Dependencies

Let \(w = n - 3\).

• For every odd \(n \geq 15\), four sign changes fail for integer frieze patterns of width \(w\) in a row of the printed range \(1 \leq k \leq \lfloor w/2 \rfloor \).

• For every \(n \geq 14\), they fail in a row of the range \(1 \leq k \leq \lceil w/2 \rceil \).

Insert ears after the positions \(j\) and \(j + h\). The diameters of the new polygon are the old ones, in the same cyclic order, together with one new diameter between the two new vertices, of value

Hence if \(D_i(U) {\lt} D_i(V)\) for all \(i\) and \(S(U, j) {\lt} S(V, j)\), the strict order survives the insertion.

For \(w \leq 10\), every pair of distinct integer frieze patterns of width \(w\) has, in every row \(k \leq \lceil w/2 \rceil \), \(Z + c \geq 4\); hence \(s^+ \geq 4\) and \(E \geq 4\). For \(w = 11\) the same holds in rows \(k \leq 5\), and in the middle row there are \(672\) pairs \((a, b)\) with \(d_i {\gt} 0\) for every \(i\) (each such unordered pair counted once, in the order that makes \(d\) positive). For \(w = 12\) the same holds in rows \(k \leq 5\); in row \(6\) there are \(480\) unordered pairs with no zero and exactly two sign changes, and \(60\) more with one zero and \(s^+ = 2\). Frieze patterns are labelled: a rotated or reflected triangulation counts as a different frieze pattern.

• In which rows \(k \leq \lfloor w/2 \rfloor \) do four sign changes fail, besides the row \(\lfloor w/2 \rfloor \) of Theorem B? For \(w \leq 12\) in no other (in the readings \(s^+\) and \(E\)).

In particular, is there an even \(n\) with a counterexample inside the printed range, and is there, for each fixed \(k \geq 3\), a width beyond which four sign changes fail in row \(k\)? Both families have long runs of \(2\)s in their quiddities.

Is there a structural description of the pairs with fewer than four sign changes? The infinitesimal version holds for every \(k\) [ , around the constant frieze pattern with quiddity \(q_n^* = (2\cos \frac{\pi }{n}, \dots , 2\cos \frac{\pi }{n})\).

How large is

(when such pairs exist), the distance from the constant frieze pattern at which four sign changes first fail for integer frieze patterns?

In the reading \(s^-\) (zeros deleted) the answer is negative much earlier: there are integer pairs with \(s^- = 2\) in row \(3\) of width \(6\), and pairs whose non-zero differences all have the same sign in widths \(7\) and \(8\); for example \(a = (1,2,2,3,3,2,1,3,4,1,5)\) and \(b = (1,3,1,5,4,1,2,2,3,2,3)\) give, in row \(3\) of width \(8\), \(d = (0,2,1,0,0,0,0,1,0,0,0)\), with \(s^- = 0\), \(s^+ = 8\) and \(E = 10\). These examples all have zeros.

For every even \(n \geq 14\) there are two integer frieze patterns of width \(n - 3\) whose middle rows (\(k = n/2 - 1\), the diameters of the \(n\)-gon) are strictly ordered: \(d_i {\gt} 0\) for all \(i\).

For every odd \(n \geq 15\) there are two integer frieze patterns of width \(w = n - 3\) whose rows \(k = \lfloor w/2 \rfloor = (n-3)/2\) differ with no zero and exactly two sign changes.