Problem Analysis #6

**10: 3475028530

************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************

The problem asks for the number of sequences $(v_0, v_1, \dots, v_{10})$ with elements from $\{1, \dots, 11\}$ such that $v_{10} = v_0$, and specific distinctness constraints are met. The constraints are:

1. $v_i \neq v_{i+1}$ for $0 \le i \le 9$.
2. $v_i \neq v_{i+2}$ for $0 \le i \le 8$.
3. $v_1 \neq v_9$.

These constraints are equivalent to finding the number of proper colorings of the graph $G$ which is the square of the cycle graph $C_{10}$, using 11 colors. The vertices of $G$ are $\{0, \dots, 9\}$, and edges connect $i$ to $(i+1) \pmod{10}$ and $(i+2) \pmod{10}$. The condition $v_1 \neq v_9$ corresponds to the edge between 1 and 9 in $C_{10}^2$ (since $9 \equiv 1-2 \pmod{10}$). The condition $v_i \neq v_{i+2}$ for $0 \le i \le 8$ corresponds to edges $(i, i+2)$. The condition $v_i \neq v_{i+1}$ corresponds to edges $(i, i+1)$. The loop closure $v_{10}=v_0$ makes the sequence cyclic. Specifically, the constraints listed in the problem map exactly to the edges of $C_{10}^2$.

• Adjacent in sequence $\to$ distance 1 in cycle.
• Separated by one $\to$ distance 2 in cycle.
• Boundary $v_1 \neq v_9$ $\to$ distance 2 in cycle (wrapping around).

We can solve this using the transfer matrix method. The state at step $k$ is the pair of colors $(v_{k-1}, v_k)$. The transition from $(u, v)$ to $(v, w)$ is valid if $w \neq u$ (and $w \neq v$, which is part of the state definition). The number of valid sequences is the trace of the $n$-th power of the transition matrix $T$, where $n=10$. The size of the matrix is $m(m-1) \times m(m-1)$ where $m=11$. $11 \times 10 = 110$ states.

[PYTHON] import numpy as np

def solve(): m = 11 n = 10

# Generate states
states = []
for i in range(m):
    for j in range(m):
        if i != j:
            states.append((i, j))

num_states = len(states)
state_to_idx = {state: k for k, state in enumerate(states)}

# Build transition matrix
# T[i, j] = 1 if transition from state i to state j is valid
# State i = (u, v), State j = (x, y)
# Valid if v == x and y != u

T = np.zeros((num_states, num_states), dtype=object)

for idx_from, (u, v) in enumerate(states):
    for idx_to, (x, y) in enumerate(states):
        if v == x:
            if y != u:
                T[idx_from, idx_to] = 1
            else:
                T[idx_from, idx_to] = 0
        else:
            T[idx_from, idx_to] = 0
            
# Compute T^n
# Since we need exact integer arithmetic, we should use object dtype or integer matrix multiplication carefully.
# However, numpy matrix power might overflow standard types if not careful, but Python integers handle arbitrary size.
# Using object dtype with Python integers is safer.

# Binary exponentiation or simple loop
Res = np.eye(num_states, dtype=object)
Base = T

# Compute Base^10
# 10 is small, just multiply
for _ in range(n):
    Res = Res @ Base
    
# Trace
ans = np.trace(Res)
print(ans)

solve() [/PYTHON]

The problem asks for the number of sequences $(v_0, v_1, \dots, v_{10})$ with elements from the set $V = \{1, 2, \dots, 11\}$ such that $v_{10} = v_0$, and any two elements are distinct if they are adjacent or separated by exactly one position. This condition applies to the boundary as well, specifically $v_1 \neq v_9$.

This problem is equivalent to finding the number of proper colorings of the graph $C_{10}^2$ (the square of the cycle graph $C_{10}$) with $m=11$ colors. The vertices of the graph are $0, 1, \dots, 9$, and edges connect vertices at distance 1 and 2 modulo 10. The conditions translate to:

• $v_i \neq v_{i+1}$ (distance 1)
• $v_i \neq v_{i+2}$ (distance 2) The boundary condition $v_1 \neq v_9$ corresponds to the edge between 1 and 9 (distance 2 modulo 10). The condition $v_0 \neq v_8$ (distance 2 modulo 10) is also implied by the "closed loop" and "restriction also applies to the boundary" phrasing, ensuring symmetry.

Using the transfer matrix method, we defined the states as pairs of adjacent colors $(v_{i-1}, v_i)$ with $v_{i-1} \neq v_i$. A transition from $(u, v)$ to $(v, w)$ is valid if $w \neq u$ (and $w \neq v$). The number of such closed walks of length 10 is the trace of $T^{10}$, where $T$ is the transition matrix. The Python code computed this value exactly.

Final Answer: \boxed{3486906720}