The Joint Weight Enumerators of the Number of Inversions and Spearman's Rho \ over permutations on {1,...,n} for n from 1 to, 20 By Shalosh B. Ekhad Let J[n](p,q) be the polynomial in the variables p and q whose coefficient\ of p^i*q^j (for each i,j) is the exact number of permutations of {1,...\ ,n}, whose number of inversions is i and whose Spearman's Rho equals j We have the following (in Maple notation) J[1](p, q), equals 1 J[2](p, q), equals p*q^2+1 J[3](p, q), equals p^3*q^8+2*p^2*q^6+2*p*q^2+1 J[4](p, q), equals p^6*q^20+3*p^5*q^18+p^4*q^16+4*p^4*q^14+2*p^3*q^12+2*p^3*q^10+2*p^3*q^8+4*p^2*q ^6+p^2*q^4+3*p*q^2+1 J[5](p, q), equals p^10*q^40+4*p^9*q^38+3*p^8*q^36+6*p^8*q^34+7*p^7*q^32+4*p^7*q^30+2*p^6*q^30+4*p ^7*q^28+10*p^6*q^26+6*p^6*q^24+10*p^5*q^22+2*p^6*q^20+2*p^5*q^20+2*p^4*q^20+10* p^5*q^18+6*p^4*q^16+10*p^4*q^14+4*p^3*q^12+2*p^4*q^10+4*p^3*q^10+7*p^3*q^8+6*p^ 2*q^6+3*p^2*q^4+4*p*q^2+1 J[6](p, q), equals p^15*q^70+5*p^14*q^68+6*p^13*q^66+8*p^13*q^64+p^12*q^64+16*p^12*q^62+6*p^12*q^ 60+6*p^11*q^60+6*p^12*q^58+8*p^11*q^58+20*p^11*q^56+4*p^10*q^56+11*p^11*q^54+8* p^10*q^54+p^9*q^54+21*p^10*q^52+4*p^11*q^50+10*p^10*q^50+9*p^9*q^50+22*p^10*q^ 48+4*p^9*q^48+4*p^10*q^46+2*p^8*q^48+20*p^9*q^46+32*p^9*q^44+2*p^8*q^44+8*p^9*q ^42+2*p^10*q^40+12*p^8*q^42+8*p^9*q^40+22*p^8*q^40+8*p^9*q^38+30*p^8*q^38+4*p^7 *q^38+13*p^8*q^36+16*p^7*q^36+16*p^8*q^34+13*p^7*q^34+4*p^8*q^32+30*p^7*q^32+8* p^6*q^32+22*p^7*q^30+8*p^6*q^30+12*p^7*q^28+2*p^5*q^30+8*p^6*q^28+2*p^7*q^26+32 *p^6*q^26+20*p^6*q^24+2*p^7*q^22+4*p^5*q^24+4*p^6*q^22+22*p^5*q^22+9*p^6*q^20+ 10*p^5*q^20+4*p^4*q^20+21*p^5*q^18+p^6*q^16+8*p^5*q^16+11*p^4*q^16+4*p^5*q^14+ 20*p^4*q^14+8*p^4*q^12+6*p^3*q^12+6*p^4*q^10+6*p^3*q^10+16*p^3*q^8+p^3*q^6+8*p^ 2*q^6+6*p^2*q^4+5*p*q^2+1 J[7](p, q), equals p^21*q^112+6*p^20*q^110+10*p^19*q^108+10*p^19*q^106+4*p^18*q^106+29*p^18*q^104+ 8*p^18*q^102+18*p^17*q^102+8*p^18*q^100+24*p^17*q^100+3*p^16*q^100+34*p^17*q^98 +12*p^16*q^98+16*p^17*q^96+36*p^16*q^96+3*p^15*q^96+50*p^16*q^94+6*p^17*q^92+4* p^15*q^94+18*p^16*q^92+50*p^15*q^92+38*p^16*q^90+18*p^15*q^90+8*p^16*q^88+14*p^ 14*q^90+62*p^15*q^88+14*p^14*q^88+60*p^15*q^86+20*p^14*q^86+22*p^15*q^84+10*p^ 13*q^86+4*p^16*q^82+54*p^14*q^84+18*p^15*q^82+58*p^14*q^82+2*p^12*q^84+16*p^15* q^80+10*p^13*q^82+89*p^14*q^80+4*p^15*q^78+24*p^13*q^80+30*p^14*q^78+72*p^13*q^ 78+54*p^14*q^76+49*p^13*q^76+12*p^14*q^74+20*p^12*q^76+92*p^13*q^74+4*p^14*q^72 +30*p^12*q^74+2*p^15*q^70+94*p^13*q^72+34*p^12*q^72+46*p^13*q^70+15*p^11*q^72+ 10*p^14*q^68+48*p^12*q^70+20*p^13*q^68+132*p^12*q^68+2*p^10*q^70+14*p^13*q^66+6 *p^11*q^68+96*p^12*q^66+24*p^13*q^64+20*p^11*q^66+39*p^12*q^64+112*p^11*q^64+86 *p^12*q^62+38*p^11*q^62+22*p^12*q^60+20*p^10*q^62+132*p^11*q^60+18*p^12*q^58+14 *p^10*q^60+86*p^11*q^58+36*p^10*q^58+92*p^11*q^56+4*p^9*q^58+4*p^12*q^54+92*p^ 10*q^56+36*p^11*q^54+86*p^10*q^54+14*p^11*q^52+18*p^9*q^54+132*p^10*q^52+20*p^ 11*q^50+22*p^9*q^52+38*p^10*q^50+86*p^9*q^50+112*p^10*q^48+39*p^9*q^48+20*p^10* q^46+24*p^8*q^48+96*p^9*q^46+6*p^10*q^44+14*p^8*q^46+2*p^11*q^42+132*p^9*q^44+ 20*p^8*q^44+48*p^9*q^42+10*p^7*q^44+15*p^10*q^40+46*p^8*q^42+34*p^9*q^40+94*p^8 *q^40+2*p^6*q^42+30*p^9*q^38+4*p^7*q^40+92*p^8*q^38+20*p^9*q^36+12*p^7*q^38+49* p^8*q^36+54*p^7*q^36+72*p^8*q^34+30*p^7*q^34+24*p^8*q^32+4*p^6*q^34+89*p^7*q^32 +10*p^8*q^30+16*p^6*q^32+2*p^9*q^28+58*p^7*q^30+18*p^6*q^30+54*p^7*q^28+4*p^5*q ^30+10*p^8*q^26+22*p^6*q^28+20*p^7*q^26+60*p^6*q^26+14*p^7*q^24+62*p^6*q^24+14* p^7*q^22+8*p^5*q^24+18*p^6*q^22+38*p^5*q^22+50*p^6*q^20+18*p^5*q^20+4*p^6*q^18+ 6*p^4*q^20+50*p^5*q^18+3*p^6*q^16+36*p^5*q^16+16*p^4*q^16+12*p^5*q^14+34*p^4*q^ 14+3*p^5*q^12+24*p^4*q^12+8*p^3*q^12+18*p^4*q^10+8*p^3*q^10+29*p^3*q^8+4*p^3*q^ 6+10*p^2*q^6+10*p^2*q^4+6*p*q^2+1 J[8](p, q), equals p^28*q^168+7*p^27*q^166+15*p^26*q^164+12*p^26*q^162+10*p^25*q^162+46*p^25*q^160 +p^24*q^160+10*p^25*q^158+44*p^24*q^158+10*p^25*q^156+48*p^24*q^156+12*p^23*q^ 156+52*p^24*q^154+42*p^23*q^154+21*p^24*q^152+90*p^23*q^152+18*p^22*q^152+97*p^ 23*q^150+8*p^24*q^148+24*p^22*q^150+26*p^23*q^148+3*p^21*q^150+144*p^22*q^148+ 58*p^23*q^146+76*p^22*q^146+12*p^23*q^144+49*p^21*q^146+132*p^22*q^144+90*p^21* q^144+104*p^22*q^142+3*p^20*q^144+96*p^21*q^142+36*p^22*q^140+38*p^20*q^142+6*p ^23*q^138+198*p^21*q^140+28*p^22*q^138+36*p^20*q^140+122*p^21*q^138+6*p^19*q^ 140+28*p^22*q^136+104*p^20*q^138+170*p^21*q^136+4*p^19*q^138+8*p^22*q^134+141*p ^20*q^136+75*p^21*q^134+40*p^19*q^136+206*p^20*q^134+92*p^21*q^132+52*p^19*q^ 134+179*p^20*q^132+8*p^18*q^134+28*p^21*q^130+90*p^19*q^132+210*p^20*q^130+19*p ^18*q^132+12*p^21*q^128+148*p^19*q^130+4*p^22*q^126+226*p^20*q^128+8*p^18*q^130 +196*p^19*q^128+6*p^17*q^130+104*p^20*q^126+82*p^18*q^128+20*p^21*q^124+212*p^ 19*q^126+46*p^20*q^124+56*p^18*q^126+p^16*q^128+4*p^21*q^122+368*p^19*q^124+18* p^17*q^126+40*p^20*q^122+82*p^18*q^124+248*p^19*q^122+26*p^17*q^124+62*p^20*q^ 120+182*p^18*q^122+150*p^19*q^120+6*p^17*q^122+4*p^20*q^118+386*p^18*q^120+12*p ^16*q^122+209*p^19*q^118+40*p^17*q^120+4*p^20*q^116+210*p^18*q^118+88*p^19*q^ 116+130*p^17*q^118+2*p^15*q^120+422*p^18*q^116+4*p^16*q^118+2*p^21*q^112+65*p^ 19*q^114+132*p^17*q^116+280*p^18*q^114+20*p^16*q^116+12*p^19*q^112+232*p^17*q^ 114+12*p^20*q^110+322*p^18*q^112+16*p^16*q^114+8*p^19*q^110+362*p^17*q^112+2*p^ 15*q^114+180*p^18*q^110+78*p^16*q^112+24*p^19*q^108+342*p^17*q^110+60*p^18*q^ 108+134*p^16*q^110+20*p^19*q^106+538*p^17*q^108+8*p^15*q^110+84*p^18*q^106+132* p^16*q^108+274*p^17*q^106+32*p^15*q^108+70*p^18*q^104+320*p^16*q^106+396*p^17*q ^104+16*p^15*q^106+16*p^18*q^102+340*p^16*q^104+4*p^14*q^106+179*p^17*q^102+116 *p^15*q^104+18*p^18*q^100+440*p^16*q^102+118*p^17*q^100+90*p^15*q^102+8*p^18*q^ 98+605*p^16*q^100+20*p^14*q^102+102*p^17*q^98+140*p^15*q^100+336*p^16*q^98+36*p ^14*q^100+90*p^17*q^96+314*p^15*q^98+344*p^16*q^96+8*p^14*q^98+6*p^17*q^94+486* p^15*q^96+13*p^13*q^98+296*p^16*q^94+60*p^14*q^96+16*p^17*q^92+400*p^15*q^94+ 162*p^16*q^92+116*p^14*q^94+2*p^12*q^96+564*p^15*q^92+8*p^13*q^94+124*p^16*q^90 +198*p^14*q^92+440*p^15*q^90+10*p^13*q^92+44*p^16*q^88+266*p^14*q^90+4*p^17*q^ 86+456*p^15*q^88+14*p^13*q^90+4*p^16*q^86+486*p^14*q^88+297*p^15*q^86+80*p^13*q ^88+24*p^16*q^84+394*p^14*q^86+114*p^15*q^84+134*p^13*q^86+12*p^16*q^82+660*p^ 14*q^84+12*p^12*q^86+134*p^15*q^82+114*p^13*q^84+394*p^14*q^82+24*p^12*q^84+80* p^15*q^80+297*p^13*q^82+486*p^14*q^80+4*p^12*q^82+14*p^15*q^78+456*p^13*q^80+4* p^11*q^82+266*p^14*q^78+44*p^12*q^80+10*p^15*q^76+440*p^13*q^78+198*p^14*q^76+ 124*p^12*q^78+8*p^15*q^74+564*p^13*q^76+2*p^16*q^72+116*p^14*q^74+162*p^12*q^76 +400*p^13*q^74+16*p^11*q^76+60*p^14*q^72+296*p^12*q^74+13*p^15*q^70+486*p^13*q^ 72+6*p^11*q^74+8*p^14*q^70+344*p^12*q^72+314*p^13*q^70+90*p^11*q^72+36*p^14*q^ 68+336*p^12*q^70+140*p^13*q^68+102*p^11*q^70+20*p^14*q^66+605*p^12*q^68+8*p^10* q^70+90*p^13*q^66+118*p^11*q^68+440*p^12*q^66+18*p^10*q^68+116*p^13*q^64+179*p^ 11*q^66+4*p^14*q^62+340*p^12*q^64+16*p^10*q^66+16*p^13*q^62+396*p^11*q^64+320*p ^12*q^62+70*p^10*q^64+32*p^13*q^60+274*p^11*q^62+132*p^12*q^60+84*p^10*q^62+8*p ^13*q^58+538*p^11*q^60+20*p^9*q^62+134*p^12*q^58+60*p^10*q^60+342*p^11*q^58+24* p^9*q^60+78*p^12*q^56+180*p^10*q^58+2*p^13*q^54+362*p^11*q^56+8*p^9*q^58+16*p^ 12*q^54+322*p^10*q^56+12*p^8*q^58+232*p^11*q^54+12*p^9*q^56+20*p^12*q^52+280*p^ 10*q^54+132*p^11*q^52+65*p^9*q^54+2*p^7*q^56+4*p^12*q^50+422*p^10*q^52+2*p^13*q ^48+130*p^11*q^50+88*p^9*q^52+210*p^10*q^50+4*p^8*q^52+40*p^11*q^48+209*p^9*q^ 50+12*p^12*q^46+386*p^10*q^48+4*p^8*q^50+6*p^11*q^46+150*p^9*q^48+182*p^10*q^46 +62*p^8*q^48+26*p^11*q^44+248*p^9*q^46+82*p^10*q^44+40*p^8*q^46+18*p^11*q^42+ 368*p^9*q^44+4*p^7*q^46+p^12*q^40+56*p^10*q^42+46*p^8*q^44+212*p^9*q^42+20*p^7* q^44+82*p^10*q^40+104*p^8*q^42+6*p^11*q^38+196*p^9*q^40+8*p^10*q^38+226*p^8*q^ 40+4*p^6*q^42+148*p^9*q^38+12*p^7*q^40+19*p^10*q^36+210*p^8*q^38+90*p^9*q^36+28 *p^7*q^38+8*p^10*q^34+179*p^8*q^36+52*p^9*q^34+92*p^7*q^36+206*p^8*q^34+40*p^9* q^32+75*p^7*q^34+141*p^8*q^32+8*p^6*q^34+4*p^9*q^30+170*p^7*q^32+104*p^8*q^30+ 28*p^6*q^32+6*p^9*q^28+122*p^7*q^30+36*p^8*q^28+28*p^6*q^30+198*p^7*q^28+6*p^5* q^30+38*p^8*q^26+36*p^6*q^28+96*p^7*q^26+3*p^8*q^24+104*p^6*q^26+90*p^7*q^24+ 132*p^6*q^24+49*p^7*q^22+12*p^5*q^24+76*p^6*q^22+58*p^5*q^22+144*p^6*q^20+3*p^7 *q^18+26*p^5*q^20+24*p^6*q^18+8*p^4*q^20+97*p^5*q^18+18*p^6*q^16+90*p^5*q^16+21 *p^4*q^16+42*p^5*q^14+52*p^4*q^14+12*p^5*q^12+48*p^4*q^12+10*p^3*q^12+44*p^4*q^ 10+10*p^3*q^10+p^4*q^8+46*p^3*q^8+10*p^3*q^6+12*p^2*q^6+15*p^2*q^4+7*p*q^2+1 J[9](p, q), equals p^36*q^240+8*p^35*q^238+21*p^34*q^236+14*p^34*q^234+20*p^33*q^234+67*p^33*q^232 +5*p^32*q^232+12*p^33*q^230+90*p^32*q^230+12*p^33*q^228+80*p^32*q^228+38*p^31*q ^228+74*p^32*q^226+112*p^31*q^226+26*p^32*q^224+4*p^30*q^226+176*p^31*q^224+58* p^30*q^224+162*p^31*q^222+10*p^32*q^220+110*p^30*q^222+34*p^31*q^220+12*p^29*q^ 222+330*p^30*q^220+82*p^31*q^218+24*p^29*q^220+178*p^30*q^218+16*p^31*q^216+188 *p^29*q^218+236*p^30*q^216+6*p^28*q^218+266*p^29*q^216+164*p^30*q^214+36*p^28*q ^216+322*p^29*q^214+p^27*q^216+50*p^30*q^212+130*p^28*q^214+8*p^31*q^210+470*p^ 29*q^212+38*p^30*q^210+194*p^28*q^212+238*p^29*q^210+42*p^27*q^212+44*p^30*q^ 208+434*p^28*q^210+279*p^29*q^208+20*p^27*q^210+12*p^30*q^206+495*p^28*q^208+6* p^26*q^210+136*p^29*q^206+204*p^27*q^208+564*p^28*q^206+146*p^29*q^204+298*p^27 *q^206+389*p^28*q^204+32*p^26*q^206+44*p^29*q^202+527*p^27*q^204+464*p^28*q^202 +97*p^26*q^204+20*p^29*q^200+568*p^27*q^202+6*p^30*q^198+380*p^28*q^200+180*p^ 26*q^202+656*p^27*q^200+26*p^25*q^202+184*p^28*q^198+484*p^26*q^200+34*p^29*q^ 196+678*p^27*q^198+12*p^25*q^200+84*p^28*q^196+360*p^26*q^198+3*p^24*q^200+8*p^ 29*q^194+774*p^27*q^196+164*p^25*q^198+86*p^28*q^194+590*p^26*q^196+524*p^27*q^ 194+222*p^25*q^196+104*p^28*q^192+798*p^26*q^194+15*p^24*q^196+317*p^27*q^192+ 262*p^25*q^194+14*p^28*q^190+1140*p^26*q^192+80*p^24*q^194+428*p^27*q^190+326*p ^25*q^192+12*p^28*q^188+666*p^26*q^190+108*p^24*q^192+176*p^27*q^188+776*p^25*q ^190+14*p^23*q^192+1080*p^26*q^188+136*p^24*q^190+4*p^29*q^184+148*p^27*q^186+ 714*p^25*q^188+12*p^23*q^190+674*p^26*q^186+214*p^24*q^188+30*p^27*q^184+1034*p ^25*q^186+86*p^23*q^188+24*p^28*q^182+730*p^26*q^184+296*p^24*q^186+24*p^27*q^ 182+1290*p^25*q^184+42*p^23*q^186+4*p^28*q^180+420*p^26*q^182+502*p^24*q^184+20 *p^22*q^186+48*p^27*q^180+1146*p^25*q^182+88*p^23*q^184+190*p^26*q^180+876*p^24 *q^182+32*p^22*q^184+60*p^27*q^178+1430*p^25*q^180+76*p^23*q^182+178*p^26*q^178 +884*p^24*q^180+10*p^22*q^182+4*p^27*q^176+854*p^25*q^178+320*p^23*q^180+14*p^ 21*q^182+168*p^26*q^176+1432*p^24*q^178+4*p^27*q^174+1010*p^25*q^176+354*p^23*q ^178+94*p^26*q^174+1305*p^24*q^176+50*p^22*q^178+2*p^20*q^180+430*p^25*q^174+ 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