Linear Recurrences from Counting Schreier-Type Multisets

A nonempty set $F$ is Schreier if $\min F\ge \mathrm{\mid }F\mathrm{\mid }$. 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 $\min F\ge p\mathrm{\mid }F\mathrm{\mid }$. In particular, if we let $${\mathcal{A}}_{p,n}^{(s)}:=\{F\subset \{\underset{s}{\underbrace{1,\dots ,1}},\dots ,\underset{s}{\underbrace{n-1,\dots ,n-1}},n\}:n\in Fand\min F\ge p\mathrm{\mid }F\mathrm{\mid }\},$$ then $$\mathrm{\mid }{\mathcal{A}}_{p,n}^{(s)}\mathrm{\mid }=\sum _{i=0}^s\mathrm{\mid }{\mathcal{A}}_{p,n-1-ip}^{(s)}\mathrm{\mid }\mathrm{.}$$ If we color $s$ copies of the same integer by different colors from $1$ to $s$, i.e., ${\mathcal{B}}_{p,n}^{(s)}:=$ $$\{F\subset \{1_1,\dots ,1_s,\dots ,(n-1{)}_1,\dots ,(n-1{)}_s,n\}:n\in Fand\min F\ge p\mathrm{\mid }F\mathrm{\mid }\},$$ then $$\mathrm{\mid }{\mathcal{B}}_{p,n}^{(s)}\mathrm{\mid }=\sum _{i=0}^s\left(\genfrac{}{}{0px}{}{s}{i}\right)\mathrm{\mid }{\mathcal{B}}_{p,n-1-ip}^{(s)}\mathrm{\mid }\mathrm{.}$$ 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.