Linear Recurrences from Counting Schreier-Type Multisets

Abstract: A nonempty set $F$ is Schreier if $\min F\ge |F|$. Bird observed that counting Schreier sets in a certain way produces the Fibonacci sequence. Since then, various connections between variants of Schreier sets and well-known sequences have been discovered. Building on these works, we prove a linear recurrence for the sequence that counts multisets $F$ with $\min F\ge p|F|$. In particular, if we let $$\mathcal{A}^{(s)}_{p, n}\ :=\ {F\subset {\underbrace{1, \ldots, 1}<em>{s}, \ldots, \underbrace{n-1, \ldots, n-1}</em>{s}, n}\,:\,n\in F\mbox{ and }\min F\ge p|F|},$$ then $$|\mathcal{A}^{(s)}_{p, n}| = \sum_{i=0}^{s|\mathcal{A}^{(s)}_{p,} n-1-ip}|.$$ If we color $s$ copies of the same integer by different colors from $1$ to $s$, i.e., $\mathcal{B}^{(s)}_{p, n}:=$ $${F\subset {1_{1}, \ldots, 1_{s}, \ldots, (n-1)<em>1, \ldots, (n-1)</em>{s}, n}\,:\,n\in F\mbox{ and }\min F\ge p|F|},$$ then $$|\mathcal{B}^{(s)}_{p, n}| = \sum_{i=0}^s \binom{s}{i}| \mathcal{B}^{(s)}_{p, n-1-ip}|.$$ Lastly, we count Schreier sets that do not admit multiples of a given integer $u\ge 2$ and witness linear recurrences whose coefficients are drawn from the $u$th row of the Pascal triangle and have alternating signs, except possibly the last one.