The first 6 terms in the asymptotic expansion, in t, for alpha_{2r} (the 2r-th moment divided by the r-th power of the second moment (about the mean, of course), for the r.v. NumberOfInversions on words with a*t 1's and b*t 2's Note that r, a,b, and t are symbolic! / 2 2 | (a + a b + b ) (-1 + r) r 3 2 2 2 2 (2 r)! |1 - -------------------------- + (-1 + r) (42 a b r + 63 a b r \ 5 a b (a + b) t 4 2 4 2 3 2 2 2 3 4 + 21 b r + 21 a r + 42 a b r - 75 a b r - 50 a b r - 25 a r 4 3 4 3 4 3 2 2 - 25 b r - 50 a b r - 34 a - 68 a b - 34 b - 68 a b - 312 a b ) r / 2 2 2 2 2 4 2 2 4 3 / (1050 a b (a + b) t ) - (-1 + r) (-408 a b r - 108 a b r / 2 4 4 2 5 3 5 4 5 3 3 2 + 42 a b r - 129 r a b - 54 r a b + 21 r a b - 391 a b r 3 3 3 3 3 4 2 4 2 5 3 5 - 126 a b r + 49 a b r + 762 a b r - 129 r a b - 54 r a b 4 5 4 2 4 2 2 4 2 3 4 2 4 + 21 r a b + 762 a b r - 408 a b r - 108 a b r + 42 a b r 3 3 6 5 5 3 6 2 6 + 764 a b r + 102 r b + 306 r a b + 306 r a b - 18 r b - 43 r b 4 6 3 6 2 6 4 6 6 4 2 + 7 r b - 18 r a - 43 r a + 7 r a + 102 r a + 300 a b 3 3 2 4 / 3 3 3 3 - 790 a b + 300 a b ) r / (5250 a b (a + b) t ) + (-1 + r) ( / 8 4 8 3 8 4 8 3 8 -25465 b r + 125615 b r - 25465 a r + 125615 a r + 268080 a 6 2 2 6 2 3 6 2 4 6 2 5 + 18700 b a r + 1317470 b a r - 296230 b a r - 66990 b a r 7 7 2 7 3 7 4 - 1466864 b a r - 21248 b a r + 502460 b a r - 101860 b a r 5 3 5 3 2 5 3 3 5 3 4 - 2997176 b a r - 1158512 b a r + 1944320 b a r - 435160 b a r 5 3 5 6 2 5 3 5 3 2 - 107184 b a r - 3412640 b a r - 2997176 a b r - 1158512 a b r 5 3 3 5 3 4 5 3 5 4 4 + 1944320 a b r - 435160 a b r - 107184 a b r - 4882304 a b r 4 4 2 4 4 3 4 4 4 4 4 5 - 340328 a b r + 2382485 a b r - 553135 a b r - 127281 a b r 7 7 2 7 3 7 4 - 1466864 a b r - 21248 a b r + 502460 a b r - 101860 a b r 6 2 6 2 2 7 5 6 2 3 - 3412640 a b r + 18700 a b r - 26796 b a r + 1317470 a b r 6 2 4 6 2 5 7 5 8 2 - 296230 a b r - 66990 a b r - 26796 a b r - 5312 b r 8 8 5 8 2 8 8 5 - 366716 b r - 6699 b r - 5312 a r - 366716 a r - 6699 a r 7 7 4 4 6 2 + 1072320 b a + 1072320 a b - 1828080 a b + 2059200 a b 5 3 5 3 6 2 6 2 6 + 3810480 a b + 3810480 b a + 2059200 b a + 16170 b a r 5 3 6 4 4 6 5 3 6 7 6 + 25872 b a r + 30723 a b r + 25872 a b r + 6468 a b r 6 2 6 7 6 8 8 6 8 6 + 16170 a b r + 6468 b a r + 268080 b + 1617 b r + 1617 a r ) r / 4 4 4 4 10 / (24255000 a b (a + b) t ) - (-1 + r) (-89579520 a / 3 7 8 2 8 2 5 - 1297029600 b a - 1032724800 a b + 77526540 b a r 8 2 6 9 9 2 10 - 10549110 b a r + 139171920 b a r + 418033420 b a r - 89579520 b 9 3 9 4 9 5 9 6 - 262499980 b a r + 2028145 b a r + 26129480 b a r - 3416270 b a r 7 3 5 7 3 6 8 2 + 144620580 b a r - 19596720 b a r + 298304160 b a r 8 2 2 8 2 3 8 2 4 + 1085319060 b a r - 691690440 b a r - 1234065 b a r 6 4 6 4 2 6 4 3 + 195514880 b a r + 2026004480 b a r - 1310647820 b a r 6 4 4 6 4 5 6 4 6 - 101999195 b a r + 210852720 b a r - 28944630 b a r 7 3 7 3 2 7 3 3 - 35826680 b a r + 1849511620 b a r - 1008533380 b a r 7 3 4 8 2 3 8 2 4 - 63723130 b a r - 691690440 a b r - 1234065 a b r 8 2 5 8 2 6 3 7 + 77526540 a b r - 10549110 a b r - 35826680 b a r 3 7 2 3 7 3 3 7 4 + 1849511620 b a r - 1008533380 b a r - 63723130 b a r 3 7 5 3 7 6 4 6 + 144620580 b a r - 19596720 b a r + 195514880 b a r 4 6 2 4 6 3 4 6 4 + 2026004480 b a r - 1310647820 b a r - 101999195 b a r 4 6 5 4 6 6 5 5 + 210852720 b a r - 28944630 b a r - 1337616816 b a r 5 5 2 5 5 3 5 5 4 + 2782275084 b a r - 1189648296 b a r - 178838421 b a r 5 5 5 5 5 6 9 + 231775596 b a r - 31542654 b a r + 139171920 b a r 9 2 9 3 9 4 + 418033420 b a r - 262499980 b a r + 2028145 b a r 9 5 9 6 8 2 + 26129480 b a r - 3416270 b a r + 298304160 a b r 8 2 2 8 2 9 + 1085319060 a b r - 1032724800 b a - 447897600 b a 10 2 10 10 4 10 3 + 83606684 b r + 27834384 b r + 405629 b r - 52499996 b r 10 6 10 5 6 4 7 3 - 683254 b r + 5225896 b r - 799916400 b a - 1297029600 b a 5 5 4 6 10 10 3 - 2514098520 b a - 799916400 b a + 27834384 a r - 52499996 a r 10 2 10 5 10 4 10 6 + 83606684 a r + 5225896 a r + 405629 a r - 683254 a r 9 7 8 2 7 7 3 7 6 4 - 447897600 b a - 1861860 r b a - 3723720 r b a - 5585580 r b a 7 7 3 7 5 5 7 6 4 7 9 - 3723720 r a b - 6330324 r a b - 5585580 r a b - 620620 r b a 7 8 2 7 9 8 8 2 8 9 - 1861860 r a b - 620620 r a b + 315315 r b a + 105105 r b a 8 7 3 8 6 4 8 8 2 8 3 7 + 630630 r b a + 945945 r b a + 315315 r a b + 630630 r b a 8 5 5 8 4 6 8 9 7 10 + 1072071 r b a + 945945 r b a + 105105 r b a - 124124 r b 7 10 8 10 8 10 / 5 5 - 124124 r a + 21021 r b + 21021 r a ) r / (7882875000 a b / 5 5 12 12 (a + b) t ) + r (-1 + r) (-3536467200 b - 3536467200 a 11 2 11 4 10 2 4 - 655795687344 b a r + 82842361140 b a r + 261621407040 b a r 10 2 5 10 2 6 10 2 7 - 115018447845 b a r + 759129693 b a r + 5546330790 b a r 10 2 8 10 2 9 11 - 531621090 b a r - 72837765 b a r + 597063303360 b a r 11 9 11 8 11 7 - 20810790 b a r - 154594440 b a r + 1633384740 b a r 11 5 11 3 11 6 - 37144872870 b a r + 139994659560 b a r + 498775998 b a r 9 3 4 9 3 5 9 3 6 + 545385801100 b a r - 218678995850 b a r - 3117903950 b a r 9 3 7 9 3 8 9 3 9 + 12367969900 b a r - 1174773600 b a r - 173423250 b a r 10 2 10 2 2 + 1678492690560 b a r - 1847416985904 b a r 10 2 3 8 4 8 8 4 9 + 374090200260 b a r - 1997295300 b a r - 312161850 b a r 11 9 3 9 3 2 - 21218803200 b a + 2931958173600 b a r - 3068595563600 b a r 2 10 9 3 3 7 5 5 - 60010675200 b a + 482151619600 b a r - 356083162470 b a r 7 5 6 7 5 7 7 5 8 - 24549508242 b a r + 28203788340 b a r - 2641078440 b a r 7 5 9 8 4 8 4 2 - 437026590 b a r + 3545963553600 b a r - 3551606283960 b a r 8 4 3 8 4 4 8 4 5 + 367747487100 b a r + 839897358300 b a r - 319005744750 b a r 8 4 6 8 4 7 6 6 7 - 11192624730 b a r + 21111093900 b a r + 31214196390 b a r 6 6 8 6 6 9 7 5 - 2923510590 b a r - 489053565 b a r + 4335834731760 b a r 7 5 2 7 5 3 7 5 4 - 3303586821024 b a r - 279361409040 b a r + 1106513464140 b a r 6 6 6 6 2 6 6 3 + 2786680533960 b a r - 2519470222584 b a r - 228596277840 b a r 6 6 4 6 6 5 6 6 6 + 1133005193040 b a r - 379752639945 b a r - 28090354047 b a r 4 8 2 4 8 3 4 8 4 - 3551606283960 b a r + 367747487100 b a r + 839897358300 b a r 4 8 5 4 8 6 4 8 7 - 319005744750 b a r - 11192624730 b a r + 21111093900 b a r 4 8 8 4 8 9 5 7 - 1997295300 b a r - 312161850 b a r + 4335834731760 b a r 5 7 2 5 7 3 5 7 4 - 3303586821024 b a r - 279361409040 b a r + 1106513464140 b a r 5 7 5 5 7 6 5 7 7 - 356083162470 b a r - 24549508242 b a r + 28203788340 b a r 5 7 8 5 7 9 4 8 - 2641078440 b a r - 437026590 b a r + 3545963553600 b a r 2 10 3 2 10 4 2 10 5 + 374090200260 b a r + 261621407040 b a r - 115018447845 b a r 2 10 6 2 10 7 2 10 8 + 759129693 b a r + 5546330790 b a r - 531621090 b a r 2 10 9 3 9 3 9 2 - 72837765 b a r + 2931958173600 b a r - 3068595563600 b a r 3 9 3 3 9 4 3 9 5 + 482151619600 b a r + 545385801100 b a r - 218678995850 b a r 3 9 6 3 9 7 3 9 8 - 3117903950 b a r + 12367969900 b a r - 1174773600 b a r 3 9 9 2 10 2 10 2 - 173423250 b a r + 1678492690560 b a r - 1847416985904 b a r 11 11 9 11 8 + 597063303360 b a r - 20810790 b a r - 154594440 b a r 11 7 11 5 11 3 + 1633384740 b a r - 37144872870 b a r + 139994659560 b a r 11 6 11 2 11 4 + 498775998 b a r - 655795687344 b a r + 82842361140 b a r 3 9 11 10 2 - 191813328000 b a - 21218803200 b a - 60010675200 b a 9 3 8 4 12 3 - 191813328000 b a - 413096544000 b a + 23332443260 b r 12 5 12 2 12 6 - 6190812145 b r - 109299281224 b r + 83129333 b r 12 12 4 12 7 + 99510550560 b r + 13807060190 b r + 272230790 b r 12 8 12 9 7 5 - 25765740 b r - 3468465 b r - 727984303200 b a 6 6 5 7 4 8 - 1898508175200 b a - 727984303200 b a - 413096544000 b a 12 8 12 9 12 5 12 7 - 25765740 a r - 3468465 a r - 6190812145 a r + 272230790 a r 12 6 12 3 12 4 + 83129333 a r + 23332443260 a r + 13807060190 a r 12 2 12 10 2 10 - 109299281224 a r + 99510550560 a r + 9270261 b a r 11 10 9 3 10 8 4 10 + 2648646 b a r + 22072050 b a r + 39729690 b a r 12 10 7 5 10 12 10 + 441441 b r + 55621566 b a r + 441441 a r 6 6 10 4 8 10 5 7 10 + 62243181 b a r + 39729690 b a r + 55621566 b a r 2 10 10 3 9 10 11 10 / + 9270261 b a r + 22072050 b a r + 2648646 b a r ) / ( / \ 6 6 6 6 | / r 4966211250000 a b (a + b) t )| / (r! 2 ) / / This all things too, 98.327, seconds of CPU time