Reuse-Growth Sequences: Insertion Bijections and Pattern Avoidance

We introduce reuse-growth sequences, a class of statistic-controlled restricted-growth words in which the admissible range increases whenever a previously seen value is reused. The class is singled out by a simple insertion property. If a prefix has length m and contains d distinct values, then it has exactly d extensions by old values and m + 2 − 2d extensions by new values. These are precisely the two insertion counts in the standard generating tree for simsun permutations. Matching the two types of extensions gives a direct recursive bijection between reuse-growth sequences and simsun permutations, with des = d − 1, asc = rep. A second recursive bijection maps reuse-growth sequences to 000-avoiding inversion sequences. Consequently, |Uₙ| = Eₙ₊₁, with the bijections providing statistic refinements rather than only an equinumeracy. We then develop the length-three pattern-avoidance theory of the class. All thirteen packed patterns are treated structurally. Six single-pattern classes have closed enumerations. For 102, the ordinary generating function satisfies A = 1 + x(1 − x)A² + x²(1 − x)A³, from which we obtain a coefficient formula and square-root asymptotics. Moreover, U_n(102) coincides with the corresponding class of level sequences, so this result also enumerates 102-avoiding level sequences. Four further single-pattern classes admit exact succession rules and are superexponential, while 120, 201, and 210 require ordered used/unused states. For the pair {201, 210}, these states reduce to three parameters, leading to an explicit algebraic generating function and asymptotics. The resulting avoidance theory differs substantially from that of inversion sequences; in particular, the inversion-sequence Wilf equivalences 101 ~ 110 and 201 ~ 210 do not persist for reuse-growth sequences.