July 2nd, 2014, 11:51 am
Let [$]a(w, m)[$] is number of such pairs formed with w women and m men if a woman is in the leftmost seat, and [$]b(w, m)[$] - number of pairs if a man is in the leftmost seat. Then we have [$]a(w,m)=\frac{(w+m-2)!}{(w-1)!(m-1)!}+b(w-1,m)+a(w-1,m)\,,\quad b(w,m)=\frac{(w+m-2)!}{(w-1)!(m-1)!}+a(w,m-1)+b(w,m-1)[$]. Introduce the transform[$]A(x,y)=\sum_{w=1}^\infty\sum_{m=1}^\infty a(w,m)x^wy^m[$] and same for [$]B(x,y)[$]. Then [$]A, B[$] satisfy[$]A=\frac{xy}{(1-x - y)}+xA+xB\,,\quad B=\frac{xy}{(1-x-y)}+yA+yB[$]. This can be solved, for the quantity of interest [$]A+B=\frac{2xy}{(1-x-y)^2}[$]. All that remains to do is to find the coefficient in front of [$]x^7y^8[$].
Last edited by
EBal on July 1st, 2014, 10:00 pm, edited 1 time in total.