A nonempty set F is Schreier if F |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 F p|F|. In particular, if we let A^ (s) , ₍\: =\ \F \{1, , 1ₒ, , n-1, , n-1ₒ, n\\,: \, n F and F p|F|\}, then |A^ (s) , ₍| = ₈=₀ˢ|A^ (s) , ₍-₁-₈|. If we color s copies of the same integer by different colors from 1 to s, i. e. , B^ (s) , ₍: = \F \{1₁, , 1ₒ, , (n-1) ₁, , (n-1) ₒ, n\\,: \, n F and F p|F|\}, then |B^ (s) , ₍| = ₈=₀ˢ si| B^ (s) , ₍-₁-₈|. Lastly, we count Schreier sets that do not admit multiples of a given integer u 2 and witness linear recurrences whose coefficients are drawn from the uth row of the Pascal triangle and have alternating signs, except possibly the last one.
Chu et al. (Fri,) studied this question.