The first, 5, bi-weighted-generating functions for the number of tilings of a\ n m by n rectangle by rectangles, according to the bi-weight [number of \ horiz. edges, number of vertical edges is [-(w1^2*w2^2*x-w1^2*w2*x-w1^2*x+1)/(w1^2*w2*x+w1^2*x-1), -(2*w1^5*w2^5*x^2+2*w1 ^5*w2^4*x^2-2*w1^5*w2^3*x^2-2*w1^5*w2^2*x^2-2*w1^5*w2*x^2-w1^3*w2^4*x-w1^5*x^2- w1^2*w2^4*x+w1^3*w2^2*x+2*w1^3*w2*x+w1^2*w2^2*x+w1^3*x+w1^2*x-1)/(2*w1^5*w2^3*x ^2+2*w1^5*w2^2*x^2+2*w1^5*w2*x^2+w1^5*x^2-w1^3*w2^2*x-2*w1^3*w2*x-w1^2*w2^2*x- w1^3*x-w1^2*x+1), -(w1^9*w2^10*x^3+6*w1^9*w2^9*x^3+10*w1^9*w2^8*x^3+10*w1^9*w2^ 7*x^3-2*w1^9*w2^6*x^3-10*w1^9*w2^5*x^3-3*w1^7*w2^8*x^2-12*w1^9*w2^4*x^3-7*w1^7* w2^7*x^2-3*w1^6*w2^8*x^2-10*w1^9*w2^3*x^3-3*w1^7*w2^6*x^2-3*w1^6*w2^7*x^2-w1^5* w2^8*x^2-7*w1^9*w2^2*x^3+3*w1^7*w2^5*x^2-2*w1^6*w2^6*x^2-w1^5*w2^7*x^2-4*w1^9* w2*x^3+8*w1^7*w2^4*x^2+3*w1^6*w2^5*x^2-3*w1^5*w2^6*x^2-w1^9*x^3+9*w1^7*w2^3*x^2 +3*w1^6*w2^4*x^2+w1^5*w2^5*x^2+7*w1^7*w2^2*x^2+2*w1^6*w2^3*x^2+w1^5*w2^4*x^2+w1 ^4*w2^6*x+4*w1^7*w2*x^2+3*w1^6*w2^2*x^2+3*w1^5*w2^3*x^2+2*w1^3*w2^6*x+w1^7*x^2+ 3*w1^6*w2*x^2+w1^5*w2^2*x^2+w1^2*w2^6*x+w1^6*x^2+w1^5*w2*x^2-w1^4*w2^3*x+w1^5*x ^2-3*w1^4*w2^2*x-2*w1^3*w2^3*x-3*w1^4*w2*x-w1^3*w2^2*x-w1^2*w2^3*x-w1^4*x-w1^3* w2*x-w1^3*x-w1^2*x+1)/(w1^9*w2^7*x^3+6*w1^9*w2^6*x^3+10*w1^9*w2^5*x^3+12*w1^9* w2^4*x^3+10*w1^9*w2^3*x^3+7*w1^9*w2^2*x^3-3*w1^7*w2^5*x^2+4*w1^9*w2*x^3-8*w1^7* w2^4*x^2-3*w1^6*w2^5*x^2+w1^9*x^3-9*w1^7*w2^3*x^2-3*w1^6*w2^4*x^2-w1^5*w2^5*x^2 -7*w1^7*w2^2*x^2-2*w1^6*w2^3*x^2-w1^5*w2^4*x^2-4*w1^7*w2*x^2-3*w1^6*w2^2*x^2-3* w1^5*w2^3*x^2-w1^7*x^2-3*w1^6*w2*x^2-w1^5*w2^2*x^2-w1^6*x^2-w1^5*w2*x^2+w1^4*w2 ^3*x-w1^5*x^2+3*w1^4*w2^2*x+2*w1^3*w2^3*x+3*w1^4*w2*x+w1^3*w2^2*x+w1^2*w2^3*x+ w1^4*x+w1^3*w2*x+w1^3*x+w1^2*x-1), -(14*w1^21*w2^20*x^6+78*w1^21*w2^19*x^6+246* w1^21*w2^18*x^6+560*w1^21*w2^17*x^6+964*w1^21*w2^16*x^6+1276*w1^21*w2^15*x^6-15 *w1^19*w2^18*x^5+1260*w1^21*w2^14*x^6-98*w1^19*w2^17*x^5-27*w1^18*w2^18*x^5+784 *w1^21*w2^13*x^6-302*w1^19*w2^16*x^5-134*w1^18*w2^17*x^5-16*w1^17*w2^18*x^5-38* w1^21*w2^12*x^6-593*w1^19*w2^15*x^5-348*w1^18*w2^16*x^5-62*w1^17*w2^17*x^5-4*w1 ^16*w2^18*x^5-904*w1^21*w2^11*x^6-812*w1^19*w2^14*x^5-637*w1^18*w2^15*x^5-160* w1^17*w2^16*x^5-12*w1^16*w2^17*x^5-1519*w1^21*w2^10*x^6-761*w1^19*w2^13*x^5-872 *w1^18*w2^14*x^5-320*w1^17*w2^15*x^5-36*w1^16*w2^16*x^5-1746*w1^21*w2^9*x^6-363 *w1^19*w2^12*x^5-873*w1^18*w2^13*x^5-479*w1^17*w2^14*x^5+14*w1^16*w2^16*x^4-72* w1^16*w2^15*x^5-1622*w1^21*w2^8*x^6+264*w1^19*w2^11*x^5-539*w1^18*w2^12*x^5-534 *w1^17*w2^13*x^5+90*w1^16*w2^15*x^4-103*w1^16*w2^14*x^5+24*w1^15*w2^16*x^4-1289 *w1^21*w2^7*x^6+880*w1^19*w2^10*x^5+44*w1^18*w2^11*x^5-388*w1^17*w2^12*x^5+256* w1^16*w2^14*x^4-118*w1^16*w2^13*x^5+133*w1^15*w2^15*x^4+22*w1^14*w2^16*x^4-897* w1^21*w2^6*x^6+1270*w1^19*w2^9*x^5+640*w1^18*w2^10*x^5-63*w1^17*w2^11*x^5+440* w1^16*w2^13*x^4-92*w1^16*w2^12*x^5+318*w1^15*w2^14*x^4+103*w1^14*w2^15*x^4+10* w1^13*w2^16*x^4-545*w1^21*w2^5*x^6+1348*w1^19*w2^8*x^5+1052*w1^18*w2^9*x^5+309* w1^17*w2^10*x^5+493*w1^16*w2^12*x^4-29*w1^16*w2^11*x^5+479*w1^15*w2^13*x^4+222* w1^14*w2^14*x^4+32*w1^13*w2^15*x^4+2*w1^12*w2^16*x^4-283*w1^21*w2^4*x^6+1168*w1 ^19*w2^7*x^5+1208*w1^18*w2^8*x^5+585*w1^17*w2^9*x^5+312*w1^16*w2^11*x^4+51*w1^ 16*w2^10*x^5+486*w1^15*w2^12*x^4+359*w1^14*w2^13*x^4+64*w1^13*w2^14*x^4+6*w1^12 *w2^15*x^4-121*w1^21*w2^3*x^6+857*w1^19*w2^6*x^5+1140*w1^18*w2^7*x^5+683*w1^17* w2^8*x^5-67*w1^16*w2^10*x^4+115*w1^16*w2^9*x^5+270*w1^15*w2^11*x^4+429*w1^14*w2 ^12*x^4-2*w1^13*w2^14*x^3+114*w1^13*w2^13*x^4+18*w1^12*w2^14*x^4-40*w1^21*w2^2* x^6+536*w1^19*w2^5*x^5+929*w1^18*w2^6*x^5+624*w1^17*w2^7*x^5-502*w1^16*w2^9*x^4 +153*w1^16*w2^8*x^5-100*w1^15*w2^10*x^4+314*w1^14*w2^11*x^4-16*w1^13*w2^13*x^3+ 134*w1^13*w2^12*x^4-7*w1^12*w2^14*x^3+32*w1^12*w2^13*x^4-9*w1^21*w2*x^6+282*w1^ 19*w2^4*x^5+656*w1^18*w2^5*x^5+493*w1^17*w2^6*x^5-830*w1^16*w2^8*x^4+155*w1^16* w2^7*x^5-452*w1^15*w2^9*x^4+50*w1^14*w2^10*x^4-44*w1^13*w2^12*x^3+109*w1^13*w2^ 11*x^4-56*w1^12*w2^13*x^3+40*w1^12*w2^12*x^4-9*w1^11*w2^14*x^3-w1^21*x^6+121*w1 ^19*w2^3*x^5+393*w1^18*w2^4*x^5+348*w1^17*w2^5*x^5-949*w1^16*w2^7*x^4+127*w1^16 *w2^6*x^5-654*w1^15*w2^8*x^4-250*w1^14*w2^9*x^4-62*w1^13*w2^11*x^3+42*w1^13*w2^ 10*x^4-141*w1^12*w2^12*x^3+47*w1^12*w2^11*x^4-58*w1^11*w2^13*x^3-5*w1^10*w2^14* x^3+40*w1^19*w2^2*x^5+191*w1^18*w2^3*x^5+216*w1^17*w2^4*x^5-859*w1^16*w2^6*x^4+ 91*w1^16*w2^5*x^5-698*w1^15*w2^7*x^4-475*w1^14*w2^8*x^4-48*w1^13*w2^10*x^3-59* w1^13*w2^9*x^4-185*w1^12*w2^11*x^3+20*w1^12*w2^10*x^4-115*w1^11*w2^12*x^3-32*w1 ^10*w2^13*x^3-w1^9*w2^14*x^3+9*w1^19*w2*x^5+70*w1^18*w2^2*x^5+114*w1^17*w2^3*x^ 5-639*w1^16*w2^5*x^4+57*w1^16*w2^4*x^5-637*w1^15*w2^6*x^4-556*w1^14*w2^7*x^4-6* w1^13*w2^9*x^3-138*w1^13*w2^8*x^4-143*w1^12*w2^10*x^3-9*w1^12*w2^9*x^4-135*w1^ 11*w2^11*x^3-52*w1^10*w2^12*x^3-12*w1^9*w2^13*x^3+w1^19*x^5+17*w1^18*w2*x^5+48* w1^17*w2^2*x^5-391*w1^16*w2^4*x^4+31*w1^16*w2^3*x^5-506*w1^15*w2^5*x^4-518*w1^ 14*w2^6*x^4+52*w1^13*w2^8*x^3-173*w1^13*w2^7*x^4-17*w1^12*w2^9*x^3-33*w1^12*w2^ 8*x^4-111*w1^11*w2^10*x^3-77*w1^10*w2^11*x^3-14*w1^9*w2^12*x^3-2*w1^8*w2^13*x^3 +2*w1^18*x^5+14*w1^17*w2*x^5-191*w1^16*w2^3*x^4+14*w1^16*w2^2*x^5-340*w1^15*w2^ 4*x^4-422*w1^14*w2^5*x^4+116*w1^13*w2^7*x^3-165*w1^13*w2^6*x^4+143*w1^12*w2^8*x ^3-59*w1^12*w2^7*x^4-11*w1^11*w2^9*x^3-73*w1^10*w2^10*x^3-29*w1^9*w2^11*x^3+2* w1^17*x^5-70*w1^16*w2^2*x^4+5*w1^16*w2*x^5-186*w1^15*w2^3*x^4-308*w1^14*w2^4*x^ 4+162*w1^13*w2^6*x^3-139*w1^13*w2^5*x^4+270*w1^12*w2^7*x^3-55*w1^12*w2^6*x^4+ 105*w1^11*w2^8*x^3-17*w1^10*w2^9*x^3+4*w1^9*w2^11*x^2-29*w1^9*w2^10*x^3-4*w1^8* w2^11*x^3-17*w1^16*w2*x^4+w1^16*x^5-78*w1^15*w2^2*x^4-193*w1^14*w2^3*x^4+164*w1 ^13*w2^5*x^3-98*w1^13*w2^4*x^4+325*w1^12*w2^6*x^3-45*w1^12*w2^5*x^4+167*w1^11* w2^7*x^3+34*w1^10*w2^8*x^3+16*w1^9*w2^10*x^2-13*w1^9*w2^9*x^3+11*w1^8*w2^11*x^2 -6*w1^8*w2^10*x^3-2*w1^16*x^4-22*w1^15*w2*x^4-94*w1^14*w2^2*x^4+125*w1^13*w2^4* x^3-62*w1^13*w2^3*x^4+310*w1^12*w2^5*x^3-33*w1^12*w2^4*x^4+183*w1^11*w2^6*x^3+ 83*w1^10*w2^7*x^3+16*w1^9*w2^9*x^2-2*w1^9*w2^8*x^3+26*w1^8*w2^10*x^2+11*w1^7*w2 ^11*x^2-3*w1^15*x^4-31*w1^14*w2*x^4+72*w1^13*w2^3*x^3-34*w1^13*w2^2*x^4+248*w1^ 12*w2^4*x^3-22*w1^12*w2^3*x^4+181*w1^11*w2^5*x^3+99*w1^10*w2^6*x^3+5*w1^9*w2^8* x^2+29*w1^9*w2^7*x^3+19*w1^8*w2^9*x^2-4*w1^8*w2^8*x^3+20*w1^7*w2^10*x^2+5*w1^6* w2^11*x^2-5*w1^14*x^4+30*w1^13*w2^2*x^3-14*w1^13*w2*x^4+162*w1^12*w2^3*x^3-12* w1^12*w2^2*x^4+155*w1^11*w2^4*x^3+93*w1^10*w2^5*x^3-4*w1^9*w2^7*x^2+31*w1^9*w2^ 6*x^3+8*w1^8*w2^8*x^2+4*w1^8*w2^7*x^3+19*w1^7*w2^9*x^2+8*w1^6*w2^10*x^2+w1^5*w2 ^11*x^2+8*w1^13*w2*x^3-3*w1^13*x^4+80*w1^12*w2^2*x^3-6*w1^12*w2*x^4+114*w1^11* w2^3*x^3+74*w1^10*w2^4*x^3-20*w1^9*w2^6*x^2+37*w1^9*w2^5*x^3-11*w1^8*w2^7*x^2+6 *w1^8*w2^6*x^3+14*w1^7*w2^8*x^2+5*w1^6*w2^9*x^2+2*w1^5*w2^10*x^2+w1^13*x^3+26* 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