The Generating Function for Lecture-Hall partitions with, j, parts By Shalosh B. Ekhad We will derive generating functions for the so-called Lecture Hall Partitions first defined (and proved!) by Mireille Bousquet-Melou and Kimmo Eriksson Theorem No. , 1, : The GLOBAL generating function for for Lecture Hall Partitions with, 2, parts is: 1 -------------------------------- 3 2 (t x[1] x[2] - 1) (t x[2] - 1) and in Maple input notation: 1/(t^3*x[1]*x[2]^2-1)/(t*x[2]-1) where the weight of a partition [b[j],b[j-1], ..., b[1]] is x[j]^b[j]*...*x[1]^b[1] the generating function according to the weight t^n*y^(b[j]-b[j-1]+b[j-2]+ ... ) is: 1 -------------------- 3 (t y - 1) (t y - 1) and in Maple input notation: 1/(t^3*y-1)/(t*y-1) Theorem No. , 2, : The GLOBAL generating function for for Lecture Hall Partitions with, 3, parts is: 3 2 t x[2] x[3] + 1 - ------------------------------------------------------------ 6 2 3 5 3 2 (t x[2] x[3] x[1] - 1) (-1 + t x[3] x[2] ) (t x[3] - 1) and in Maple input notation: -(t^3*x[2]*x[3]^2+1)/(t^6*x[2]^2*x[3]^3*x[1]-1)/(-1+t^5*x[3]^3*x[2]^2)/(t*x[3]-\ 1) where the weight of a partition [b[j],b[j-1], ..., b[1]] is x[j]^b[j]*...*x[1]^b[1] the generating function according to the weight t^n*y^(b[j]-b[j-1]+b[j-2]+ ... ) is: 1 - -------------------------------- 5 3 (t y - 1) (-1 + t y) (t y - 1) and in Maple input notation: -1/(t*y-1)/(-1+t^5*y)/(t^3*y-1) Theorem No. , 3, : The GLOBAL generating function for for Lecture Hall Partitions with, 4, parts is: 3 2 8 3 4 11 4 6 (t x[3] x[4] + t x[2] x[3] x[4] + t x[2] x[3] x[4] + 1 5 2 3 6 2 3 / 9 2 3 4 + t x[3] x[4] + t x[2] x[3] x[4] ) / ((-1 + t x[2] x[3] x[4] ) / 10 4 2 3 7 3 4 (t x[4] x[2] x[3] x[1] - 1) (t x[3] x[4] - 1) (t x[4] - 1)) and in Maple input notation: (t^3*x[3]*x[4]^2+t^8*x[2]*x[3]^3*x[4]^4+t^11*x[2]*x[3]^4*x[4]^6+1+t^5*x[3]^2*x[ 4]^3+t^6*x[2]*x[3]^2*x[4]^3)/(-1+t^9*x[2]^2*x[3]^3*x[4]^4)/(t^10*x[4]^4*x[2]^2* x[3]^3*x[1]-1)/(t^7*x[3]^3*x[4]^4-1)/(t*x[4]-1) where the weight of a partition [b[j],b[j-1], ..., b[1]] is x[j]^b[j]*...*x[1]^b[1] the generating function according to the weight t^n*y^(b[j]-b[j-1]+b[j-2]+ ... ) is: 1 -------------------------------------------- 7 5 3 (t y - 1) (-1 + t y) (-1 + t y) (t y - 1) and in Maple input notation: 1/(t*y-1)/(-1+t^7*y)/(-1+t^5*y)/(t^3*y-1) Theorem No. , 4, : The GLOBAL generating function for for Lecture Hall Partitions with, 5, parts is: 12 2 4 5 17 2 6 8 - (1 + t x[2] x[3] x[4] x[5] + t x[2] x[3] x[4] x[5] 14 2 5 7 15 2 5 7 5 2 3 + t x[3] x[4] x[5] + t x[2] x[3] x[4] x[5] + t x[4] x[5] 10 4 5 21 4 7 9 + t x[3] x[4] x[5] + t x[2] x[3] x[4] x[5] 16 3 5 7 26 4 9 12 + t x[2] x[3] x[4] x[5] + t x[2] x[3] x[4] x[5] 20 3 7 9 6 2 3 3 2 + t x[2] x[3] x[4] x[5] + t x[3] x[4] x[5] + t x[4] x[5] 23 4 8 10 19 4 6 8 + t x[2] x[3] x[4] x[5] + t x[2] x[3] x[4] x[5] 13 5 7 18 3 6 8 + t x[3] x[4] x[5] + t x[2] x[3] x[4] x[5] 11 2 4 5 16 2 6 8 9 2 3 4 + t x[3] x[4] x[5] + t x[3] x[4] x[5] + t x[3] x[4] x[5] 10 2 3 4 13 3 4 5 7 3 4 + t x[2] x[3] x[4] x[5] + t x[2] x[3] x[4] x[5] + t x[4] x[5] 8 3 4 / 9 4 5 + t x[3] x[4] x[5] ) / ((t x[4] x[5] - 1) / 15 5 3 4 2 14 5 4 2 3 (-1 + t x[5] x[3] x[4] x[2] x[1]) (t x[5] x[4] x[2] x[3] - 1) 12 5 3 4 (-1 + t x[5] x[3] x[4] ) (t x[5] - 1)) and in Maple input notation: -(1+t^12*x[2]*x[3]^2*x[4]^4*x[5]^5+t^17*x[2]*x[3]^2*x[4]^6*x[5]^8+t^14*x[3]^2*x [4]^5*x[5]^7+t^15*x[2]*x[3]^2*x[4]^5*x[5]^7+t^5*x[4]^2*x[5]^3+t^10*x[3]*x[4]^4* x[5]^5+t^21*x[2]*x[3]^4*x[4]^7*x[5]^9+t^16*x[2]*x[3]^3*x[4]^5*x[5]^7+t^26*x[2]* x[3]^4*x[4]^9*x[5]^12+t^20*x[2]*x[3]^3*x[4]^7*x[5]^9+t^6*x[3]*x[4]^2*x[5]^3+t^3 *x[4]*x[5]^2+t^23*x[2]*x[3]^4*x[4]^8*x[5]^10+t^19*x[2]*x[3]^4*x[4]^6*x[5]^8+t^ 13*x[3]*x[4]^5*x[5]^7+t^18*x[2]*x[3]^3*x[4]^6*x[5]^8+t^11*x[3]^2*x[4]^4*x[5]^5+ t^16*x[3]^2*x[4]^6*x[5]^8+t^9*x[3]^2*x[4]^3*x[5]^4+t^10*x[2]*x[3]^2*x[4]^3*x[5] ^4+t^13*x[2]*x[3]^3*x[4]^4*x[5]^5+t^7*x[4]^3*x[5]^4+t^8*x[3]*x[4]^3*x[5]^4)/(t^ 9*x[4]^4*x[5]^5-1)/(-1+t^15*x[5]^5*x[3]^3*x[4]^4*x[2]^2*x[1])/(t^14*x[5]^5*x[4] ^4*x[2]^2*x[3]^3-1)/(-1+t^12*x[5]^5*x[3]^3*x[4]^4)/(t*x[5]-1) where the weight of a partition [b[j],b[j-1], ..., b[1]] is x[j]^b[j]*...*x[1]^b[1] the generating function according to the weight t^n*y^(b[j]-b[j-1]+b[j-2]+ ... ) is: 1 - ------------------------------------------------------- 9 3 5 7 (t y - 1) (t y - 1) (t y - 1) (-1 + t y) (-1 + t y) and in Maple input notation: -1/(t*y-1)/(t^9*y-1)/(t^3*y-1)/(-1+t^5*y)/(-1+t^7*y)