,1,5,2,6,3]=>624 [4,1,5,3,2,6]=>352 [4,1,5,3,6,2]=>2384 [4,1,5,6,2,3]=>3468 [4,1,5,6,3,2]=>12608 [4,1,6,2,3,5]=>624 [4,1,6,2,5,3]=>3154 [4,1,6,3,2,5]=>2384 [4,1,6,3,5,2]=>15232 [4,1,6,5,2,3]=>14904 [4,1,6,5,3,2]=>62040 [4,2,1,3,5,6]=>18 [4,2,1,3,6,5]=>86 [4,2,1,5,3,6]=>86 [4,2,1,5,6,3]=>504 [4,2,1,6,3,5]=>504 [4,2,1,6,5,3]=>2406 [4,2,3,1,5,6]=>73 [4,2,3,1,6,5]=>418 [4,2,3,5,1,6]=>418 [4,2,3,5,6,1]=>2854 [4,2,3,6,1,5]=>2854 [4,2,3,6,5,1]=>15410 [4,2,5,1,3,6]=>462 [4,2,5,1,6,3]=>3154 [4,2,5,3,1,6]=>1958 [4,2,5,3,6,1]=>15232 [4,2,5,6,1,3]=>20356 [4,2,5,6,3,1]=>90002 [4,2,6,1,3,5]=>3154 [4,2,6,1,5,3]=>17574 [4,2,6,3,1,5]=>15232 [4,2,6,3,5,1]=>109358 [4,2,6,5,1,3]=>96902 [4,2,6,5,3,1]=>490008 [4,3,1,2,5,6]=>62 [4,3,1,2,6,5]=>352 [4,3,1,5,2,6]=>352 [4,3,1,5,6,2]=>2384 [4,3,1,6,2,5]=>2384 [4,3,1,6,5,2]=>12800 [4,3,2,1,5,6]=>210 [4,3,2,1,6,5]=>1382 [4,3,2,5,1,6]=>1382 [4,3,2,5,6,1]=>10632 [4,3,2,6,1,5]=>10632 [4,3,2,6,5,1]=>63510 [4,3,5,1,2,6]=>1926 [4,3,5,1,6,2]=>14904 [4,3,5,2,1,6]=>7160 [4,3,5,2,6,1]=>62040 [4,3,5,6,1,2]=>100380 [4,3,5,6,2,1]=>400224 [4,3,6,1,2,5]=>14904 [4,3,6,1,5,2]=>96902 [4,3,6,2,1,5]=>62040 [4,3,6,2,5,1]=>490008 [4,3,6,5,1,2]=>526176 [4,3,6,5,2,1]=>2358416 [4,5,1,2,3,6]=>508 [4,5,1,2,6,3]=>3468 [4,5,1,3,2,6]=>1926 [4,5,1,3,6,2]=>14904 [4,5,1,6,2,3]=>21736 [4,5,1,6,3,2]=>87676 [4,5,2,1,3,6]=>1926 [4,5,2,1,6,3]=>14904 [4,5,2,3,1,6]=>10544 [4,5,2,3,6,1]=>92128 [4,5,2,6,1,3]=>110764 [4,5,2,6,3,1]=>598728 [4,5,3,1,2,6]=>8022 [4,5,3,1,6,2]=>69482 [4,5,3,2,1,6]=>34266 [4,5,3,2,6,1]=>329304 [4,5,3,6,1,2]=>512684 [4,5,3,6,2,1]=>2300762 [4,5,6,1,2,3]=>136706 [4,5,6,1,3,2]=>606384 [4,5,6,2,1,3]=>606384 [4,5,6,2,3,1]=>4050488 [4,5,6,3,1,2]=>2845672 [4,5,6,3,2,1]=>14292656 [4,6,1,2,3,5]=>3468 [4,6,1,2,5,3]=>20356 [4,6,1,3,2,5]=>14904 [4,6,1,3,5,2]=>108326 [4,6,1,5,2,3]=>110764 [4,6,1,5,3,2]=>506992 [4,6,2,1,3,5]=>14904 [4,6,2,1,5,3]=>96902 [4,6,2,3,1,5]=>92128 [4,6,2,3,5,1]=>753904 [4,6,2,5,1,3]=>692928 [4,6,2,5,3,1]=>4022640 [4,6,3,1,2,5]=>69482 [4,6,3,1,5,2]=>556686 [4,6,3,2,1,5]=>329304 [4,6,3,2,5,1]=>3035178 [4,6,3,5,1,2]=>3833302 [4,6,3,5,2,1]=>18816048 [4,6,5,1,2,3]=>606384 [4,6,5,1,3,2]=>3001406 [4,6,5,2,1,3]=>3019208 [4,6,5,2,3,1]=>22018344 [4,6,5,3,1,2]=>15762128 [4,6,5,3,2,1]=>86110376 [5,1,2,3,4,6]=>24 [5,1,2,3,6,4]=>120 [5,1,2,4,3,6]=>86 [5,1,2,4,6,3]=>504 [5,1,2,6,3,4]=>624 [5,1,2,6,4,3]=>2384 [5,1,3,2,4,6]=>86 [5,1,3,2,6,4]=>504 [5,1,3,4,2,6]=>418 [5,1,3,4,6,2]=>2854 [5,1,3,6,2,4]=>3154 [5,1,3,6,4,2]=>15232 [5,1,4,2,3,6]=>352 [5,1,4,2,6,3]=>2384 [5,1,4,3,2,6]=>1382 [5,1,4,3,6,2]=>10632 [5,1,4,6,2,3]=>14904 [5,1,4,6,3,2]=>62040 [5,1,6,2,3,4]=>3468 [5,1,6,2,4,3]=>14904 [5,1,6,3,2,4]=>14904 [5,1,6,3,4,2]=>92128 [5,1,6,4,2,3]=>69482 [5,1,6,4,3,2]=>329304 [5,2,1,3,4,6]=>86 [5,2,1,3,6,4]=>504 [5,2,1,4,3,6]=>354 [5,2,1,4,6,3]=>2406 [5,2,1,6,3,4]=>2986 [5,2,1,6,4,3]=>12800 [5,2,3,1,4,6]=>418 [5,2,3,1,6,4]=>2854 [5,2,3,4,1,6]=>2381 [5,2,3,4,6,1]=>18610 [5,2,3,6,1,4]=>20374 [5,2,3,6,4,1]=>110878 [5,2,4,1,3,6]=>1958 [5,2,4,1,6,3]=>15232 [5,2,4,3,1,6]=>9234 [5,2,4,3,6,1]=>80582 [5,2,4,6,1,3]=>108326 [5,2,4,6,3,1]=>521224 [5,2,6,1,3,4]=>20356 [5,2,6,1,4,3]=>96902 [5,2,6,3,1,4]=>108326 [5,2,6,3,4,1]=>753904 [5,2,6,4,1,3]=>556686 [5,2,6,4,3,1]=>3035178 [5,3,1,2,4,6]=>352 [5,3,1,2,6,4]=>2384 [5,3,1,4,2,6]=>1958 [5,3,1,4,6,2]=>15232 [5,3,1,6,2,4]=>16832 [5,3,1,6,4,2]=>90456 [5,3,2,1,4,6]=>1382 [5,3,2,1,6,4]=>10632 [5,3,2,4,1,6]=>9234 [5,3,2,4,6,1]=>80582 [5,3,2,6,1,4]=>85802 [5,3,2,6,4,1]=>527840 [5,3,4,1,2,6]=>10544 [5,3,4,1,6,2]=>92128 [5,3,4,2,1,6]=>44062 [5,3,4,2,6,1]=>426328 [5,3,4,6,1,2]=>683144 [5,3,4,6,2,1]=>3000024 [5,3,6,1,2,4]=>110764 [5,3,6,1,4,2]=>692928 [5,3,6,2,1,4]=>506992 [5,3,6,2,4,1]=>4022640 [5,3,6,4,1,2]=>3833302 [5,3,6,4,2,1]=>18816048 [5,4,1,2,3,6]=>1634 [5,4,1,2,6,3]=>12608 [5,4,1,3,2,6]=>7160 [5,4,1,3,6,2]=>62040 [5,4,1,6,2,3]=>87676 [5,4,1,6,3,2]=>399248 [5,4,2,1,3,6]=>7160 [5,4,2,1,6,3]=>62040 [5,4,2,3,1,6]=>44062 [5,4,2,3,6,1]=>426328 [5,4,2,6,1,3]=>506992 [5,4,2,6,3,1]=>3013576 [5,4,3,1,2,6]=>34266 [5,4,3,1,6,2]=>329304 [5,4,3,2,1,6]=>162482 [5,4,3,2,6,1]=>1718598 [5,4,3,6,1,2]=>2651262 [5,4,3,6,2,1]=>12996712 [5,4,6,1,2,3]=>606384 [5,4,6,1,3,2]=>3019208 [5,4,6,2,1,3]=>3001406 [5,4,6,2,3,1]=>22018344 [5,4,6,3,1,2]=>15762128 [5,4,6,3,2,1]=>86110376 [5,6,1,2,3,4]=>20944 [5,6,1,2,4,3]=>100380 [5,6,1,3,2,4]=>99744 [5,6,1,3,4,2]=>683144 [5,6,1,4,2,3]=>512684 [5,6,1,4,3,2]=>2651262 [5,6,2,1,3,4]=>100380 [5,6,2,1,4,3]=>526176 [5,6,2,3,1,4]=>683144 [5,6,2,3,4,1]=>5308656 [5,6,2,4,1,3]=>3833302 [5,6,2,4,3,1]=>23875224 [5,6,3,1,2,4]=>512684 [5,6,3,1,4,2]=>3833302 [5,6,3,2,1,4]=>2651262 [5,6,3,2,4,1]=>23875224 [5,6,3,4,1,2]=>25948726 [5,6,3,4,2,1]=>138388118 [5,6,4,1,2,3]=>2845672 [5,6,4,1,3,2]=>15762128 [5,6,4,2,1,3]=>15762128 [5,6,4,2,3,1]=>125228144 [5,6,4,3,1,2]=>91769546 [5,6,4,3,2,1]=>542024608 [6,1,2,3,4,5]=>120 [6,1,2,3,5,4]=>504 [6,1,2,4,3,5]=>504 [6,1,2,4,5,3]=>2854 [6,1,2,5,3,4]=>2384 [6,1,2,5,4,3]=>10632 [6,1,3,2,4,5]=>504 [6,1,3,2,5,4]=>2406 [6,1,3,4,2,5]=>2854 [6,1,3,4,5,2]=>18610 [6,1,3,5,2,4]=>15232 [6,1,3,5,4,2]=>80582 [6,1,4,2,3,5]=>2384 [6,1,4,2,5,3]=>15232 [6,1,4,3,2,5]=>10632 [6,1,4,3,5,2]=>80582 [6,1,4,5,2,3]=>92128 [6,1,4,5,3,2]=>426328 [6,1,5,2,3,4]=>12608 [6,1,5,2,4,3]=>62040 [6,1,5,3,2,4]=>62040 [6,1,5,3,4,2]=>426328 [6,1,5,4,2,3]=>329304 [6,1,5,4,3,2]=>1718598 [6,2,1,3,4,5]=>504 [6,2,1,3,5,4]=>2406 [6,2,1,4,3,5]=>2406 [6,2,1,4,5,3]=>15410 [6,2,1,5,3,4]=>12800 [6,2,1,5,4,3]=>63510 [6,2,3,1,4,5]=>2854 [6,2,3,1,5,4]=>15410 [6,2,3,4,1,5]=>18610 [6,2,3,4,5,1]=>136081 [6,2,3,5,1,4]=>110878 [6,2,3,5,4,1]=>651730 [6,2,4,1,3,5]=>15232 [6,2,4,1,5,3]=>109358 [6,2,4,3,1,5]=>80582 [6,2,4,3,5,1]=>681938 [6,2,4,5,1,3]=>753904 [6,2,4,5,3,1]=>3979998 [6,2,5,1,3,4]=>90002 [6,2,5,1,4,3]=>490008 [6,2,5,3,1,4]=>521224 [6,2,5,3,4,1]=>3979998 [6,2,5,4,1,3]=>3035178 [6,2,5,4,3,1]=>17788450 [6,3,1,2,4,5]=>2384 [6,3,1,2,5,4]=>12800 [6,3,1,4,2,5]=>15232 [6,3,1,4,5,2]=>110878 [6,3,1,5,2,4]=>90456 [6,3,1,5,4,2]=>527840 [6,3,2,1,4,5]=>10632 [6,3,2,1,5,4]=>63510 [6,3,2,4,1,5]=>80582 [6,3,2,4,5,1]=>651730 [6,3,2,5,1,4]=>527840 [6,3,2,5,4,1]=>3457350 [6,3,4,1,2,5]=>92128 [6,3,4,1,5,2]=>753904 [6,3,4,2,1,5]=>426328 [6,3,4,2,5,1]=>3979998 [6,3,4,5,1,2]=>5308656 [6,3,4,5,2,1]=>25359304 [6,3,5,1,2,4]=>598728 [6,3,5,1,4,2]=>4022640 [6,3,5,2,1,4]=>3013576 [6,3,5,2,4,1]=>25618632 [6,3,5,4,1,2]=>23875224 [6,3,5,4,2,1]=>126450110 [6,4,1,2,3,5]=>12608 [6,4,1,2,5,3]=>90002 [6,4,1,3,2,5]=>62040 [6,4,1,3,5,2]=>521224 [6,4,1,5,2,3]=>598728 [6,4,1,5,3,2]=>3013576 [6,4,2,1,3,5]=>62040 [6,4,2,1,5,3]=>490008 [6,4,2,3,1,5]=>426328 [6,4,2,3,5,1]=>3979998 [6,4,2,5,1,3]=>4022640 [6,4,2,5,3,1]=>25618632 [6,4,3,1,2,5]=>329304 [6,4,3,1,5,2]=>3035178 [6,4,3,2,1,5]=>1718598 [6,4,3,2,5,1]=>17788450 [6,4,3,5,1,2]=>23875224 [6,4,3,5,2,1]=>126450110 [6,4,5,1,2,3]=>4050488 [6,4,5,1,3,2]=>22018344 [6,4,5,2,1,3]=>22018344 [6,4,5,2,3,1]=>174937994 [6,4,5,3,1,2]=>125228144 [6,4,5,3,2,1]=>737498144 [6,5,1,2,3,4]=>74024 [6,5,1,2,4,3]=>400224 [6,5,1,3,2,4]=>398456 [6,5,1,3,4,2]=>3000024 [6,5,1,4,2,3]=>2300762 [6,5,1,4,3,2]=>12996712 [6,5,2,1,3,4]=>400224 [6,5,2,1,4,3]=>2358416 [6,5,2,3,1,4]=>3000024 [6,5,2,3,4,1]=>25359304 [6,5,2,4,1,3]=>18816048 [6,5,2,4,3,1]=>126450110 [6,5,3,1,2,4]=>2300762 [6,5,3,1,4,2]=>18816048 [6,5,3,2,1,4]=>12996712 [6,5,3,2,4,1]=>126450110 [6,5,3,4,1,2]=>138388118 [6,5,3,4,2,1]=>791921586 [6,5,4,1,2,3]=>14292656 [6,5,4,1,3,2]=>86110376 [6,5,4,2,1,3]=>86110376 [6,5,4,2,3,1]=>737498144 [6,5,4,3,1,2]=>542024608
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Description
The number of strong Bruhat factorizations of a permutation.
This is, the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that $t_1 \cdots t_i$ has more inversions than $t_1 \cdots t_{i-1}$ for all $1 \leq i \leq \ell$.
Code
@cached_function
def Transpositions(n):
    return Permutations(n).conjugacy_class([2]+[1]*(n-2))

@cached_function
def strong_Bruhat_factorizations(pi):
    inv = pi.number_of_inversions()
    if inv == 0:
        return 1
    else:
        facs = 0
        for t in Transpositions(len(pi)):
            tau = pi*t
            if tau.number_of_inversions() < inv:
                facs += strong_Bruhat_factorizations(tau)
        return facs

def statistic(pi):
    return strong_Bruhat_factorizations(pi)
Created
Aug 28, 2017 at 16:39 by Christian Stump
Updated
Aug 28, 2017 at 16:39 by Christian Stump