Question 87
Let $T=\{1,2,\ldots,n\}$. For $S=\{s_1,\ldots,s_k\}\subseteq T$, with $s_1\gt s_2\gt \cdots\gt s_k$, define
$$f(S)=s_1-s_2+s_3-s_4+\cdots,$$
$$g(S)=s_1-s_3+s_5-s_7+\cdots.$$
Thus $g$ uses the cyclic weights $1,0,-1,0$, while $f$ uses $1,-1$. Clearly $f(\emptyset)=g(\emptyset)=0$.
(a) Find $\sum_{S\subseteq T}f(S)$.
(b) Find $\sum_{S\subseteq T}g(S)$.
(c) If you cannot do (b), find the latter sum for $T=\{1,2,\ldots,8\}$.
$$f(S)=s_1-s_2+s_3-s_4+\cdots,$$
$$g(S)=s_1-s_3+s_5-s_7+\cdots.$$
Thus $g$ uses the cyclic weights $1,0,-1,0$, while $f$ uses $1,-1$. Clearly $f(\emptyset)=g(\emptyset)=0$.
(a) Find $\sum_{S\subseteq T}f(S)$.
(b) Find $\sum_{S\subseteq T}g(S)$.
(c) If you cannot do (b), find the latter sum for $T=\{1,2,\ldots,8\}$.