Linear Recurrences of Generalized Schreier Sets Revisited

For $p, q\in \mathbb{N}$, a finite nonempty set $F$ is said to be $(p,q)$-Schreier (or maximal $(p,q)$-Schreier, respectively) if $q\min F\ge p|F|$ (or $q\min F = p|F|$, respectively). For $n\in \mathbb{N}$, let $$\mathcal{S}^{p/q}_{n}\ :=\ |\{F\subset\{1, 2, \ldots, n\}\,:\, q\min F\ge p|F|\mbox{ and }n\in F\}|.$$ Using the Inclusion-Exclusion Principle, Beanland et al. proved the recurrence $$|\mathcal{S}^{p/q}_{n}|\ =\ \sum_{k=1}^q(-1)^{k+1}\binom{q}{k}|\mathcal{S}^{p/q}_{n-k}| + |\mathcal{S}^{p/q}_{n-(p+q)}|.$$ We show that $(|\mathcal{S}^{p/q}_n|)_{n=1}^\infty$ is a subsequence with terms taken periodically from Padovan-like sequences which satisfy simple recurrence relations. As an application, we obtain an alternative proof of the above linear recurrence. Furthermore, a similar result holds for the sequence $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$ that counts maximal $(p,q)$-Schreier sets. We end with a discussion of the relation between $(|\mathcal{S}^{p/q}_{n}|)_{n=1}^\infty$ and $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$.