Generating functions and Limiting Densitiy for maximal sitting arrangements \ in 1 row where the Social Distancing prohibits b adjacent seats for b f\ rom 2 to, 8 By Shalosh B. Ekhad Let A[b](m,n) be the number of ways that m people can be seated in a row of \ n chairs such that is is forbidden to have more than b consecutive occup\ ied seatgs and such that no emtpy seat can be occupied without violating this restriction. Let f[b](z\ ,x) be the bi-variate generating function infinity / n \ ----- |----- | \ | \ m| n f[b](z, x) = ) | ) A[b](m, n) z | x / | / | ----- |----- | n = 0 \m = 0 / We have 2 x z + x z + 1 f[2](z, x) = - --------------- 3 2 x z + x z - 1 and in Maple notation f[2](z,x) = -(x^2*z+x*z+1)/(x^3*z+x^2*z-1) The limiting average density of occupied chairs, as n goes to infinity is .4114955887 5 3 3 2 2 2 2 x z - x z - x z + x z - x z - 1 f[3](z, x) = - -------------------------------------- 6 3 4 2 3 2 2 x z - x z - x z - x z + 1 and in Maple notation f[3](z,x) = -(x^5*z^3-x^3*z^2-x^2*z^2+x^2*z-x*z-1)/(x^6*z^3-x^4*z^2-x^3*z^2-x^2 *z+1) The limiting average density of occupied chairs, as n goes to infinity is .5772029462 f[4](z, x) = 9 6 7 5 5 4 5 3 4 3 3 3 3 2 2 2 x z + x z - x z + x z - 2 x z - x z + x z - x z - x z - 1 - ------------------------------------------------------------------------- 10 6 8 5 6 4 5 3 4 3 3 2 x z + x z - x z - 2 x z - x z - x z + 1 and in Maple notation f[4](z,x) = -(x^9*z^6+x^7*z^5-x^5*z^4+x^5*z^3-2*x^4*z^3-x^3*z^3+x^3*z^2-x^2*z^2 -x*z-1)/(x^10*z^6+x^8*z^5-x^6*z^4-2*x^5*z^3-x^4*z^3-x^3*z^2+1) The limiting average density of occupied chairs, as n goes to infinity is .6686427921 14 10 11 8 9 7 9 6 8 6 6 5 6 4 f[5](z, x) = - (x z - x z - 2 x z + x z - 2 x z + x z - x z 5 4 4 4 4 3 3 3 3 2 2 2 / + 2 x z + x z - 2 x z + x z - x z + x z + x z + 1) / ( / 15 10 12 8 10 7 9 6 7 5 6 4 5 4 4 3 x z - x z - 2 x z - 2 x z + x z + 2 x z + x z + x z 3 2 + x z - 1) and in Maple notation f[5](z,x) = -(x^14*z^10-x^11*z^8-2*x^9*z^7+x^9*z^6-2*x^8*z^6+x^6*z^5-x^6*z^4+2* x^5*z^4+x^4*z^4-2*x^4*z^3+x^3*z^3-x^3*z^2+x^2*z^2+x*z+1)/(x^15*z^10-x^12*z^8-2* x^10*z^7-2*x^9*z^6+x^7*z^5+2*x^6*z^4+x^5*z^4+x^4*z^3+x^3*z^2-1) The limiting average density of occupied chairs, as n goes to infinity is .7269949175 20 15 17 13 14 11 14 10 13 10 11 9 f[6](z, x) = - (x z + x z - 2 x z + x z - 3 x z - 2 x z 11 8 10 8 8 7 8 6 7 6 7 5 6 5 + x z - 2 x z + x z - 2 x z + 2 x z - 2 x z + 3 x z 5 5 5 4 4 4 4 3 3 3 2 2 / 21 15 + x z - 2 x z + x z - x z + x z + x z + x z + 1) / (x z / 18 13 15 11 14 10 12 9 11 8 9 7 8 6 + x z - 2 x z - 3 x z - 2 x z - 2 x z + x z + 2 x z 7 5 6 5 5 4 4 3 + 3 x z + x z + x z + x z - 1) and in Maple notation f[6](z,x) = -(x^20*z^15+x^17*z^13-2*x^14*z^11+x^14*z^10-3*x^13*z^10-2*x^11*z^9+ x^11*z^8-2*x^10*z^8+x^8*z^7-2*x^8*z^6+2*x^7*z^6-2*x^7*z^5+3*x^6*z^5+x^5*z^5-2*x ^5*z^4+x^4*z^4-x^4*z^3+x^3*z^3+x^2*z^2+x*z+1)/(x^21*z^15+x^18*z^13-2*x^15*z^11-\ 3*x^14*z^10-2*x^12*z^9-2*x^11*z^8+x^9*z^7+2*x^8*z^6+3*x^7*z^5+x^6*z^5+x^5*z^4+x ^4*z^3-1) The limiting average density of occupied chairs, as n goes to infinity is .7675902978 27 21 23 18 20 16 20 15 19 15 16 13 f[7](z, x) = - (x z - x z - 3 x z + x z - 3 x z + 2 x z 16 12 15 12 13 11 13 10 12 10 12 9 - x z + 3 x z + 3 x z - 3 x z + 4 x z - 2 x z 11 9 9 8 9 7 8 7 8 6 7 6 6 6 + 3 x z - x z + 2 x z - 2 x z + 2 x z - 3 x z - x z 6 5 5 5 5 4 4 4 4 3 3 3 2 2 + 3 x z - x z + 2 x z - x z + x z - x z - x z - x z - 1) / 28 21 24 18 21 16 20 15 17 13 16 12 / (x z - x z - 3 x z - 3 x z + 2 x z + 3 x z / 14 11 13 10 12 9 10 8 9 7 8 6 7 6 + 3 x z + 4 x z + 3 x z - x z - 2 x z - 3 x z - x z 6 5 5 4 4 3 - x z - x z - x z + 1) and in Maple notation f[7](z,x) = -(x^27*z^21-x^23*z^18-3*x^20*z^16+x^20*z^15-3*x^19*z^15+2*x^16*z^13 -x^16*z^12+3*x^15*z^12+3*x^13*z^11-3*x^13*z^10+4*x^12*z^10-2*x^12*z^9+3*x^11*z^ 9-x^9*z^8+2*x^9*z^7-2*x^8*z^7+2*x^8*z^6-3*x^7*z^6-x^6*z^6+3*x^6*z^5-x^5*z^5+2*x ^5*z^4-x^4*z^4+x^4*z^3-x^3*z^3-x^2*z^2-x*z-1)/(x^28*z^21-x^24*z^18-3*x^21*z^16-\ 3*x^20*z^15+2*x^17*z^13+3*x^16*z^12+3*x^14*z^11+4*x^13*z^10+3*x^12*z^9-x^10*z^8 -2*x^9*z^7-3*x^8*z^6-x^7*z^6-x^6*z^5-x^5*z^4-x^4*z^3+1) The limiting average density of occupied chairs, as n goes to infinity is .7975140257 35 28 31 25 27 22 27 21 26 21 23 19 f[8](z, x) = - (x z + x z - 3 x z + x z - 4 x z - 3 x z 23 18 22 18 19 16 19 15 18 15 18 14 + x z - 3 x z + 3 x z - 3 x z + 6 x z - 3 x z 17 14 15 13 15 12 14 12 14 11 13 11 + 6 x z + 3 x z - 3 x z + 4 x z - 2 x z + 3 x z 11 10 11 9 10 9 10 8 9 8 9 7 8 7 - x z + 3 x z - 2 x z + 4 x z - 3 x z + 3 x z - 4 x z 7 7 7 6 6 6 6 5 5 5 5 4 4 4 3 3 - x z + 3 x z - x z + 2 x z - x z + x z - x z - x z 2 2 / 36 28 32 25 28 22 27 21 - x z - x z - 1) / (x z + x z - 3 x z - 4 x z / 24 19 23 18 20 16 19 15 18 14 16 13 - 3 x z - 3 x z + 3 x z + 6 x z + 6 x z + 3 x z 15 12 14 11 12 10 11 9 10 8 9 7 8 7 + 4 x z + 3 x z - x z - 2 x z - 3 x z - 4 x z - x z 7 6 6 5 5 4 - x z - x z - x z + 1) and in Maple notation f[8](z,x) = -(x^35*z^28+x^31*z^25-3*x^27*z^22+x^27*z^21-4*x^26*z^21-3*x^23*z^19 +x^23*z^18-3*x^22*z^18+3*x^19*z^16-3*x^19*z^15+6*x^18*z^15-3*x^18*z^14+6*x^17*z ^14+3*x^15*z^13-3*x^15*z^12+4*x^14*z^12-2*x^14*z^11+3*x^13*z^11-x^11*z^10+3*x^ 11*z^9-2*x^10*z^9+4*x^10*z^8-3*x^9*z^8+3*x^9*z^7-4*x^8*z^7-x^7*z^7+3*x^7*z^6-x^ 6*z^6+2*x^6*z^5-x^5*z^5+x^5*z^4-x^4*z^4-x^3*z^3-x^2*z^2-x*z-1)/(x^36*z^28+x^32* z^25-3*x^28*z^22-4*x^27*z^21-3*x^24*z^19-3*x^23*z^18+3*x^20*z^16+6*x^19*z^15+6* x^18*z^14+3*x^16*z^13+4*x^15*z^12+3*x^14*z^11-x^12*z^10-2*x^11*z^9-3*x^10*z^8-4 *x^9*z^7-x^8*z^7-x^7*z^6-x^6*z^5-x^5*z^4+1) The limiting average density of occupied chairs, as n goes to infinity is .8205096203 ------------------------------- To sum up the list of generating functions from b=1 to b=, 8, is [-(x^2*z+x*z+1)/(x^3*z+x^2*z-1), -(x^5*z^3-x^3*z^2-x^2*z^2+x^2*z-x*z-1)/(x^6*z^ 3-x^4*z^2-x^3*z^2-x^2*z+1), -(x^9*z^6+x^7*z^5-x^5*z^4+x^5*z^3-2*x^4*z^3-x^3*z^3 +x^3*z^2-x^2*z^2-x*z-1)/(x^10*z^6+x^8*z^5-x^6*z^4-2*x^5*z^3-x^4*z^3-x^3*z^2+1), -(x^14*z^10-x^11*z^8-2*x^9*z^7+x^9*z^6-2*x^8*z^6+x^6*z^5-x^6*z^4+2*x^5*z^4+x^4* z^4-2*x^4*z^3+x^3*z^3-x^3*z^2+x^2*z^2+x*z+1)/(x^15*z^10-x^12*z^8-2*x^10*z^7-2*x ^9*z^6+x^7*z^5+2*x^6*z^4+x^5*z^4+x^4*z^3+x^3*z^2-1), -(x^20*z^15+x^17*z^13-2*x^ 14*z^11+x^14*z^10-3*x^13*z^10-2*x^11*z^9+x^11*z^8-2*x^10*z^8+x^8*z^7-2*x^8*z^6+ 2*x^7*z^6-2*x^7*z^5+3*x^6*z^5+x^5*z^5-2*x^5*z^4+x^4*z^4-x^4*z^3+x^3*z^3+x^2*z^2 +x*z+1)/(x^21*z^15+x^18*z^13-2*x^15*z^11-3*x^14*z^10-2*x^12*z^9-2*x^11*z^8+x^9* z^7+2*x^8*z^6+3*x^7*z^5+x^6*z^5+x^5*z^4+x^4*z^3-1), -(x^27*z^21-x^23*z^18-3*x^ 20*z^16+x^20*z^15-3*x^19*z^15+2*x^16*z^13-x^16*z^12+3*x^15*z^12+3*x^13*z^11-3*x ^13*z^10+4*x^12*z^10-2*x^12*z^9+3*x^11*z^9-x^9*z^8+2*x^9*z^7-2*x^8*z^7+2*x^8*z^ 6-3*x^7*z^6-x^6*z^6+3*x^6*z^5-x^5*z^5+2*x^5*z^4-x^4*z^4+x^4*z^3-x^3*z^3-x^2*z^2 -x*z-1)/(x^28*z^21-x^24*z^18-3*x^21*z^16-3*x^20*z^15+2*x^17*z^13+3*x^16*z^12+3* x^14*z^11+4*x^13*z^10+3*x^12*z^9-x^10*z^8-2*x^9*z^7-3*x^8*z^6-x^7*z^6-x^6*z^5-x ^5*z^4-x^4*z^3+1), -(x^35*z^28+x^31*z^25-3*x^27*z^22+x^27*z^21-4*x^26*z^21-3*x^ 23*z^19+x^23*z^18-3*x^22*z^18+3*x^19*z^16-3*x^19*z^15+6*x^18*z^15-3*x^18*z^14+6 *x^17*z^14+3*x^15*z^13-3*x^15*z^12+4*x^14*z^12-2*x^14*z^11+3*x^13*z^11-x^11*z^ 10+3*x^11*z^9-2*x^10*z^9+4*x^10*z^8-3*x^9*z^8+3*x^9*z^7-4*x^8*z^7-x^7*z^7+3*x^7 *z^6-x^6*z^6+2*x^6*z^5-x^5*z^5+x^5*z^4-x^4*z^4-x^3*z^3-x^2*z^2-x*z-1)/(x^36*z^ 28+x^32*z^25-3*x^28*z^22-4*x^27*z^21-3*x^24*z^19-3*x^23*z^18+3*x^20*z^16+6*x^19 *z^15+6*x^18*z^14+3*x^16*z^13+4*x^15*z^12+3*x^14*z^11-x^12*z^10-2*x^11*z^9-3*x^ 10*z^8-4*x^9*z^7-x^8*z^7-x^7*z^6-x^6*z^5-x^5*z^4+1)] and the list of limiting average densities from b=1 to b=, 8, is [.4114955887, .5772029462, .6686427921, .7269949175, .7675902978, .7975140257, .8205096203] ------------------------------ This ends this article that took, 99.282, second to generate