The Exact Probability Distirbution of the occupants of cell, [1, 5], in a random Standard Young Tableau of shape, [50, 50, 50], using the exact symbolic expression Versus the approximation obtained by simulation via the Greene-Nijenhuis-Wil\ f algorithm, 1000, times By Shalosh B. Ekhad The occupants of cell, [1, 5], in a standard Young tableau of symbolic shape, [n, n, n], are all the integers from, 5, to , 13 The probability generating function, using the variable x (i.e. the polyno\ mial whose coeff. of x^j is the probability that Y[1,i]=j) is 154/27*x^5*((x^8+8*x^7+360/11*x^6+972/11*x^5+1890/11*x^4+2754/11*x^3+1944/7*x^2 +2430/11*x+2187/22)*n^8+(-24*x^7-2220/11*x^6-9072/11*x^5-23220/11*x^4-41040/11* x^3-372762/77*x^2-48600/11*x-24786/11)*n^7+(-6*x^8-72*x^7-120/11*x^6+19656/11*x ^5+93300/11*x^4+225576/11*x^3+2570724/77*x^2+404460/11*x+22113)*n^6+(120*x^7+ 15960/11*x^6+33840/11*x^5-88080/11*x^4-502500/11*x^3-8536860/77*x^2-1784160/11* x-1349460/11)*n^5+(9*x^8+192*x^7-5160/11*x^6-125892/11*x^5-21570*x^4+9906*x^3+ 11939376/77*x^2+4310670/11*x+9254763/22)*n^4+(-96*x^7-29580/11*x^6+6912/11*x^5+ 350340/11*x^4+1064820/11*x^3+2817222/77*x^2-4842600/11*x-10039554/11)*n^3+(-4*x ^8-128*x^7+4920/11*x^6+105264/11*x^5+222720/11*x^4-639696/11*x^3-22837404/77*x^ 2-483960/11*x+13453722/11)*n^2+(1440*x^6-2880*x^5-14400*x^4-74880*x^3+8874480/ 77*x^2+6473760/11*x-10177560/11)*n+144000*x^2-403200*x+302400)/(-1+3*n)/(-2+3*n )/(-4+3*n)/(-5+3*n)/(-7+3*n)/(-8+3*n)/(3*n-10)/(-11+3*n) BTW this took, 0.379, seconds to compute We are interested in what happens when n=, 50, i.e. for the specific shape, [50, 50, 50] Then the prob. gen. function is 13 12 11 10 0.001204863919 x + 0.009048773921 x + 0.03466040306 x + 0.08769145033 x 9 8 7 6 + 0.1605419929 x + 0.2217263503 x + 0.2335632561 x + 0.1763533799 x 5 + 0.07520952968 x Note that there are, 30029983483935083858438698423851117882968874317657169412268673840, Standard Young Tableaux of shape, [50, 50, 50], so it would be impractica\ l to compute this directly. Let's do it by simulation Now let's compare it with the approximation that you get by generating, 1000, standard Young tableau of that shape, namely, [50, 50, 50], using the Green-Nijenhuis-Algorithm 12 11 10 9 0.01000000000 x + 0.03900000000 x + 0.09000000000 x + 0.1460000000 x 8 7 6 5 + 0.2210000000 x + 0.2460000000 x + 0.1690000000 x + 0.07900000000 x BTW this took, 24.976, seconds The difference is 13 12 11 10 0.001204863919 x - 0.000951226079 x - 0.00433959694 x - 0.00230854967 x 9 8 7 6 + 0.0145419929 x + 0.0007263503 x - 0.0124367439 x + 0.0073533799 x 5 - 0.00379047032 x ----------------------- The Exact Probability Distirbution of the occupants of cell, [1, 7], in a random Standard Young Tableau of shape, [200, 100, 100], using the exact symbolic expression Versus the approximation obtained by simulation via the Greene-Nijenhuis-Wil\ f algorithm, 1000, times By Shalosh B. Ekhad The occupants of cell, [1, 7], in a standard Young tableau of symbolic shape, [2 n, n, n], are all the integers from, 7, to , 19 The probability generating function, using the variable x (i.e. the polyno\ mial whose coeff. of x^j is the probability that Y[1,i]=j) is 21879/64*((8151040/7293*x+x^12+9*x^11+734/17*x^10+7340/51*x^9+6248/17*x^8+12820 /17*x^7+847976/663*x^6+404032/221*x^5+5400960/2431*x^4+16528384/7293*x^3+ 13576192/7293*x^2+8224768/21879)*n^13+(-140389888/2431*x^2-266893312/7293*x-\ 284917760/21879-31/2*x^12-151*x^11-26855/34*x^10-146740/51*x^9-136066/17*x^8-\ 301606/17*x^7-21308710/663*x^6-10737840/221*x^5-151163840/2431*x^4-486775808/ 7293*x^3)*n^12+(492252800/561*x^3+5885918464/7293*x^2+3972342784/7293*x+359/4*x ^12+3859/4*x^11+191423/34*x^10+4695055/204*x^9+2438025/34*x^8+11980651/68*x^7+ 461412077/1326*x^6+124734292/221*x^5+1870177040/2431*x^4+348256256/1683)*n^11+( -13385077400/2431*x^4-49327816960/7293*x^3-48828393280/7293*x^2-11837236480/ 2431*x-1805/8*x^12-11125/4*x^11-2568805/136*x^10-37027375/408*x^9-45157575/136* x^8-128365005/136*x^7-11138703685/5304*x^6-824970805/221*x^5-43713095680/21879) *n^10+(6750742017/442*x^5+121404733635/4862*x^4+81897579484/2431*x^3+6828533144 /187*x^2+70697444640/2431*x+291/2*x^12+10413/4*x^11+1685253/68*x^10+22090945/ 136*x^9+214132869/272*x^8+193119645/68*x^7+26788317841/3536*x^6+5613816704/429) *n^9+(-570531399/136*x^7-56286042155/3536*x^6-661533450/17*x^5-713858117355/ 9724*x^4-272683141228/2431*x^3-2322874738/17*x^2-296430342952/2431*x+3063/8*x^ 12+15663/4*x^11+2226015/136*x^10-713885/68*x^9-165153243/272*x^8-448061841280/ 7293)*n^8+(-78675515/68*x^8+3360977/17*x^7+805964701/51*x^6+25241371927/442*x^5 +1314812379105/9724*x^4+1803527983175/7293*x^3+2577814465982/7293*x^2+ 2670957244352/7293*x-652*x^12-37353/4*x^11-5045267/68*x^10-39506755/102*x^9+ 4657925198984/21879)*n^7+(158812625/408*x^9+86463125/34*x^8+1137666485/136*x^7+ 1953599560/663*x^6-427938660/13*x^5-651362507225/4862*x^4-2450133340465/7293*x^ 3-2985943175925/4862*x^2-337093694770/429*x+145/8*x^12+7345/4*x^11+4917785/136* x^10-12073389813400/21879)*n^6+(3582311/68*x^10+78047695/408*x^9-159220287/272* x^8-632529475/68*x^7-212134554289/10608*x^6-10339742981/442*x^5+51344987410/ 2431*x^4+1538626935514/7293*x^3+4715411527667/7293*x^2+8510898305990/7293*x+ 1007/2*x^12+27851/4*x^11+23432568706678/21879)*n^5+(-14159/4*x^11-5514095/136*x ^10-38032505/102*x^9-519253551/272*x^8-43259553/136*x^7+91898623475/10608*x^6+ 7798122340/221*x^5+804261446465/9724*x^4+530401473142/7293*x^3-1866669677234/ 7293*x^2-2655751365394/2431*x-1639/8*x^12-33670632673550/21879)*n^4+(-351/4*x^ 12-2403/2*x^11-608079/68*x^10+354805/34*x^9+152222655/136*x^8+389515191/68*x^7+ 15601130721/1768*x^6+1473666189/442*x^5-421492794015/9724*x^4-437360927675/2431 *x^3-573771770657/2431*x^2+1068334019238/2431*x+11618761400746/7293)*n^3+(45*x^ 12+720*x^11+7695*x^10+87750*x^9+5324625/17*x^8-49137210/17*x^7-3306860235/442*x ^6-190143225/13*x^5-59973964425/2431*x^4+79119568725/2431*x^3+1448569797645/ 4862*x^2+2758394160/11*x-2738258622730/2431)*n^2+(-236250*x^8+359100*x^7+ 1367100*x^6+3534300*x^5+10111500*x^4+46147500*x^3-89473315050/2431*x^2-\ 945385975800/2431*x+1186270833300/2431)*n-55566000*x^2+142884000*x-97902000)*x^ 7/(-1+4*n)/(2*n-1)/(-3+4*n)/(-5+4*n)/(2*n-3)/(-7+4*n)/(-9+4*n)/(-5+2*n)/(-11+4* n)/(-13+4*n)/(-7+2*n)/(-15+4*n)/(4*n-17) BTW this took, 25.946, seconds to compute We are interested in what happens when n=, 100, i.e. for the specific shape, [200, 100, 100] Then the prob. gen. function is 19 18 17 0.00009268672854 x + 0.0008233200562 x + 0.003889018392 x 16 15 14 + 0.01274123799 x + 0.03195444536 x + 0.06442251181 x 13 12 11 + 0.1075188373 x + 0.1514666073 x + 0.1814706415 x 10 9 8 7 + 0.1824080542 x + 0.1474737303 x + 0.08702196666 x + 0.02871694228 x Note that there are, 23419141635244396333531228415321970631336329309270232771\ 437214952405849760049194522010906235008599574425264115534857673820986061\ 455102421738911608857162892944591669826375038400, Standard Young Tableaux of shape, [200, 100, 100], so it would be impract\ ical to compute this directly. Let's do it by simulation Now let's compare it with the approximation that you get by generating, 1000, standard Young tableau of that shape, namely, [200, 100, 100], using the Green-Nijenhuis-Algorithm 18 17 16 15 0.001000000000 x + 0.005000000000 x + 0.01100000000 x + 0.03100000000 x 14 13 12 + 0.07800000000 x + 0.1080000000 x + 0.1540000000 x 11 10 9 8 + 0.1870000000 x + 0.1640000000 x + 0.1460000000 x + 0.08600000000 x 7 + 0.02900000000 x BTW this took, 101.772, seconds The difference is 10 9 15 7 0.0184080542 x + 0.0014737303 x + 0.00095444536 x - 0.00028305772 x 19 14 12 + 0.00009268672854 x - 0.01357748819 x - 0.0025333927 x 8 13 16 + 0.00102196666 x - 0.0004811627 x + 0.00174123799 x 17 18 11 - 0.001110981608 x - 0.0001766799438 x - 0.0055293585 x ----------------------- The Exact Probability Distirbution of the occupants of cell, [1, 3], in a random Standard Young Tableau of shape, [100, 100, 100, 100], using the exact symbolic expression Versus the approximation obtained by simulation via the Greene-Nijenhuis-Wil\ f algorithm, 1000, times By Shalosh B. Ekhad The occupants of cell, [1, 3], in a standard Young tableau of symbolic shape, [n, n, n, n], are all the integers from, 3, to , 9 The probability generating function, using the variable x (i.e. the polyno\ mial whose coeff. of x^j is the probability that Y[1,i]=j) is 7*((x^6+15*x^5+720/7*x^4+2880/7*x^3+7296/7*x^2+11520/7*x+10240/7)*n^6+(9*x^6+75 *x^5+900/7*x^4-5520/7*x^3-33216/7*x^2-74880/7*x-84480/7)*n^5+(31*x^6+45*x^5-\ 5220/7*x^4-15480/7*x^3+29016/7*x^2+169200/7*x+284800/7)*n^4+(51*x^6-255*x^5-\ 6120/7*x^4+29040/7*x^3+56436/7*x^2-117000/7*x-501600/7)*n^3+(40*x^6-420*x^5+ 9360/7*x^4+18360/7*x^3-80340/7*x^2-105120/7*x+486160/7)*n^2+(12*x^6-180*x^5+ 1440*x^4-27840/7*x^3-21888/7*x^2+191880/7*x-245520/7)*n-1440*x^3+6480*x^2-10800 *x+7200)*x^3/(32768*n^6-196608*n^5+462848*n^4-540672*n^3+325760*n^2-94464*n+ 10080) BTW this took, 128.229, seconds to compute We are interested in what happens when n=, 100, i.e. for the specific shape, [100, 100, 100, 100] Then the prob. gen. function is 9 8 7 6 0.0002480595928 x + 0.003574976485 x + 0.02361513970 x + 0.09152332937 x 5 4 3 + 0.2258451878 x + 0.3497185650 x + 0.3054747422 x Note that there are, 91744305057222844133424691204107544786175718823135304608\ 623623089796733069181510950920939466336776560602931742685880381526412044\ 422727974681906771734078414745843524323495659756902517895956550501414136\ 93647293390999808013888000, Standard Young Tableaux of shape, [100, 100, 100, 100], so it would be impractical to compute this directly\ . Let's do it by simulation Now let's compare it with the approximation that you get by generating, 1000, standard Young tableau of that shape, namely, [100, 100, 100, 100], using the Green-Nijenhuis-Algorithm 7 6 5 4 0.01600000000 x + 0.09800000000 x + 0.2510000000 x + 0.3120000000 x 3 + 0.3230000000 x BTW this took, 96.358, seconds The difference is 9 8 7 6 0.0002480595928 x + 0.003574976485 x + 0.00761513970 x - 0.00647667063 x 5 4 3 - 0.0251548122 x + 0.0377185650 x - 0.0175252578 x