Reuse-Growth Sequences: Insertion Bijections and Pattern Avoidance

We introduce reuse-growth sequences, a new class of 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 mm and contains dd distinct values, then it has exactly dd extensions by old values and m+2−2dm+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\operatorname{des}=d-1 and asc⁡=rep⁡\operatorname{asc}=\operatorname{rep}. A second recursive bijection maps reuse-growth sequences to 000000-avoiding inversion sequences. Consequently, ∣Un∣=En+1|\mathcal U_n|=E_{n+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 102102, the ordinary generating function satisfies A=1+x(1−x)A2+x2(1−x)A3A=1+x(1-x)A^2+x^2(1-x)A^3, from which we obtain a coefficient formula and square-root asymptotics. Moreover, Un(102)\mathcal U_n(102) coincides with the corresponding class of level sequences, so this result also enumerates 102102-avoiding level sequences. Four further single-pattern classes admit exact succession rules and are superexponential, while 120,201,120,201, and 210210 require ordered used/unused states. For the pair {201,210}\{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∼110101\sim110 and 201∼210201\sim210 fail for reuse-growth sequences.