Linear Recurrences from Counting Schreier-Type Multisets

A nonempty set F is Schreier if min 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 min F≥ p|F|. In particular, if we let A⁽ˢ⁾p, n\ :=\ ⊂ \1, …, 1ₛ, …, n-1, …, n-1ₛ, n\\,:\,n∈ F and min F≥ p|F|\, then |A⁽ˢ⁾p, n| = ∑ᵢ₌₀ˢ|A⁽ˢ⁾p, n-1-ip|. If we color s copies of the same integer by different colors from $1$ to s, i.e., B⁽ˢ⁾p, n:= ⊂ \1₁, …, 1ₛ, …, (n-1)₁, …, (n-1)ₛ, n\\,:\,n∈ F and min F≥ p|F|\, then |B⁽ˢ⁾p, n| = ∑ᵢ₌₀ˢ si| B⁽ˢ⁾p, n-1-ip|. 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.