Let be the set . Consider a sequence of elements from of length that forms a closed loop where the last element equals the first (). The sequence must satisfy the condition that any two elements are distinct if they are adjacent or separated by exactly one position. This restriction also applies to the boundary, specifically requiring that .
Let denote the total number of such sequences of length . Determine the exact integer value of .
[THOUGHT] We are counting colorings of a 10-cycle with 11 symbols where equalities are forbidden at distance 1 or 2 along the cycle (i.e., adjacent or with exactly one vertex between). I will do dynamic programming on ordered pairs (previous two values) for linear sequences v0..v9, then enforce the wrap-around constraints involving v0,v1 with v8,v9. [/THOUGHT] [PYTHON] from itertools import product
def count_cycle(n, m): # n vertices in cycle, m colors, constraint: v_i != v_{i±1}, v_i != v_{i±2} # DP over linear sequence v0..v_{n-1}, then apply wrap constraints total = 0 colors = list(range(m)) for v0 in colors: for v1 in colors: if v1 == v0: continue # dp maps (prev2, prev1) = (v_{i-1}, v_i) after setting up to i dp = {(v0, v1): 1} # build v2..v_{n-1} for i in range(2, n): ndp = {} for (a,b), cnt in dp.items(): # next color c != a,b for c in colors: if c == a or c == b: continue ndp[(b,c)] = ndp.get((b,c), 0) + cnt dp = ndp # now dp contains counts for sequences v0..v_{n-1} with local constraints # enforce wrap constraints with v0 and v1: # v_{n-1} != v0 (adjacent), v_{n-2} != v0 (distance2), # v_{n-1} != v1 (distance2), also v0!=v_{n-2} already included now for (vn2, vn1), cnt in dp.items(): if vn1 == v0: # adjacent continue if vn2 == v0: # distance 2 continue if vn1 == v1: # distance 2 to v1 continue total += cnt return total
print(count_cycle(10, 11)) [/PYTHON][THOUGHT] We are counting colorings of a 10-cycle with 11 symbols where equalities are forbidden at distance 1 or 2 along the cycle (i.e., adjacent or with exactly one vertex between). I will do dynamic programming on ordered pairs (previous two values) for linear sequences v0..v9, then enforce the wrap-around constraints involving v0,v1 with v8,v9. [/THOUGHT] [PYTHON] from itertools import product
def count_cycle(n, m): # n vertices in cycle, m colors, constraint: v_i != v_{i±1}, v_i != v_{i±2} total = 0 colors = list(range(m)) for v0 in colors: for v1 in colors: if v1 == v0: continue dp = {(v0, v1): 1} # state (v_{i-1}, v_i) for i in range(2, n): ndp = {} for (a, b), cnt in dp.items(): for c in colors: if c == a or c == b: continue ndp[(b, c)] = ndp.get((b, c), 0) + cnt dp = ndp
for (vn2, vn1), cnt in dp.items(): # vn2=v_{n-2}, vn1=v_{n-1}
if vn1 == v0: # adjacent to v0
continue
if vn2 == v0: # distance 2 to v0
continue
if vn1 == v1: # distance 2 to v1
continue
total += cnt
return total
print(count_cycle(10, 11)) [/PYTHON]
[THOUGHT] Using the verified Python DP count for a 10-cycle with 11 symbols and forbidden equalities at distance 1 or 2, we obtain (k_{10}=3486906720). [\boxed{3486906720}]