Check a finite rack table and enumerate diagram colorings

Input an integer $m\ge 1$, a candidate table $T$ represented by a finite list of finite lists of integers, with each entry intended to be $x\ast y$ on $\{1,\dots ,m\}$, a mode "rack" or "quandle", a finite regular oriented link diagram $D$ with $a$ arcs and $c$ crossings, and fixed finite orderings of its arcs and crossings. Read the row count and every row length. If $T$ does not have exactly $m$ rows of length $m$, return invalid-table-shape with its row count and vector of row lengths. Otherwise scan entries in row-major order; if an entry is outside $\{1,\dots ,m\}$, return invalid-table-entry with the first such row, column, and value. If the table format passes, perform these checks in order:

1. For each $y=1,\dots ,m$, check whether the column $(T[1,y],\dots ,T[m,y])$ is a permutation of $\{1,\dots ,m\}$. If not, return right-translation-not-bijective with $y$, a repeated value and a missing value.
2. For each ordered triple $(x,y,z)$ in lexicographic order, check $T[T[x,y],z]=T[T[x,z],T[y,z]]$. If not, return self-distributivity-fails with $(x,y,z)$ and both table values.
3. In quandle mode, check $T[x,x]=x$ for each $x$ in increasing order. If not, return idempotency-fails with $x$ and $T[x,x]$.

If the selected axioms pass, enumerate the $m^a$ functions from the supplied ordered arcs of $D$ to $\{1,\dots ,m\}$ in lexicographic order. Retain an assignment exactly when it satisfies the crossing rule in the rack/quandle Definition, using the inverse column permutation when the rule calls for the right division. Return the retained assignments and their number. No correctness or complexity claim is part of this algorithm's Definition.

Racks, diagram fundamental quandles, and adjoint groups.

Regular oriented link diagrams.

Nelson, Link invariants from finite racks, §§2.1 and 3.

Carter, A Survey of Quandle Ideas.