Question 28
a.
Let $A_{n} = \{a_{1}, a_{2}, a_{3}, \ldots, a_{n}, b\}$, for $n \geq 3$, and let $C_{n}$ be the 2-configuration consisting of $\{a_{i}, a_{i+1}\}$ for all $1 \leq i \leq n-1$, $\{a_{1}, a_{n}\}$, and $\{a_{i}, b\}$ for $1 \leq i \leq n$. Let $S_{e}(n)$ be the number of subsets of $C_{n}$ that are consistent of order $e$. Find $S_{e}(101)$ for $e=1,2$, and $3$.
b.
Let $A = \{V, W, X, Y, Z, v, w, x, y, z\}$. Find the number of subsets of the 2-configuration
$$
\begin{gathered}
\{\{V, W\},\{W, X\},\{X, Y\},\{Y, Z\},\{Z, V\},\{v, x\},\{v, y\},\{w, y\},\{w, z\},\{x, z\}, \\
\{V, v\},\{W, w\},\{X, x\},\{Y, y\},\{Z, z\}\}
\end{gathered}
$$
that are consistent of order 1.
c.
Let $A = \{a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{10}, b_{10}\}$, and consider the 2-configuration $C$ consisting of $\{a_{i}, b_{i}\}$ for all $1 \leq i \leq 10$, $\{a_{i}, a_{i+1}\}$ for all $1 \leq i \leq 9$, and $\{b_{i}, b_{i+1}\}$ for all $1 \leq i \leq 9$. Find the number of subsets of $C$ that are consistent of order 1.
Let $A_{n} = \{a_{1}, a_{2}, a_{3}, \ldots, a_{n}, b\}$, for $n \geq 3$, and let $C_{n}$ be the 2-configuration consisting of $\{a_{i}, a_{i+1}\}$ for all $1 \leq i \leq n-1$, $\{a_{1}, a_{n}\}$, and $\{a_{i}, b\}$ for $1 \leq i \leq n$. Let $S_{e}(n)$ be the number of subsets of $C_{n}$ that are consistent of order $e$. Find $S_{e}(101)$ for $e=1,2$, and $3$.
b.
Let $A = \{V, W, X, Y, Z, v, w, x, y, z\}$. Find the number of subsets of the 2-configuration
$$
\begin{gathered}
\{\{V, W\},\{W, X\},\{X, Y\},\{Y, Z\},\{Z, V\},\{v, x\},\{v, y\},\{w, y\},\{w, z\},\{x, z\}, \\
\{V, v\},\{W, w\},\{X, x\},\{Y, y\},\{Z, z\}\}
\end{gathered}
$$
that are consistent of order 1.
c.
Let $A = \{a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{10}, b_{10}\}$, and consider the 2-configuration $C$ consisting of $\{a_{i}, b_{i}\}$ for all $1 \leq i \leq 10$, $\{a_{i}, a_{i+1}\}$ for all $1 \leq i \leq 9$, and $\{b_{i}, b_{i+1}\}$ for all $1 \leq i \leq 9$. Find the number of subsets of $C$ that are consistent of order 1.