The, 2, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is a b (b + a + 1) --------------- 12 The, 4, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 2 2 2 2 a b (b + a + 1) (-2 a + 5 a b - 2 a + 3 a b + 5 a b - 2 b - 2 b ) -------------------------------------------------------------------- 240 The, 6, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 4 4 4 2 3 3 3 3 a b (b + a + 1) (-42 a b + 16 a + 35 a b + 70 a b - 52 a b + 32 a 3 2 2 2 4 2 2 2 2 3 2 - 14 a b + 8 a + 35 a b - 43 a b + 14 a b - 14 a b + 14 a b 4 3 4 2 3 + 16 a b - 42 b a - 52 a b - 8 a + 16 b + 8 b + 32 b - 8 b)/4032 The, 8, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 5 5 5 4 3 a b (b + a + 1) (-96 a - 96 b + 780 a b + 780 b a - 432 a - 395 a b 2 2 5 3 3 3 4 + 96 a + 192 a b + 96 b - 432 b + 240 a + 240 b - 395 a b 2 2 3 2 2 4 3 + 224 a b + 224 a b - 540 a b - 600 a b - 240 b - 540 a b 6 5 2 5 2 4 6 2 5 3 + 404 a b - 48 a b - 48 b a - 240 a - 420 a b - 735 a b 6 2 6 6 4 3 2 2 3 - 420 b a - 144 a - 144 b - 100 a b - 200 a b - 200 a b 4 4 2 3 3 2 4 6 3 5 4 - 100 b a + 900 a b + 1095 a b + 900 a b + 175 b a + 525 b a 4 4 3 5 4 5 3 6 6 - 630 a b - 735 a b + 525 b a + 175 b a + 404 b a)/34560 The, 10, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 7 5 6 4 5 a b (b + a + 1) (-1152 a - 1152 b + 1540 b a + 3762 b a + 7456 a b 5 5 7 7 4 3 2 + 7456 b a - 2688 a + 3072 a + 3072 b - 2028 a b + 1152 a 2 5 3 3 3 4 + 2304 a b + 1152 b - 2688 b + 2688 a + 2688 b - 2028 a b 2 2 3 2 2 4 3 + 3168 a b + 3168 a b - 6560 a b - 7744 a b - 2688 b - 6560 a b 6 5 2 5 2 4 6 2 7 - 128 a b - 256 a b - 256 b a - 2688 a - 8712 a b - 6080 a b 7 5 3 6 2 6 6 8 - 6080 b a - 12952 a b - 8712 b a + 2688 a + 2688 b + 768 b 4 3 2 2 3 4 4 2 - 2480 a b - 4304 a b - 4304 a b - 2480 b a + 12812 a b 3 3 2 4 6 3 5 4 8 + 16088 a b + 12812 a b + 6336 b a + 8888 b a + 768 a 4 4 3 5 4 5 3 6 3 8 - 14511 a b - 12952 a b + 8888 b a + 6336 b a - 1540 b a 6 7 2 8 7 2 8 7 3 - 128 b a + 1584 a b - 2288 a b + 1584 b a - 2288 b a + 4576 b a 4 6 3 7 8 2 8 2 5 5 + 3762 b a + 4576 b a + 2684 a b + 2684 b a + 3740 b a 8 3 7 4 6 5 5 6 4 7 - 1540 b a - 4620 b a - 6160 b a - 6160 b a - 4620 b a 8 4 6 6 5 7 4 8 + 385 b a + 2310 b a + 1540 b a + 385 b a )/101376 The, 12, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 8 6 7 5 a b (b + a + 1) (-5306880 a - 5306880 b - 8058050 b a - 1927926 b a 9 6 6 4 7 7 9 + 875875 b a + 45544086 b a - 9459450 b a + 11331040 b a 5 5 5 6 7 + 34837152 a b + 34837152 b a - 11144448 a + 16846830 b a 9 5 8 7 10 5 7 7 - 4379375 b a + 1751750 b a + 175175 b a + 6368256 a + 6368256 b 5 9 3 9 4 9 4 3 - 4379375 b a - 6946940 b a + 9059050 b a + 6151656 a b 2 2 5 3 + 5306880 a + 10613760 a b + 5306880 b - 11144448 b + 12205824 a 3 3 4 2 2 + 12205824 b + 6151656 a b + 15959808 a b + 15959808 a b 3 2 2 6 9 4 - 30642496 a b - 36873344 a b + 875875 b a - 12205824 b 3 6 5 2 5 2 - 30642496 a b + 4258464 a b + 8082088 a b + 8082088 b a 2 10 5 8 10 4 4 - 4252248 b a + 14034020 b a - 1051050 b a - 12205824 a 6 2 7 7 5 3 - 48127576 a b - 21363264 a b - 21363264 b a - 79333324 a b 5 10 3 10 7 8 10 + 175175 b a + 2862860 b a + 1751750 b a - 1061376 b 9 4 10 6 2 6 + 11331040 b a - 1051050 b a - 48127576 b a + 11144448 a 6 8 4 3 2 + 11144448 b - 6368256 b - 15566144 a b - 26827680 a b 2 3 4 9 6 8 - 26827680 a b - 15566144 b a - 5306880 b - 8058050 b a 10 9 4 2 3 3 - 1061376 a - 5306880 a + 64851584 a b + 82317760 a b 2 4 6 3 5 4 10 3 + 64851584 a b + 21594860 b a + 32847514 b a + 2862860 b a 9 4 8 5 8 4 4 + 9059050 b a + 14034020 b a - 6368256 a - 92401512 a b 3 5 4 5 3 6 3 8 - 79333324 a b + 32847514 b a + 21594860 b a - 17879680 b a 6 7 2 8 7 2 + 4258464 b a + 7078480 a b + 2103360 a b + 7078480 b a 8 10 9 2 10 + 2103360 b a + 3327584 b a - 4623320 b a + 3327584 a b 9 2 7 3 4 6 3 7 - 4623320 a b + 32712840 b a + 45544086 b a + 32712840 b a 8 2 8 2 5 5 8 3 + 18764080 a b + 18764080 b a + 51136113 b a - 17879680 b a 7 4 6 5 5 6 4 7 - 32083116 b a - 43661709 b a - 43661709 b a - 32083116 b a 8 4 6 6 5 7 4 8 - 4328324 b a - 588588 b a - 1927926 b a - 4328324 b a 10 2 9 3 7 6 - 4252248 b a - 6946940 b a + 16846830 b a )/50319360 The, 14, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 3 12 4 11 2 11 a b (b + a + 1) (-1849848 b a - 6174168 b a - 12736512 a + 3967680 b a 11 2 12 12 - 12736512 b + 3967680 b a - 1810176 a b - 1810176 b a 8 6 11 4 7 5 9 6 - 11308089 b a - 6174168 b a - 55984260 b a - 14444430 b a 6 4 7 7 9 5 + 166957380 b a - 14229306 b a + 20012160 b a + 85239936 a b 5 5 6 7 9 5 + 85239936 b a - 26155008 a + 74762550 b a - 4889040 b a 8 7 10 5 7 7 - 16780764 b a - 10624614 b a + 13381632 a + 13381632 b 4 12 5 11 5 9 3 9 + 820820 b a + 3663660 b a - 4889040 b a - 27036640 b a 4 9 12 2 2 12 11 3 + 34503600 b a + 2471456 b a + 2471456 b a + 3729648 b a 3 11 4 3 2 + 3729648 b a + 46686912 a b + 12736512 a + 25473024 a b 2 5 3 3 3 4 + 12736512 b - 26155008 b + 29196288 a + 29196288 b + 46686912 a b 2 2 3 2 2 + 40349184 a b + 40349184 a b - 74113536 a b - 89834496 a b 6 9 4 3 6 - 14444430 b a - 29196288 b - 74113536 a b + 18549120 a b 5 2 5 2 2 10 5 8 + 39087648 a b + 39087648 b a - 13819136 b a + 58112178 b a 10 4 4 6 2 7 + 562588 b a - 29196288 a - 128903744 a b - 51971328 a b 7 5 3 5 10 3 10 - 51971328 b a - 219989760 a b - 10624614 b a + 15968280 b a 7 8 10 9 4 10 - 16780764 b a + 5068800 b + 20012160 b a + 562588 b a 6 2 6 6 8 - 128903744 b a + 26155008 a + 26155008 b - 13381632 b 4 3 2 2 3 4 - 43973376 a b - 76210560 a b - 76210560 a b - 43973376 b a 11 5 12 4 9 7 10 6 + 3663660 b a + 820820 b a + 11411400 b a + 7812805 b a 9 6 8 12 10 11 - 5068800 b - 11308089 b a + 552960 b + 5068800 a - 7543296 a b 7 10 11 7 7 11 11 6 - 2612610 a b + 150150 a b + 150150 a b - 1111110 a b 9 4 2 3 3 2 4 - 5068800 a + 163041728 a b + 207913600 a b + 163041728 a b 11 6 3 5 4 10 3 - 7543296 b a + 24389160 b a + 41684160 b a + 15968280 b a 9 4 8 5 11 8 + 34503600 b a + 58112178 b a + 3317760 a - 13381632 a 4 4 3 5 6 11 4 5 - 259351424 a b - 219989760 a b - 1111110 a b + 41684160 b a 3 6 3 8 6 11 + 24389160 b a - 37032840 b a + 18549120 b a + 3317760 b 8 8 7 2 8 7 2 + 12882870 b a + 4247040 a b - 550656 a b + 4247040 b a 8 6 12 5 12 10 - 550656 b a + 25025 a b - 210210 a b - 2730624 b a 9 2 10 9 2 12 - 11238240 b a - 2730624 a b - 11238240 a b + 552960 a 7 3 4 6 3 7 8 2 + 110644080 b a + 166957380 b a + 110644080 b a + 54510432 a b 8 2 5 5 8 3 7 4 + 54510432 b a + 191760744 b a - 37032840 b a - 72977112 b a 6 5 5 6 4 7 9 8 - 102092874 b a - 102092874 b a - 72977112 b a - 3753750 a b 9 9 10 7 8 9 10 8 + 500500 a b - 2612610 a b - 3753750 a b + 375375 a b 8 10 12 5 12 6 8 4 + 375375 a b - 210210 a b + 25025 a b - 43084788 b a 6 6 5 7 4 8 10 2 - 60685969 b a - 55984260 b a - 43084788 b a - 13819136 b a 9 3 7 6 12 3 6 10 - 27036640 b a + 74762550 b a - 1849848 b a + 7812805 b a 7 9 + 11411400 b a )/6635520 The, 16, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 3 12 4 11 a b (b + a + 1) (-8906861680 b a - 22314701296 b a - 28000788480 a 2 11 13 11 2 + 8845100672 b a - 1400039424 a - 28000788480 b + 8845100672 b a 12 12 8 6 + 1835938560 a b + 1835938560 b a + 36737160280 b a 11 4 7 5 9 6 - 22314701296 b a - 227575553928 b a - 68738553168 b a 6 4 13 7 7 + 450025893376 b a - 1400039424 b + 37942239045 b a 9 5 5 14 + 44512472832 b a + 189558497280 a b + 189558497280 b a - 200005632 a 5 6 7 9 5 - 57201610752 a + 181042942960 b a + 32229941128 b a 8 7 10 5 7 7 - 85283107885 b a - 44291312168 b a + 28600805376 a + 28600805376 b 4 12 5 11 5 9 + 995296416 b a + 6384424392 b a + 32229941128 b a 3 9 4 9 12 2 - 88567911408 b a + 78280540240 b a + 6607614080 b a 2 12 11 3 3 11 + 6607614080 b a + 13979570800 b a + 13979570800 b a 4 3 2 2 + 155935717632 a b + 28000788480 a + 56001576960 a b + 28000788480 b 5 3 3 3 4 - 57201610752 b + 64135139328 a + 64135139328 b + 155935717632 a b 2 2 3 + 91608662016 a b + 91608662016 a b - 163579020288 a b 2 2 6 9 4 - 198887761920 a b - 68738553168 b a - 64135139328 b 3 6 5 2 - 163579020288 a b + 53558403072 a b + 118748840704 a b 5 2 2 10 5 8 + 118748840704 b a - 37118185344 b a + 137670949896 b a 14 10 4 4 6 2 - 200005632 b + 23791592576 b a - 64135139328 a - 300084836096 a b 7 7 5 3 - 116840997120 a b - 116840997120 b a - 519795460864 a b 5 10 3 10 7 8 - 44291312168 b a + 32687822832 b a - 85283107885 b a 13 5 14 4 10 9 - 1901139240 b a - 397517120 b a + 9533601792 b + 44512472832 b a 4 10 6 2 6 + 23791592576 b a - 300084836096 b a + 57201610752 a 6 8 4 3 2 + 57201610752 b - 28600805376 b - 106338315264 a b - 185384020992 a b 2 3 4 4 13 - 185384020992 a b - 106338315264 b a + 2765219184 b a 11 5 12 4 9 7 + 6384424392 b a + 995296416 b a + 24258452073 b a 10 6 9 6 8 12 + 15263636456 b a - 9533601792 b + 36737160280 b a - 2600073216 b 10 11 7 10 + 9533601792 a - 11808221952 a b + 9225925007 a b 11 7 7 11 11 6 - 7802379585 a b - 7802379585 a b + 7440567888 a b 9 4 2 3 3 - 9533601792 a + 366671766528 a b + 468629760000 a b 2 4 11 6 3 + 366671766528 a b - 11808221952 b a - 4001307392 b a 5 4 10 3 9 4 + 9957740480 b a + 32687822832 b a + 78280540240 b a 8 5 11 8 + 137670949896 b a + 2600073216 a - 28600805376 a 4 4 3 5 6 11 - 615901633024 a b - 519795460864 a b + 7440567888 a b 4 5 3 6 3 8 + 9957740480 b a - 4001307392 b a - 55733063120 b a 3 13 13 4 3 14 - 1302707280 b a + 2765219184 b a + 792548432 b a 13 13 6 11 + 3355726080 a b + 3355726080 b a + 53558403072 b a + 2600073216 b 8 8 14 3 7 2 + 28119771570 b a + 792548432 b a - 17330876032 a b 8 7 2 8 - 10095806208 a b - 17330876032 b a - 10095806208 b a 14 13 2 2 13 + 679395072 a b - 2094780800 b a - 2094780800 b a 14 2 13 3 14 2 - 978649472 a b - 1302707280 b a - 978649472 b a 6 12 5 12 10 - 4568043480 a b + 5109723320 a b - 2047454976 b a 9 2 10 9 2 - 14476920192 b a - 2047454976 a b - 14476920192 a b 12 7 3 4 6 - 2600073216 a + 290161744080 b a + 450025893376 b a 3 7 8 2 8 2 + 290161744080 b a + 135705269120 a b + 135705269120 b a 5 5 14 8 3 + 521181094752 b a + 679395072 b a - 55733063120 b a 7 4 6 5 5 6 - 118182728336 b a - 170038453760 b a - 170038453760 b a 4 7 9 8 9 9 - 118182728336 b a + 10074058917 a b - 11678937270 a b 10 7 8 9 10 8 + 9225925007 a b + 10074058917 a b - 10577426860 a b 8 10 12 5 12 6 - 10577426860 a b + 5109723320 a b - 4568043480 a b 8 4 6 6 5 7 - 160946786784 b a - 254573283888 b a - 227575553928 b a 4 8 10 2 9 3 - 160946786784 b a - 37118185344 b a - 88567911408 b a 7 6 12 3 6 10 + 181042942960 b a - 8906861680 b a + 15263636456 b a 7 9 4 14 5 13 + 24258452073 b a - 397517120 b a - 1901139240 b a 12 8 12 9 13 7 11 9 - 434784350 a b + 44669625 a b - 151876725 a b - 753427675 a b 13 8 11 10 12 7 + 14889875 a b + 74449375 a b + 1916539625 a b 6 13 8 13 7 13 + 714714000 a b + 14889875 a b - 151876725 a b 10 10 14 7 10 11 14 6 - 893392500 a b + 2127125 a b + 74449375 a b - 23823800 a b 9 11 11 8 9 12 - 753427675 a b + 3354476125 a b + 44669625 a b 8 12 8 11 13 6 - 434784350 a b + 3354476125 a b + 714714000 a b 7 12 10 9 9 10 + 1916539625 a b + 4350396050 a b + 4350396050 a b 14 5 5 14 7 14 6 14 + 125925800 a b + 125925800 a b + 2127125 a b - 23823800 a b )/ 451215360 The, 18, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 3 12 4 11 a b (b + a + 1) (-43522514077952 b a - 122001123102208 b a 2 11 13 - 140378352943104 a + 31280567779328 b a - 2263958323200 a 11 2 12 - 140378352943104 b + 31280567779328 b a + 4617351954432 a b 12 8 6 11 4 + 4617351954432 b a + 503417759957512 b a - 122001123102208 b a 7 5 9 6 6 4 - 1507974945497600 b a - 413081000881088 b a + 2500685906076672 b a 13 7 7 9 - 2263958323200 b + 552350511837840 b a + 228639366365184 b a 5 5 14 + 956180843458560 a b + 956180843458560 b a + 2263958323200 a 5 6 7 9 5 - 286410133241856 a + 790386929034240 b a + 379007836807936 b a 8 7 10 5 7 - 522028084637560 b a - 256303971265216 b a + 142448257695744 a 7 4 12 5 11 + 142448257695744 b - 20087365469472 b a - 23489615121280 b a 5 9 3 9 4 9 + 379007836807936 b a - 545848657038080 b a + 326426014602240 b a 12 2 2 12 11 3 + 40879720408576 b a + 40879720408576 b a + 110846807534848 b a 3 11 4 3 2 + 110846807534848 b a + 976694061794304 a b + 140378352943104 a 2 5 + 280756705886208 a b + 140378352943104 b - 286410133241856 b 3 3 3 4 + 321469144989696 a + 321469144989696 b + 976694061794304 a b 2 2 3 + 469042993225728 a b + 469042993225728 a b - 821732612677632 a b 2 2 6 9 4 - 1000526935375872 a b - 413081000881088 b a - 321469144989696 b 3 6 5 2 - 821732612677632 a b + 312962514038784 a b + 714757468214272 a b 5 2 2 10 5 8 + 714757468214272 b a - 213702188035072 b a + 592383597235136 b a 14 10 4 4 + 2263958323200 b + 231084469326144 b a - 321469144989696 a 6 2 7 7 - 1549409655009280 a b - 593991069511680 a b - 593991069511680 b a 5 3 5 10 3 10 - 2703314823580672 a b - 256303971265216 b a + 128031761088000 b a 15 2 7 8 16 + 1838152177664 a b - 522028084637560 b a - 453757851648 a b 15 2 16 13 5 + 1838152177664 b a - 453757851648 b a - 9411778528768 b a 14 4 15 10 - 1874671017920 b a - 2595110436864 a b + 46249434316800 b 9 4 10 6 2 + 228639366365184 b a + 231084469326144 b a - 1549409655009280 b a 6 6 8 + 286410133241856 a + 286410133241856 b - 142448257695744 b 15 4 3 2 - 2595110436864 b a - 566002566918144 a b - 991379926278144 a b 2 3 4 4 13 - 991379926278144 a b - 566002566918144 b a + 23409100431360 b a 11 5 12 4 9 7 - 23489615121280 b a - 20087365469472 b a - 4821471510768 b a 10 6 9 6 8 - 17048530539648 b a - 46249434316800 b + 503417759957512 b a 12 10 11 - 10931684474880 b + 46249434316800 a - 59747371905024 a b 7 10 11 7 7 11 + 138984312618000 a b - 46657300754128 a b - 46657300754128 a b 11 6 9 15 3 + 95250071646400 a b - 46249434316800 a + 754988076032 a b 15 3 16 2 4 2 + 754988076032 b a + 683526873856 b a + 1859308166016000 a b 3 3 2 4 11 + 2379074911598592 a b + 1859308166016000 a b - 59747371905024 b a 6 3 5 4 10 3 - 254998492637440 b a - 289551395141120 b a + 128031761088000 b a 9 4 8 5 11 + 326426014602240 b a + 592383597235136 b a + 10931684474880 a 8 4 4 3 5 - 142448257695744 a - 3210895960774656 a b - 2703314823580672 a b 6 11 4 5 3 6 + 95250071646400 a b - 289551395141120 b a - 254998492637440 b a 3 8 3 13 13 4 - 143629270360448 b a - 11914594304256 b a + 23409100431360 b a 3 14 13 13 + 8269277960960 b a + 11616007753728 a b + 11616007753728 b a 16 6 11 + 129369047040 b + 312962514038784 b a + 10931684474880 b 8 8 14 3 15 + 1461912009065 b a + 8269277960960 b a + 1034952376320 a 7 2 8 7 2 - 193627901186048 a b - 82712872728576 a b - 193627901186048 b a 8 14 13 2 - 82712872728576 b a - 1917542854656 a b - 10539619514368 b a 2 13 14 2 13 3 - 10539619514368 b a - 5380907875328 a b - 11914594304256 b a 14 2 6 12 5 12 - 5380907875328 b a - 24862128140912 a b + 53035676989120 a b 10 9 2 10 + 3968993009664 b a - 21310580139008 b a + 3968993009664 a b 9 2 12 7 3 - 21310580139008 a b - 10931684474880 a + 1592478869505792 b a 4 6 3 7 8 2 + 2500685906076672 b a + 1592478869505792 b a + 727831074333440 a b 8 2 5 5 14 + 727831074333440 b a + 2907296357711872 b a - 1917542854656 b a 15 8 3 7 4 + 1034952376320 b - 143629270360448 b a - 346921462356096 b a 6 5 5 6 4 7 - 523747887654592 b a - 523747887654592 b a - 346921462356096 b a 9 8 9 9 10 7 + 167196298119040 a b - 75135712644200 a b + 138984312618000 a b 8 9 10 8 8 10 + 167196298119040 a b - 66828675243988 a b - 66828675243988 a b 12 5 12 6 8 4 + 53035676989120 a b - 24862128140912 a b - 1037149878101104 b a 6 6 5 7 4 8 - 1703574447659296 b a - 1507974945497600 b a - 1037149878101104 b a 10 2 9 3 7 6 - 213702188035072 b a - 545848657038080 b a + 790386929034240 b a 12 3 6 10 7 9 - 43522514077952 b a - 17048530539648 b a - 4821471510768 b a 15 5 16 4 15 5 + 1646658878592 b a + 323159810064 b a + 1646658878592 a b 4 16 4 14 5 13 + 323159810064 b a - 1874671017920 b a - 9411778528768 b a 15 4 16 3 16 3 - 2127948434304 b a - 589153364352 a b - 589153364352 b a 12 8 12 9 13 7 + 13099112430950 a b - 6751035971680 a b + 8430263705136 a b 11 9 13 8 11 10 + 16891700215208 a b - 4354876285760 a b - 8343466411280 a b 12 7 6 13 8 13 - 7087977354528 a b - 6041521944192 a b - 4354876285760 a b 7 13 10 10 14 7 + 8430263705136 a b + 18355158285828 a b - 2143941880080 a b 10 11 14 6 9 11 - 8343466411280 a b + 4368771415008 a b + 16891700215208 a b 11 8 9 12 8 12 - 6826205032976 a b - 6751035971680 a b + 13099112430950 a b 8 11 13 6 7 12 - 6826205032976 a b - 6041521944192 a b - 7087977354528 a b 10 9 9 10 16 - 6128378964624 a b - 6128378964624 a b + 129369047040 a 14 5 4 15 16 2 - 4105493556864 a b - 2127948434304 b a + 683526873856 a b 5 14 7 14 6 14 - 4105493556864 a b - 2143941880080 a b + 4368771415008 a b 6 15 15 6 16 5 - 714828354240 a b - 714828354240 a b - 117327450240 a b 15 8 15 9 16 7 - 30327697400 a b + 2263261000 a b - 4073869800 a b 14 9 16 8 14 10 - 102299397200 a b + 282907625 a b + 7921413500 a b 15 7 9 13 11 13 + 191200289280 a b + 1264257594600 a b + 15842827000 a b 10 13 13 10 13 11 - 209125316400 a b - 209125316400 a b + 15842827000 a b 12 11 14 8 12 12 - 291508016800 a b + 614928013700 a b + 19803533750 a b 11 12 11 11 16 6 - 291508016800 a b + 2139958540720 a b + 27973905960 a b 10 12 13 9 12 10 + 1882535234580 a b + 1264257594600 a b + 1882535234580 a b 8 14 10 14 9 14 + 614928013700 a b + 7921413500 a b - 102299397200 a b 7 15 9 15 8 15 + 191200289280 a b + 2263261000 a b - 30327697400 a b 5 16 6 16 8 16 - 117327450240 a b + 27973905960 a b + 282907625 a b 7 16 - 4073869800 a b )/42361159680 The, 20, -th moment (about the mean) of the r.v. number of inversions for words with 1 a's and 2 b's is 3 12 4 11 a b (b + a + 1) (-105706001553666560 b a - 314562414376470144 b a 2 11 13 - 406606656108625920 a + 51536742008887296 b a - 5404799534432256 a 11 2 12 - 406606656108625920 b + 51536742008887296 b a + 3996571605442560 a b 12 8 6 + 3996571605442560 b a + 2153207002197407904 b a 11 4 7 5 - 314562414376470144 b a - 5031279230042482560 b a 9 6 6 4 - 1126493721384482176 b a + 7646212504250394624 b a 13 7 7 - 5404799534432256 b + 2392019512883247456 b a 9 5 + 672778363065126912 b a + 2778703569662238720 a b 5 14 5 + 2778703569662238720 b a + 5404799534432256 a - 829329458590973952 a 6 7 9 5 + 1742011891083078912 b a + 1564835702183061264 b a 8 7 10 5 - 1438435062044790480 b a - 684076083236443920 b a 7 7 + 411946572095225856 a + 411946572095225856 b 4 12 5 11 - 134633073297453696 b a - 243908691754222656 b a 5 9 3 9 + 1564835702183061264 b a - 1761452659197462528 b a 4 9 12 2 + 684552534389324832 b a + 137853613528971264 b a 2 12 11 3 + 137853613528971264 b a + 406735740396228096 b a 3 11 4 3 + 406735740396228096 b a + 3239920754937649152 a b 2 2 + 406606656108625920 a + 813213312217251840 a b + 406606656108625920 b 5 3 3 - 829329458590973952 b + 931090961195532288 a + 931090961195532288 b 3 4 2 + 3239920754937649152 a b + 1378367172163731456 a b 2 3 + 1378367172163731456 a b - 2382690293290450944 a b 2 2 6 9 - 2903198664189837312 a b - 1126493721384482176 b a 4 3 - 931090961195532288 b - 2382690293290450944 a b 6 5 2 + 997171134813757440 a b + 2320915182486091776 a b 5 2 2 10 + 2320915182486091776 b a - 668708744182110720 b a 5 8 14 18 + 1286250497419672224 b a + 5404799534432256 b + 140441050828800 a b 17 2 10 4 - 714159998968320 b a + 904984932577001760 b a 4 6 2 - 931090961195532288 a - 4558143906134747136 a b 7 7 - 1733916055460438016 a b - 1733916055460438016 b a 5 3 5 10 - 7983362804640150528 a b - 684076083236443920 b a 3 10 15 2 + 250310260478750208 b a + 5267282056943616 a b 7 8 16 - 1438435062044790480 b a + 855128998526976 a b 17 17 15 2 + 913598299164672 a b + 913598299164672 b a + 5267282056943616 b a 16 13 5 + 855128998526976 b a + 2692138896431712 b a 14 4 15 + 6593508571338432 b a - 5043033620987904 a b 10 9 + 132945462600597504 b + 672778363065126912 b a 4 10 6 2 + 904984932577001760 b a - 4558143906134747136 b a 6 6 8 + 829329458590973952 a + 829329458590973952 b - 411946572095225856 b 15 4 - 5043033620987904 b a - 1706090153372172288 a b 3 2 2 3 - 2998963734178258944 a b - 2998963734178258944 a b 4 4 13 - 1706090153372172288 b a + 77018502129084096 b a 17 11 5 - 350371158294528 b - 243908691754222656 b a 12 4 9 7 - 134633073297453696 b a - 447794406271402344 b a 10 6 9 - 360207656243900784 b a - 132945462600597504 b 6 8 12 + 2153207002197407904 b a - 30522147808149504 b 10 11 + 132945462600597504 a - 177536633723559936 a b 7 10 11 7 + 536407539507643848 a b - 46647978799819968 a b 7 11 11 6 - 46647978799819968 a b + 354509970088327680 a b 9 15 3 - 132945462600597504 a + 4410517208573952 a b 15 3 16 2 + 4410517208573952 b a + 1964166674946048 b a 4 2 3 3 + 5417734499693494272 a b + 6936720777244459008 a b 2 4 11 + 5417734499693494272 a b - 177536633723559936 b a 6 3 5 4 - 1260366586161898496 b a - 1600633240322592768 b a 10 3 9 4 + 250310260478750208 b a + 684552534389324832 b a 8 5 11 + 1286250497419672224 b a + 30522147808149504 a 8 4 4 - 411946572095225856 a - 9494376763741231104 a b 3 5 6 11 - 7983362804640150528 a b + 354509970088327680 a b 4 5 3 6 - 1600633240322592768 b a - 1260366586161898496 b a 3 8 3 13 - 77283552126105600 b a - 58395769552995840 b a 13 4 3 14 + 77018502129084096 b a + 24073968935588352 b a 13 13 16 + 33869148816900096 a b + 33869148816900096 b a - 882416250519552 b 6 11 + 997171134813757440 b a + 30522147808149504 b 8 8 14 3 - 479939995356125040 b a + 24073968935588352 b a 15 7 2 8 + 882416250519552 a - 794538136994623488 a b - 306155958320406528 a b 7 2 8 - 794538136994623488 b a - 306155958320406528 b a 14 17 13 2 - 3232474051166208 a b - 350371158294528 a - 23400337109990400 b a 2 13 14 2 - 23400337109990400 b a - 19448757729672192 a b 13 3 14 2 - 58395769552995840 b a - 19448757729672192 b a 6 12 5 12 - 14825539115332992 a b + 187151597334035712 a b 10 9 2 + 41846769185329152 b a + 61509539331491328 b a 10 9 2 + 41846769185329152 a b + 61509539331491328 a b 12 7 3 - 30522147808149504 a + 4838102557138819584 b a 4 6 3 7 + 7646212504250394624 b a + 4838102557138819584 b a 8 2 8 2 + 2185540478157459456 a b + 2185540478157459456 b a 5 5 18 14 + 8907635049609010176 b a - 38930128699392 a - 3232474051166208 b a 15 8 3 7 4 + 882416250519552 b - 77283552126105600 b a - 370275535411051776 b a 6 5 5 6 - 654853329088975104 b a - 654853329088975104 b a 4 7 9 8 - 370275535411051776 b a + 657552755735324394 a b 9 9 10 7 - 94018449467859543 a b + 536407539507643848 a b 8 9 10 8 + 657552755735324394 a b - 79723455676292142 a b 8 10 12 5 - 79723455676292142 a b + 187151597334035712 a b 12 6 8 4 - 14825539115332992 a b - 3416373743113165824 b a 6 6 5 7 - 5709327775720157952 b a - 5031279230042482560 b a 4 8 10 2 - 3416373743113165824 b a - 668708744182110720 b a 9 3 17 2 - 1761452659197462528 b a - 714159998968320 a b 18 7 6 + 140441050828800 b a + 1742011891083078912 b a 12 3 6 10 - 105706001553666560 b a - 360207656243900784 b a 7 9 15 5 - 447794406271402344 b a + 5336820985345344 b a 16 4 15 5 + 1091584399669632 b a + 5336820985345344 a b 4 16 18 4 14 + 1091584399669632 b a - 38930128699392 b + 6593508571338432 b a 5 13 15 4 + 2692138896431712 b a - 10644291102548352 b a 16 3 16 3 - 3390800993594880 a b - 3390800993594880 b a 12 8 12 9 + 56279026418000760 a b - 3227111145282112 a b 13 7 11 9 + 33493423177977840 a b + 76136187876034572 a b 13 8 11 10 - 286389264060900 a b - 5489976196787676 a b 12 7 6 13 - 89765073214255200 a b - 54291690697251840 a b 8 13 7 13 - 286389264060900 a b + 33493423177977840 a b 10 10 14 7 + 84087039833649864 a b + 1631591341882128 a b 10 11 14 6 - 5489976196787676 a b + 15588715529087712 a b 9 11 11 8 + 76136187876034572 a b - 123982242367586760 a b 9 12 8 12 - 3227111145282112 a b + 56279026418000760 a b 8 11 13 6 - 123982242367586760 a b - 54291690697251840 a b 7 12 10 9 - 89765073214255200 a b - 145158695332255737 a b 9 10 18 3 - 145158695332255737 a b + 199416835425280 a b 18 3 17 4 16 + 199416835425280 b a + 739028742601248 b a - 882416250519552 a 14 5 4 15 - 26903485287106272 a b - 10644291102548352 b a 16 2 17 3 + 1964166674946048 a b - 183377937600000 a b 5 14 7 14 - 26903485287106272 a b + 1631591341882128 a b 6 14 18 2 + 15588715529087712 a b - 219792161825280 a b 18 2 6 15 - 219792161825280 b a + 2000968933142784 a b 15 6 16 5 + 2000968933142784 a b + 1451192095118304 a b 15 8 15 9 + 2321255511174600 a b - 797775738259500 a b 16 7 14 9 + 1011768844086000 a b + 4138133010661650 a b 16 8 14 10 - 339007075407000 a b - 1400104662957000 a b 15 7 9 13 - 3884682717492192 a b - 9684443315897658 a b 11 13 10 13 - 1933326206311500 a b + 5988942693183450 a b 13 10 13 11 + 5988942693183450 a b - 1933326206311500 a b 12 11 14 8 + 7174900558725900 a b - 6702871428484308 a b 12 12 11 12 - 2148314500332000 a b + 7174900558725900 a b 11 11 16 6 - 12823458564301308 a b - 1825085153773728 a b 10 12 13 9 - 11968273207268532 a b - 9684443315897658 a b 12 10 8 14 - 11968273207268532 a b - 6702871428484308 a b 10 14 9 14 - 1400104662957000 a b + 4138133010661650 a b 7 15 9 15 - 3884682717492192 a b - 797775738259500 a b 8 15 17 3 + 2321255511174600 a b - 183377937600000 b a 5 16 6 16 + 1451192095118304 a b - 1825085153773728 a b 8 16 7 16 - 339007075407000 a b + 1011768844086000 a b 4 17 18 4 17 5 + 739028742601248 b a - 117302530691808 b a - 632677004677872 b a 17 5 4 18 18 6 - 632677004677872 a b - 117302530691808 b a - 12995644662000 a b 18 8 18 9 17 9 - 280078548750 a b + 15559919375 a b - 2380667664375 a b 17 10 18 7 12 13 + 140039274375 a b + 2422012593000 a b + 373811503065000 a b 14 13 13 13 16 10 + 1960549841250 a b - 41171546666250 a b - 9242592108750 a b 16 11 15 11 17 8 + 560157097500 a b - 21846126802500 a b + 19277406398250 a b 15 12 14 12 14 11 + 1307033227500 a b - 35289897142500 a b + 290305416901500 a b 13 12 16 9 + 373811503065000 a b + 72150901572750 a b 15 10 11 14 + 171061307917500 a b + 290305416901500 a b 13 14 12 14 10 15 + 1960549841250 a b - 35289897142500 a b + 171061307917500 a b 12 15 11 15 9 16 + 1307033227500 a b - 21846126802500 a b + 72150901572750 a b 11 16 10 16 18 5 + 560157097500 a b - 9242592108750 a b + 47005085727600 a b 17 7 17 6 6 17 - 95162688621000 a b + 306084969590400 a b + 306084969590400 a b 7 17 8 17 10 17 - 95162688621000 a b + 19277406398250 a b + 140039274375 a b 9 17 5 18 6 18 - 2380667664375 a b + 47005085727600 a b - 12995644662000 a b 7 18 9 18 8 18 + 2422012593000 a b + 15559919375 a b - 280078548750 a b )/ 1471492915200 This took, 5.441, seconds