<p>Consider a one-one function \(A_n \to A_n\) where \(A_n = \{1, 2, 3\ldots n\}\). Let \(T(n)\) be the number of functions from \(A_n \to A_n\) such that \(|f(i) - i| < 2\), then</p>
Step-by-Step Solution
Key Concept: A one-one function f: A_n → A_n where |f(i) - i| < 2 means f(i) ∈ {i-1, i, i+1} for each i. Since f is a permutation with restricted movements, we need to count derangements with only adjacent transpositions allowed.
<p><strong>Step 1: Identify the constraint</strong></p><p>For a one-one function f: A_n → A_n with |f(i) - i| < 2, each element i can only map to {i-1, i, i+1}.</p><p><strong>Step 2: Analyze valid permutations</strong></p><p>Since f is a bijection, if f(i) = i+1, then f(i+1) ≠ i+1. For f(i+1), options are {i, i+1, i+2}. But i+1 is taken by f(i), so f(i+1) ∈ {i, i+2}.</p><p><strong>Step 3: Establish recurrence</strong></p><p>Let T(n) = count of valid permutations. The analysis shows:</p><ul><li>f(1) = 1 and proceed: T(n-1) ways</li><li>f(1) = 2, f(2) = 1 (swap first two): T(n-2) ways</li><li>All other patterns violate the bijection constraint</li></ul><p><strong>Step 4: Solve recurrence</strong></p><p>T(n) = T(n-1) + T(n-2) with T(1) = 1, T(2) = 2</p><p>This gives Fibonacci-like sequence: T(n) = F(n+1) where F is Fibonacci.</p><p>∴ Answer: A</p>
Correct Answer: A