Generating functions and Limiting Densitiy for maximal sitting arrangements in 1 row where the Social Distancing requires at least b empty seats between two poeple for b from 1 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 you need at least b empty seats between people, for \ b from 1 to, 8 no emtpy seat can be occupied without violating this. Let f[b](z,x) be the b\ i-variate generating function infinity / n \ ----- |----- | \ | \ m| n f[b](z, x) = ) | ) A[b](m, n) z | x / | / | ----- |----- | n = 0 \m = 0 / It seems that a closed-form expression for SYMBOLIC (i.e. ALL) b is (2 b + 2) (b + 1) (b + 2) 2 x z - x z - x z + x z + (-1 + x) -------------------------------------------------------- (2 b + 2) (b + 1) (-1 + x) (-x z + x z + x - 1) and in Maple notation (x^(2*b+2)*z-x^(b+1)*z-x^(b+2)*z+x*z+(-1+x)^2)/(-1+x)/(-x^(2*b+2)*z+x^(b+1)*z+x -1) We have: 2 x z + x z + 1 f[1](z, x) = - --------------- 3 2 x z + x z - 1 and in Maple notation f[1](z,x) = -(x^2*z+x*z+1)/(x^3*z+x^2*z-1) The conjecture is confirmed for b=, 1 The limiting average density of occupied chairs, as n goes to infinity is .4114955887 4 3 2 x z + 2 x z + 2 x z + x z + 1 f[2](z, x) = - -------------------------------- 5 4 3 x z + x z + x z - 1 and in Maple notation f[2](z,x) = -(x^4*z+2*x^3*z+2*x^2*z+x*z+1)/(x^5*z+x^4*z+x^3*z-1) The conjecture is confirmed for b=, 2 The limiting average density of occupied chairs, as n goes to infinity is .2621257653 6 5 4 3 2 x z + 2 x z + 3 x z + 3 x z + 2 x z + x z + 1 f[3](z, x) = - -------------------------------------------------- 7 6 5 4 x z + x z + x z + x z - 1 and in Maple notation f[3](z,x) = -(x^6*z+2*x^5*z+3*x^4*z+3*x^3*z+2*x^2*z+x*z+1)/(x^7*z+x^6*z+x^5*z+x ^4*z-1) The conjecture is confirmed for b=, 3 The limiting average density of occupied chairs, as n goes to infinity is .1929946521 f[4](z, x) = 8 7 6 5 4 3 2 x z + 2 x z + 3 x z + 4 x z + 4 x z + 3 x z + 2 x z + x z + 1 - -------------------------------------------------------------------- 9 8 7 6 5 x z + x z + x z + x z + x z - 1 and in Maple notation f[4](z,x) = -(x^8*z+2*x^7*z+3*x^6*z+4*x^5*z+4*x^4*z+3*x^3*z+2*x^2*z+x*z+1)/(x^9 *z+x^8*z+x^7*z+x^6*z+x^5*z-1) The conjecture is confirmed for b=, 4 The limiting average density of occupied chairs, as n goes to infinity is .1530111140 10 9 8 7 6 5 4 f[5](z, x) = - (x z + 2 x z + 3 x z + 4 x z + 5 x z + 5 x z + 4 x z 3 2 / + 3 x z + 2 x z + x z + 1) / ( / 11 10 9 8 7 6 x z + x z + x z + x z + x z + x z - 1) and in Maple notation f[5](z,x) = -(x^10*z+2*x^9*z+3*x^8*z+4*x^7*z+5*x^6*z+5*x^5*z+4*x^4*z+3*x^3*z+2* x^2*z+x*z+1)/(x^11*z+x^10*z+x^9*z+x^8*z+x^7*z+x^6*z-1) The conjecture is confirmed for b=, 5 The limiting average density of occupied chairs, as n goes to infinity is .1269088748 12 11 10 9 8 7 6 f[6](z, x) = - (x z + 2 x z + 3 x z + 4 x z + 5 x z + 6 x z + 6 x z 5 4 3 2 / + 5 x z + 4 x z + 3 x z + 2 x z + x z + 1) / ( / 13 12 11 10 9 8 7 x z + x z + x z + x z + x z + x z + x z - 1) and in Maple notation f[6](z,x) = -(x^12*z+2*x^11*z+3*x^10*z+4*x^9*z+5*x^8*z+6*x^7*z+6*x^6*z+5*x^5*z+ 4*x^4*z+3*x^3*z+2*x^2*z+x*z+1)/(x^13*z+x^12*z+x^11*z+x^10*z+x^9*z+x^8*z+x^7*z-1 ) The conjecture is confirmed for b=, 6 The limiting average density of occupied chairs, as n goes to infinity is .1085092606 14 13 12 11 10 9 8 f[7](z, x) = - (x z + 2 x z + 3 x z + 4 x z + 5 x z + 6 x z + 7 x z 7 6 5 4 3 2 / + 7 x z + 6 x z + 5 x z + 4 x z + 3 x z + 2 x z + x z + 1) / ( / 15 14 13 12 11 10 9 8 x z + x z + x z + x z + x z + x z + x z + x z - 1) and in Maple notation f[7](z,x) = -(x^14*z+2*x^13*z+3*x^12*z+4*x^11*z+5*x^10*z+6*x^9*z+7*x^8*z+7*x^7* z+6*x^6*z+5*x^5*z+4*x^4*z+3*x^3*z+2*x^2*z+x*z+1)/(x^15*z+x^14*z+x^13*z+x^12*z+x ^11*z+x^10*z+x^9*z+x^8*z-1) The conjecture is confirmed for b=, 7 The limiting average density of occupied chairs, as n goes to infinity is .9483096439e-1 16 15 14 13 12 11 f[8](z, x) = - (x z + 2 x z + 3 x z + 4 x z + 5 x z + 6 x z 10 9 8 7 6 5 4 3 + 7 x z + 8 x z + 8 x z + 7 x z + 6 x z + 5 x z + 4 x z + 3 x z 2 / + 2 x z + x z + 1) / ( / 17 16 15 14 13 12 11 10 9 x z + x z + x z + x z + x z + x z + x z + x z + x z - 1) and in Maple notation f[8](z,x) = -(x^16*z+2*x^15*z+3*x^14*z+4*x^13*z+5*x^12*z+6*x^11*z+7*x^10*z+8*x^ 9*z+8*x^8*z+7*x^7*z+6*x^6*z+5*x^5*z+4*x^4*z+3*x^3*z+2*x^2*z+x*z+1)/(x^17*z+x^16 *z+x^15*z+x^14*z+x^13*z+x^12*z+x^11*z+x^10*z+x^9*z-1) The conjecture is confirmed for b=, 8 The limiting average density of occupied chairs, as n goes to infinity is .8425739694e-1 ------------------------------- 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^4*z+2*x^3*z+2*x^2*z+x*z+1)/(x^5*z+x^4*z+x^ 3*z-1), -(x^6*z+2*x^5*z+3*x^4*z+3*x^3*z+2*x^2*z+x*z+1)/(x^7*z+x^6*z+x^5*z+x^4*z -1), -(x^8*z+2*x^7*z+3*x^6*z+4*x^5*z+4*x^4*z+3*x^3*z+2*x^2*z+x*z+1)/(x^9*z+x^8* z+x^7*z+x^6*z+x^5*z-1), -(x^10*z+2*x^9*z+3*x^8*z+4*x^7*z+5*x^6*z+5*x^5*z+4*x^4* z+3*x^3*z+2*x^2*z+x*z+1)/(x^11*z+x^10*z+x^9*z+x^8*z+x^7*z+x^6*z-1), -(x^12*z+2* x^11*z+3*x^10*z+4*x^9*z+5*x^8*z+6*x^7*z+6*x^6*z+5*x^5*z+4*x^4*z+3*x^3*z+2*x^2*z +x*z+1)/(x^13*z+x^12*z+x^11*z+x^10*z+x^9*z+x^8*z+x^7*z-1), -(x^14*z+2*x^13*z+3* x^12*z+4*x^11*z+5*x^10*z+6*x^9*z+7*x^8*z+7*x^7*z+6*x^6*z+5*x^5*z+4*x^4*z+3*x^3* z+2*x^2*z+x*z+1)/(x^15*z+x^14*z+x^13*z+x^12*z+x^11*z+x^10*z+x^9*z+x^8*z-1), -(x ^16*z+2*x^15*z+3*x^14*z+4*x^13*z+5*x^12*z+6*x^11*z+7*x^10*z+8*x^9*z+8*x^8*z+7*x ^7*z+6*x^6*z+5*x^5*z+4*x^4*z+3*x^3*z+2*x^2*z+x*z+1)/(x^17*z+x^16*z+x^15*z+x^14* z+x^13*z+x^12*z+x^11*z+x^10*z+x^9*z-1)] and the list of limiting average densities from b=1 to b=, 8, is [.4114955887, .2621257653, .1929946521, .1530111140, .1269088748, .1085092606, .9483096439e-1, .8425739694e-1] and we also confimed the conjecture for all b from 1 to, 8 ------------------------------ This ends this article that took, 6.566, second to generate