Denote by \(x\) the Fibonacci sequence on the alphabet \(\{a, b\}\), defined as the fixed point of the morphism \(b\to ba\), \(a\to b\). Hence \(x= ba bb ab abb ab\dots\;\). Denote by \(x_1, x_2, \dots\) the sequences obtained from \(x\) by erasing the first term, the first two terms, …, for example \(x_4= ab abb ab\dots\;\).
The author addresses the problem of alignment of the \(x_i\)’s. This problem, due to D. R. Hofstadter [“Eta-lore”, first presented at the Stanford Math. Club, Stanford, California, p. 13 (1963)]has also been studied by R. J. Hendel and S. A. Monteferrante [“Hofstadter’s extraction conjecture”, Fibonacci Q. 32, 98-107 (1994)]:
one writes \(U\supset V; E\) if \(U= (v_1\dots v_{j(1)}) e_1 (v_{j(1) +1} \dots v_{j(2)} )e_2\dots\), where \(V= v_1 v_2 v_3 \dots\); \(E= e_1 e_2 e_3 \dots\); \(0= j(0)\leq j(1)\leq j(2) \dots\); \(v_i\dots v_k= \emptyset\) if \(k<i\); \(e_i\neq v_{j(i)+1}\) \(\forall i\).
Hendel and Monteferrante determined \(y_{m,0}\) where \(x_m\supset x_n; y_{m,n}\). The paper under review determines all \(y_{m,n}\). The tools are Zeckendorf representations of integers and combinatorics on words.