# The set S_n weight enumerated by # t^(# of copies of 123) for n up to, 18 L18:=[1, 2, t+5, t^4+3*t^2+6*t+14, t^10+4*t^7+6*t^5+7*t^3+9*t^4+24*t^2+27*t+42, 132+ 110*t+133*t^2+74*t^4+70*t^3+32*t^7+54*t^5+16*t^10+37*t^6+24*t^8+5*t^16+6*t^12+ 12*t^9+8*t^13+6*t^11+t^20, 429+429*t+635*t^2+507*t^4+461*t^3+320*t^7+395*t^5+ 191*t^10+387*t^6+260*t^8+74*t^16+104*t^12+232*t^9+130*t^13+162*t^11+10*t^20+13* t^22+62*t^17+32*t^19+100*t^14+24*t^15+6*t^30+10*t^25+30*t^21+18*t^18+10*t^26+8* t^23+t^35, 1430+1638*t+2807*t^2+3008*t^4+2528*t^3+2544*t^7+2570*t^5+2077*t^10+ 2864*t^6+2389*t^8+1047*t^16+1580*t^12+2182*t^9+1456*t^13+1818*t^11+546*t^20+537 *t^22+1004*t^17+656*t^19+1494*t^14+886*t^15+154*t^30+324*t^25+466*t^21+682*t^18 +252*t^26+288*t^23+38*t^35+t^56+10*t^41+12*t^45+58*t^32+7*t^50+15*t^44+40*t^39+ 18*t^38+60*t^33+47*t^36+66*t^31+156*t^27+16*t^40+68*t^34+138*t^29+228*t^24+115* t^28, 4862+6188*t+11864*t^2+16151*t^4+12525*t^3+17357*t^7+15203*t^5+17505*t^10+ 18179*t^6+18096*t^8+12472*t^16+15847*t^12+17333*t^9+15068*t^13+16605*t^11+8935* t^20+8089*t^22+12101*t^17+9588*t^19+15049*t^14+12630*t^15+3412*t^30+6072*t^25+ 8225*t^21+10837*t^18+5158*t^26+6836*t^23+1995*t^35+57*t^56+834*t^41+420*t^45+ 2524*t^32+166*t^50+488*t^44+1190*t^39+1064*t^38+2151*t^33+1583*t^36+2976*t^31+ 4541*t^27+850*t^40+1887*t^34+3462*t^29+5405*t^24+3901*t^28+t^84+10*t^62+12*t^66 +14*t^71+21*t^70+20*t^63+8*t^77+58*t^59+80*t^55+60*t^64+66*t^57+50*t^60+248*t^ 53+44*t^54+186*t^49+282*t^48+823*t^42+148*t^52+427*t^47+234*t^51+110*t^58+19*t^ 65+458*t^46+508*t^43+1312*t^37, 16796+23256*t+48756*t^2+80889*t^4+58258*t^3+ 107194*t^7+83382*t^5+129449*t^10+105082*t^6+120197*t^8+127049*t^16+132579*t^12+ 121630*t^9+130310*t^13+128712*t^11+109115*t^20+102875*t^22+121602*t^17+112240*t ^19+133945*t^14+124572*t^15+61259*t^30+84020*t^25+103134*t^21+121330*t^18+82579 *t^26+95256*t^23+41368*t^35+5451*t^56+25088*t^41+16776*t^45+53742*t^32+9812*t^ 50+18612*t^44+29182*t^39+31990*t^38+47734*t^33+39552*t^36+56574*t^31+73540*t^27 +27361*t^40+44615*t^34+64590*t^29+86372*t^24+69645*t^28+t^120+12*t^94+14*t^99+ 16*t^105+28*t^104+15*t^88+35*t^96+9*t^112+107*t^86+72*t^91+84*t^97+120*t^89+156 *t^81+64*t^87+606*t^73+378*t^78+59*t^92+60*t^85+312*t^79+22*t^98+296*t^83+603*t ^76+172*t^90+264*t^82+401*t^74+1134*t^69+608*t^75+1256*t^67+1192*t^68+3004*t^61 +192*t^80+986*t^72+326*t^84+2795*t^62+1977*t^66+1130*t^71+1480*t^70+2206*t^63+ 700*t^77+4682*t^59+5824*t^55+2678*t^64+4708*t^57+3486*t^60+9012*t^53+6788*t^54+ 11058*t^49+14304*t^48+23871*t^42+8974*t^52+14582*t^47+9188*t^51+5058*t^58+2434* t^65+15679*t^46+20574*t^43+35538*t^37, 58786+87210*t+196707*t^2+385387*t^4+ 259787*t^3+616449*t^7+431167*t^5+868607*t^10+568809*t^6+730182*t^8+1099257*t^16 +975820*t^12+778432*t^9+1000220*t^13+901457*t^11+1106169*t^20+1094214*t^22+ 1092656*t^17+1101327*t^19+1059437*t^14+1051194*t^15+867777*t^30+1005574*t^25+ 1079286*t^21+1121794*t^18+1002920*t^26+1065888*t^23+694925*t^35+191347*t^56+ 514598*t^41+402525*t^45+807892*t^32+288746*t^50+428537*t^44+567469*t^39+601738* t^38+761461*t^33+665451*t^36+833036*t^31+963317*t^27+536582*t^40+735291*t^34+ 897540*t^29+1031696*t^24+925079*t^28+150*t^120+7356*t^94+3660*t^99+1862*t^105+ 1787*t^104+11766*t^88+5607*t^96+398*t^112+14829*t^86+8690*t^91+4362*t^97+11168* t^89+25732*t^81+15964*t^87+50256*t^73+32045*t^78+8278*t^92+16286*t^85+28528*t^ 79+4878*t^98+20715*t^83+39349*t^76+11210*t^90+21425*t^82+43978*t^74+68371*t^69+ 42730*t^75+84846*t^67+74486*t^68+133790*t^61+27264*t^80+54635*t^72+19063*t^84+ 124559*t^62+89678*t^66+60868*t^71+65623*t^70+111283*t^63+35811*t^77+159312*t^59 +206056*t^55+105226*t^64+177756*t^57+140646*t^60+248644*t^53+223830*t^54+315180 *t^49+343753*t^48+487819*t^42+250591*t^52+354636*t^47+273107*t^51+167509*t^58+ 98752*t^65+377705*t^46+457925*t^43+644967*t^37+t^165+14*t^135+16*t^141+18*t^148 +166*t^121+10*t^156+112*t^139+56*t^138+222*t^130+98*t^132+36*t^147+35*t^129+148 *t^126+874*t^111+66*t^127+64*t^125+508*t^118+534*t^117+68*t^133+462*t^124+2574* t^103+1756*t^109+25*t^140+392*t^123+533*t^122+1688*t^106+867*t^114+1089*t^115+ 2283*t^107+850*t^116+224*t^131+1806*t^108+3955*t^100+746*t^113+166*t^119+3456*t ^102+8326*t^93+4146*t^101+1310*t^110+6657*t^95, 208012+326876*t+783750*t^2+ 1769705*t^4+1124704*t^3+3357324*t^7+2129734*t^5+5409361*t^10+2930602*t^6+ 4158993*t^8+8554680*t^16+6582534*t^12+4649860*t^9+7005086*t^13+5850192*t^11+ 9848663*t^20+10252212*t^22+8800138*t^17+9455254*t^19+7664290*t^14+7955082*t^15+ 10298618*t^30+10363514*t^25+9887754*t^21+9302926*t^18+10534421*t^26+10289744*t^ 23+9411088*t^35+4459522*t^56+8050836*t^41+7038964*t^45+10060255*t^32+5840485*t^ 50+7351731*t^44+8529828*t^39+8858488*t^38+9782454*t^33+9248159*t^36+10083848*t^ 31+10461734*t^27+8326384*t^40+9696595*t^34+10286752*t^29+10412901*t^24+10439795 *t^28+58398*t^120+459377*t^94+307942*t^99+195598*t^105+220097*t^104+692353*t^88 +411831*t^96+113464*t^112+789574*t^86+557190*t^91+360726*t^97+651210*t^89+ 1106488*t^81+747260*t^87+1781342*t^73+1317057*t^78+514716*t^92+831612*t^85+ 1226822*t^79+343986*t^98+964510*t^83+1520651*t^76+616455*t^90+1036376*t^82+ 1686055*t^74+2249478*t^69+1591784*t^75+2536156*t^67+2404584*t^68+3476616*t^61+ 1170509*t^80+1907813*t^72+905971*t^84+3339915*t^62+2673316*t^66+2019970*t^71+ 2143602*t^70+3128488*t^63+1410444*t^77+3856570*t^59+4648480*t^55+2987967*t^64+ 4237332*t^57+3655541*t^60+5124276*t^53+4931245*t^54+6064454*t^49+6353104*t^48+ 7878546*t^42+5315981*t^52+6520322*t^47+5557854*t^51+4065748*t^58+2809980*t^65+ 6825097*t^46+7566114*t^43+9051636*t^37+1270*t^165+15894*t^135+11348*t^141+7027* t^148+49392*t^121+3049*t^156+13684*t^139+13460*t^138+27024*t^130+25240*t^132+ 6556*t^147+25650*t^129+37987*t^126+132786*t^111+33592*t^127+43118*t^125+79584*t ^118+83976*t^117+22070*t^133+49758*t^124+246044*t^103+154968*t^109+12682*t^140+ 49646*t^123+54015*t^122+183818*t^106+97316*t^114+96054*t^115+176288*t^107+89357 *t^116+26652*t^131+162255*t^108+294353*t^100+102292*t^113+64736*t^119+264987*t^ 102+488784*t^93+280706*t^101+147435*t^110+434182*t^95+426*t^160+86*t^175+612*t^ 167+630*t^174+77*t^184+822*t^166+2658*t^157+28*t^192+1597*t^164+1452*t^163+2819 *t^154+574*t^173+928*t^162+798*t^172+296*t^182+2108*t^153+3851*t^150+6532*t^146 +4750*t^145+11956*t^136+3326*t^155+7991*t^142+2350*t^151+740*t^161+6379*t^144+ 12254*t^137+2217*t^152+6052*t^143+1878*t^158+201*t^168+5028*t^149+26591*t^128+ 17858*t^134+t^220+16*t^186+18*t^193+20*t^201+160*t^169+11*t^210+173*t^176+84*t^ 190+45*t^200+84*t^180+81*t^170+336*t^181+144*t^191+128*t^183+370*t^171+76*t^177 +1122*t^159, 742900+1225785*t+20688*t^197+7216*t^207+23647*t^195+10345*t^206+ 2366*t^217+220*t^228+3095708*t^2+7902982*t^4+4765761*t^3+17513013*t^7+10148146* t^5+31788007*t^10+14527238*t^6+22555348*t^8+61392446*t^16+41578030*t^12+ 26334141*t^9+45793201*t^13+35742354*t^11+79256093*t^20+86841291*t^22+65216222*t ^17+74159849*t^19+51617654*t^14+55613837*t^15+106264648*t^30+95364767*t^25+ 82132111*t^21+70714898*t^18+98743416*t^26+89589031*t^23+109242054*t^35+79766705 *t^56+106309300*t^41+101108234*t^45+108510167*t^32+92509865*t^50+102909182*t^44 +108100043*t^39+109263115*t^38+108598104*t^33+109535912*t^36+106913170*t^31+ 100593111*t^27+107398163*t^40+109597236*t^34+104141846*t^29+93272321*t^24+ 103107053*t^28+21504*t^194+8763*t^203+5656*t^202+4382*t^213+4220645*t^120+ 17538377*t^94+13614580*t^99+9912529*t^105+10485164*t^104+23422034*t^88+15965250 *t^96+6707782*t^112+25627820*t^86+20360834*t^91+15061766*t^97+22419747*t^89+ 32248565*t^81+24622447*t^87+44846694*t^73+36614447*t^78+19277029*t^92+26989614* t^85+35096445*t^79+12371*t^199+3416*t^209+14319433*t^98+29544910*t^83+39904659* t^76+21386087*t^90+30870791*t^82+43172948*t^74+52371569*t^69+41530642*t^75+ 56315728*t^67+54362925*t^68+68786250*t^61+33611422*t^80+46722639*t^72+28212990* t^84+66804419*t^62+58265028*t^66+48725715*t^71+50444780*t^70+64662455*t^63+ 38315583*t^77+73256353*t^59+81944477*t^55+62522023*t^64+77576749*t^57+71014543* t^60+86158442*t^53+84167053*t^54+94243658*t^49+96245699*t^48+105573955*t^42+ 88198934*t^52+97692295*t^47+90273768*t^51+75466422*t^58+60404644*t^65+99769221* t^46+104077754*t^43+109131105*t^37+18*t^248+20*t^256+205227*t^165+1673666*t^135 +1150102*t^141+724300*t^148+3960000*t^121+404781*t^156+1322758*t^139+1399816*t^ 138+2300956*t^130+2066968*t^132+778204*t^147+2427157*t^129+2968116*t^126+1260*t ^233+378*t^244+2119*t^211+1576*t^222+31*t^255+7177647*t^111+2780294*t^127+10574 *t^205+6244*t^214+3177*t^212+12034*t^204+1760*t^225+4192*t^215+2644*t^224+2290* t^223+816*t^234+3163314*t^125+4805334*t^118+5096902*t^117+1931435*t^133+3367522 *t^124+11113788*t^103+8012960*t^109+1233362*t^140+3590054*t^123+3737985*t^122+ 9323095*t^106+5970480*t^114+5702978*t^115+8923702*t^107+5395444*t^116+2207530*t ^131+8390912*t^108+12872502*t^100+6324883*t^113+4494381*t^119+11660950*t^102+ 18402179*t^93+12315934*t^101+7604672*t^110+16753863*t^95+294974*t^160+92676*t^ 175+183088*t^167+94681*t^174+41978*t^184+183338*t^166+384302*t^157+17546*t^192+ 231408*t^164+256465*t^163+480458*t^154+110594*t^173+267478*t^162+132498*t^172+ 52194*t^182+497128*t^153+612468*t^150+820946*t^146+857456*t^145+1549696*t^136+ 446604*t^155+1041404*t^142+561700*t^151+288166*t^161+924047*t^144+1456616*t^137 +530599*t^152+983450*t^143+810*t^227+84*t^236+584*t^219+1028*t^226+838*t^235+86 *t^246+5080*t^208+4008*t^216+347818*t^158+167317*t^168+663725*t^149+2572104*t^ 128+1770885*t^134+22*t^265+973*t^220+47380*t^186+485*t^221+148*t^231+17599*t^ 193+9048*t^201+155114*t^169+86*t^238+1410*t^218+3052*t^210+90642*t^176+28545*t^ 190+8642*t^200+74759*t^180+147960*t^170+60826*t^181+23908*t^191+43758*t^183+ 142488*t^171+87566*t^177+316283*t^159+12*t^275+96*t^230+126*t^242+162*t^245+212 *t^237+120*t^253+55*t^264+504*t^243+560*t^232+180*t^254+t^286+182*t^229+42206*t ^187+38435*t^188+14972*t^198+72838*t^179+83078*t^178+33464*t^189+21049*t^196+ 43358*t^185+16*t^241, 2674440+4601610*t+3207390*t^197+1745710*t^207+3566508*t^ 195+1866209*t^206+939032*t^217+430382*t^228+12152855*t^2+34539352*t^4+19873150* t^3+88211386*t^7+46993138*t^5+178382297*t^10+69819523*t^6+117666162*t^8+ 413564544*t^16+249285628*t^12+142944704*t^9+283480618*t^13+207976352*t^11+ 591941715*t^20+680508932*t^22+452230562*t^17+541319436*t^19+328398190*t^14+ 365385054*t^15+992127792*t^30+801876964*t^25+630225808*t^21+502136436*t^18+ 847368055*t^26+718681320*t^23+1126561842*t^35+1193806606*t^56+1225565538*t^41+ 1251910696*t^45+1054432886*t^32+1249751790*t^50+1252820286*t^44+1202278740*t^39 +1190961158*t^38+1077043714*t^33+1152641927*t^36+1019051018*t^31+881412134*t^27 +1217965433*t^40+1108270477*t^34+953750766*t^29+766082686*t^24+923549688*t^28+ 3709046*t^194+2094650*t^203+2278321*t^202+1092806*t^213+177196657*t^120+ 481497307*t^94+404193616*t^99+324038960*t^105+337311894*t^104+585578463*t^88+ 450611567*t^96+247115098*t^112+622702965*t^86+531317432*t^91+433791940*t^97+ 566543946*t^89+719221464*t^81+603182858*t^87+883542130*t^73+781568763*t^78+ 514925638*t^92+641134824*t^85+759837602*t^79+2865250*t^199+1522752*t^209+ 419423188*t^98+679685462*t^83+823651044*t^76+550100091*t^90+700835140*t^82+ 864432746*t^74+966079052*t^69+842261682*t^75+1005868854*t^67+988451917*t^68+ 1116426592*t^61+740767367*t^80+906790427*t^72+661099430*t^84+1101868406*t^62+ 1027825140*t^66+104*t^303+344*t^292+926165358*t^71+947629494*t^70+1082432148*t^ 63+801460744*t^77+1148873864*t^59+1202916542*t^55+1066020670*t^64+1178109916*t^ 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