Bijection Between Symmetric Walk Families

Definitions and mathematical statement

Let $\widehat S$ denote a multiset of small steps in the sense intended by the supplied source, and let $X(\widehat S)$ and $Y(\widehat S)$ denote its two symmetric walk families. The record does not define small steps, either walk family, or the meaning of symmetry. These are unresolved prerequisites, so the following is a statement of the requested construction with undefined source-dependent families, not a complete mathematical formulation.

Requested construction. For every admissible multiset $\widehat S$ of small steps, construct a direct bijection \[ \Phi_{\widehat S}\colon X(\widehat S)\longrightarrow Y(\widehat S). \] Thus the construction must assign to each walk in $X(\widehat S)$ exactly one walk in $Y(\widehat S)$, and must have a specified inverse. The admissible multisets, ambient space, starting points, endpoint conditions, allowed regions, symmetry conditions, and treatment of repeated steps cannot be specified from this record.

The supplied excerpt attributes to A. Elvey Price the assertion \[ \lvert X(\widehat S)\rvert=\lvert Y(\widehat S)\rvert \qquad\text{for every multiset $\widehat S$ of small steps}, \] and says that the existing proof uses elliptic functions. This assertion is supplied background, not independently verified here; the requested task is the direct bijective construction.

The supplied figure caption illustrates a walk in $X(\widehat S)$ with the step multiset transcribed directionally as \[ \widehat S=\{\nearrow,\ 2\times\downarrow,\ 2\times\nwarrow,\ 3\times\uparrow,\ 3\times\leftarrow,\ 4\times\swarrow,\ 5\times\rightarrow\}. \] The figure itself is not supplied. This example does not determine the missing definitions, and its directional glyphs do not specify a coordinate convention.

Short English statement

For each allowed multiset of small steps, find a direct, invertible correspondence between the source's two symmetric walk families. The supplied record reports equal cardinalities, but omits the definitions needed to state the families precisely. This statement-only draft requires human mathematical review.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005326$, source problem code catalogue entry. The supplied record labels the problem open and its statement corrected_verified; its verification note describes repairs to formula delimiters, notation, grammar, and typography while preserving mathematical scope. These are supplied registry qualifications, not independent mathematical verification.
• Supplied report citation: Enumerative Combinatorics (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/57, pp. 3215--3305. https://doi.org/10.4171/owr/2022/57. The excerpt identifies the request as Problem 1 and attributes the preceding cardinality theorem to A. Elvey Price. No proposer is specified; the registry gives proposed year 2022.
• Supplied literature source labelled "Elvey Price": https://arxiv.org/abs/2204.06847. No title or detailed bibliographic information is supplied for this reference.
• Supplied literature source labelled "original OWR report": https://doi.org/10.4171/owr/2022/57.
• Literature assessment dated exactly 2026-08-22: the supplied curation says that equality of the counted sets is known, while the requested direct bijection had not been established in the authoritative sources located by that cutoff. This is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.