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 $$\mathcal{ A}^{ (s)}_{ p, n}\ :=\ \{ F\subset \{ \underbrace{ 1, \ldots, 1}_{ s}, \ldots, \underbrace{ n-1, \ldots, n-1}_{ s}, n\}\,:\,n\in F\mbox{ and }\min F\ge p|F|\},$$ 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}:= $ $$\{ F\subset \{ 1_{ 1}, \ldots, 1_{ s}, \ldots, (n-1)_1, \ldots, (n-1)_{ s}, n\}\,:\,n\in F\mbox{ and }\min F\ge p|F|\},$$ 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.
Categories: math.CO