bibkey: recioui2026circular authors: Asma Recioui, Hacène Belbachir, Abdelhakim Ait-Zai year: 2026 title: “Circular s-choice parking functions: an exact closed formula via rotational symmetry” doi: 10.48550/arXiv.2609.23607 url: https://arxiv.org/html/2609.23607v1 claim: “Open Problem 1 asks for a canonical bijection between every fixed-increment, fixed-vacancy anchor class in the circular two-choice d=1 model and the classical parking functions.” strata_touched:

D5/S3/Combinatorics/Parking/OperationalDynamics
D5/S3/Combinatorics/Parking/FixedIncrementFiberTransport
D5/S3/Combinatorics/Parking/CutRunCorrespondence
D5/S3/Combinatorics/Parking/CircularTwoChoiceParkingBijection license: CC0-1.0 triage: anchor

Circular s-choice parking functions

Verified locator

DOI: 10.48550/arXiv.2609.23607 URL: https://arxiv.org/html/2609.23607v1

The canonical source is arXiv:2609.23607v1, submitted on 2026-09-20. The arXiv API returned exactly that version on 2026-09-25, and the current HTML still labels the bijection request as Open Problem 1. The title, author names, abstract, five numbered sections, and reference list were checked against the full HTML source. The title page orders the authors as Asma Recioui, Hacène Belbachir, and Abdelhakim Ait-Zai; the arXiv API lists the same three names but places Hacène Belbachir first. The title-page order is retained here. The HTML record declares CC Zero.

Model and established results

There are n cars and m=n+1 circular spots. An admissible s-tuple starts at an anchor and advances clockwise by s-1 positive increments, with every successive gap and the return gap at least d. A car tries the tuple entries in order and, if all are occupied, continues clockwise from its last choice. Every circular input parks all cars and leaves one spot empty.

The paper proves by rotation that the vacancy is equidistributed. It then refines the argument at a fixed increment matrix: rotating every anchor preserves the increments and moves the vacancy transitively, so every vacancy fiber has size m^(n-1). For s=2 and d=1, the total conditioned count is (n+1)^(n-1) n^n. The factor (n+1)^(n-1) is the classical parking-function count and n^n counts the independent per-car increments. These counts, equidistribution, the factorization, Pollak’s rotation argument, and the Kaplansky circular-selection count are results or acknowledged background of the source; they are not new credit for the repository construction.

Exact open problem

Open Problem 1 reads:

Give a bijective proof of the factorization of Corollary 1 for d=1, refining Theorem 2 to a canonical bijection between each anchor class {E=j} intersect C_kappa and the set of classical parking functions.

In the two-choice specialization, each car has a literal ordered pair: the anchor is tried first, and the second choice is the starting point of the clockwise fallback scan. Fixing kappa fixes the positive clockwise increment from the anchor to the second choice for every car. Thus the requested class fixes all per-car increments and the final empty spot, while its free data are the anchors. The question asks for an actual two-sided construction, not the already proved equality of finite cardinalities.

Repository correspondence

OperationalDynamics models the s=2, d=1 rule on ZMod(n+1), including the literal ordered choices and their operational run. The anchor is tried first, and its scanner includes offset zero, so a free second choice is used before any later clockwise spot. It proves prefix freshness, unique vacancy, and rotation equivariance for the run and vacancy.

FixedIncrementFiberTransport supplies the positive-increment encoding and decoding. Its vacancy-normalized rotations give the explicit equivalence between the literal fixed-increment fiber and a one-choice circular fiber, with the inverse reconstructing both entries of every ordered pair.

CutRunCorrespondence defines cut and uncut coordinates around a vacancy and proves the forward circular-to-linear scanner and run simulations.

CircularTwoChoiceParkingBijection completes the reverse feedback-state simulation against the frozen classical supplier and composes the classical bridge with the fixed-fiber transport. The public fixedFiberEquiv is the exact fixed-increment, fixed-vacancy construction, and result is the sole formal resolution statement. The separate globalParkingEquiv exposes the classical parking function, original increment matrix, and actual vacancy together; its inverse uses the matching fixed fiber and hence recovers every anchor and second choice. This global product is an auxiliary interface, not a second formulation of Open Problem 1.

The formal definitions extend to n=0; the source problem and its claimed resolution use only n>=1. At n=1, the only increment is one and the offset-zero rule still distinguishes anchor priority from second-choice priority. No quantifier over positive n, increment matrices, vacancy spots, or cars is replaced by a finite sample.

Bounded prior-art check

Two exact arXiv searches were performed on 2026-09-25: circular parking with bijection, and fixed increments with parking functions. Both returned zero records. The first twenty Crossref title results for circular s-choice parking functions contained general parking-function papers and unrelated parking-choice work, but no exact fixed-increment circular construction. An OpenAlex query returned HTTP 429 and was unread.

This is only a not-found result in the stated search scope. It does not show worldwide absence, priority, or absence from unpublished, unindexed, paywalled, or author-held material. The current paper itself is the authority for the open wording; later literature was not exhaustively read.