ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Binary words avoiding $11$ are counted by $F_{n + 2}$

Example

Let $a_{n}$ be the number of binary words of length $n$ that avoid $11$ . Then

$a_{n} = F_{n + 2}, \sum_{n \geq 0} a_{n} x^{n} = \frac{1 + x}{1 - x - x^{2}} .$

The proper-prefix states $ε, 1$ have transfer matrix

$A = (\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}) .$

Facts & Assumptions

Given: The singleton forbidden set $B = {11}$ over ${0, 1}$ .

[L1]

The Fibonacci numbers satisfy their recurrence and fixed initial values (The Fibonacci sequence $F_{0} = 0, F_{1} = 1$ and Lucas sequence $L_{0} = 2, L_{1} = 1$ ).

[L2]

The prefix automaton sends a state and letter to the longest allowed proper-prefix suffix, rejecting a completed forbidden factor (The longest-suffix prefix automaton for a finite set of forbidden factors).

[L3]

Fixed-entry walk generating functions are the corresponding cofactors of $I - x A$ divided by its determinant (Transfer-matrix theorem: weighted-walk generating functions are cofactors of $I - x A$ divided by $\det (I - x A)$ ).

Verification

technique · transfer matrix

1.1

By [L2], state $ε$ has transitions labelled $0$ to $ε$ and $1$ to $1$ , while state $1$ has only the transition labelled $0$ to $ε$ . This gives the displayed matrix.

givenL2

2.1

Every avoiding word labels one walk from $ε$ to either state. Since $I - x A = (\begin{matrix} 1 - x & - x \\ - x & 1 \end{matrix})$ , summing the first-row entries of its inverse via [L3] gives $(1 + x) / (1 - x - x^{2})$ .

step 1.1L3algebra

3.1

Its coefficients begin $a_{0} = 1, a_{1} = 2$ and satisfy $a_{n + 2} = a_{n + 1} + a_{n}$ , so comparison with [L1] gives $a_{n} = F_{n + 2}$ .

step 2.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 38 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Proposition 4.7.8 (standard reference, not scraped)

Binary words avoiding 1111 are counted by Fn+2Fn+2​