bibkey: chu2026schreiermultisets authors: Hùng Việt Chu; Yubo Geng; Julian King; Steven J. Miller; Garrett Tresch; Zachary Louis Vasseur year: 2026 title: “Linear Recurrences from Counting Schreier-Type Multisets” doi: 10.5281/zenodo.19949535 url: https://math.colgate.edu/~integers/aa53/aa53.pdf claim: “Section 5 conjectures the recurrence for the cardinalities of A^(2)_(1,2,n).” strata_touched:
- D5/S1/Recurrence/ChuSchreierMultisetRecurrenceQTwo license: citation-only triage: anchor
Linear Recurrences from Counting Schreier-Type Multisets
Section 5, “Further Investigations”, defines the multiset families and lists five conjectured recurrences. The definition on printed page 19 reads:
Define (A^{(s)}_{p,q,n} := {F \subset {1, \ldots, 1, \ldots, n-1, \ldots, n-1, n} : n \in F \text{ and } q\min F \ge p|F|}.)
The two repeated ranges are underbraced by (s) in the printed display: each of (1,\ldots,n-1) has multiplicity (s), while (n) has multiplicity one. The first item under “Below are the data and conjectured recurrences we gather” reads on the same page:
- ((|A^{(2)}{1,2,n}|)^{\infty}{n=1}: 1, 2, 4, 9, 19, 41, 88, 189, 406, 872, 1873, 4023, 8641, \ldots) with (a_n = a_{n-1} + 2a_{n-2} + a_{n-3};)
The formal reading uses sub-multisets of the displayed ground multiset, with cardinality counting multiplicity. It specializes ((p,q)=(1,2)), so the condition is (2\min F \ge |F|). Since the paper does not state a starting index for the recurrence, the formal statement starts at (n=4), the least index for which all four displayed sequence indices are at least one.
The other four recurrences on printed pages 19-20 and the preceding double-sum formula are not asserted by this formalization.
Verified locator
- DOI: https://doi.org/10.5281/zenodo.19949535
- URL: https://math.colgate.edu/~integers/aa53/aa53.pdf
- Locator: Section 5, “Further Investigations”, printed page 19.