Counting tame $SL_3$- and $SL_4$- frieze patterns over finite fields
Lucas Surmann
TL;DR
This paper develops a complete counting framework for tame $SL_3$- and $SL_4$-frieze patterns over finite fields, tying these objects to projective configurations via $C_k(n)$ and, when needed, the refined sets $C_k^*(n)$. It establishes a general counting principle: for $g=\gcd(k,n)$, the number of patterns is governed by orbit counts of $C_k(n)$ under $\mathrm{PGL}(k,q)$ when $g=1$, and by $C_k^*(n)$ with a $(q-1)^{g-1}$ factor when $g>1$, with connections to Grassmannian counts in the gcd$=1$ case. The authors work out the case $k=3$ completely, including the intricate $3\mid n$ scenario, and then perform the substantial combinatorial and algebraic work to resolve $k=4$ for both odd and even $n$, including the gcd$=2$ and gcd$=4$ subcases. Their approach blends quiddity-sequence methods, Plücker-coordinate relations, and recursion-based combinatorics to yield explicit closed forms for many counting functions $f_q(k,n)$, thus advancing the understanding of frieze-pattern enumeration over finite fields and their Grassmannian origins.
Abstract
In this article we count tame $ SL_3 $- and $ SL_4 $-frieze patterns with width $ w $ over a finite field $ K $, as well as some tame $ SL_k $-frieze patterns for higher $ k $. Let $ n = w + k + 1 $. We consider the sets $ C_k(n) $ of tuples of $ n $ points in the projective space $ \mathbb{P}^{k-1}(K) $, such that $ k $ consecutive points are always independent (the first and last point in the tuple are considered to be consecutive). First assume $ \gcd(k,n) = 1 $. In this case, we prove that the problem of counting tame $ SL_k $-frieze patterns can be reduced to counting $ C_k(n) $. We also show that $ \lvert C_k(n) \rvert $ is essentially already known as long as $ k $ and $ n $ are coprime, and we derive the number of tame $ SL_k $-frieze-patterns in that case. In the case $ \gcd(k,n) \neq 1 $, we define certain subsets $ C_k^*(n) $ and show that it is sufficient to count these sets. Afterwards, we count $ C_k^*(n)$ in the cases $ k = 3 $ and $ k = 4 $ and thus the number of tame $ SL_3 $- and $ SL_4 $-frieze patterns for any width $ w $.