A fundamental concept related to strings is that of repetitions. It has been extensively studied in many versions, from both purely combinatorial and algorithmic angles. One of the most basic questions is how many distinct squares, i.e., distinct strings of the form UU, a string of length n can contain as fragments. It turns out that this is always 𝒪(n), and the bound cannot be improved to sublinear in n [Fraenkel and Simpson, JCTA 1998]. Several similar questions about repetitions in strings have been considered, and by now we seem to have a good understanding of their repetitive structure. For higher-dimensional strings, the basic concept of periodicity has been successfully extended and applied to design efficient algorithms - it is inherently more complex than for regular strings. Extending the notion of repetitions and understanding the repetitive structure of higher-dimensional strings is however far from complete. Quartics were introduced by Apostolico and Brimkov [TCS 2000] as analogues of squares in two dimensions. Charalampopoulos, Radoszewski, Rytter, Waleń, and Zuba [ESA 2020] proved that the number of distinct quartics in an n×n 2D string is 𝒪(n²log²n) and that they can be computed in 𝒪(n²log²n) time. Gawrychowski, Ghazawi, and Landau [SPIRE 2021] constructed an infinite family of n×n 2D strings with Ω(n²log n) distinct quartics. This brings the challenge of determining asymptotically tight bounds. Here, we settle both the combinatorial and the algorithmic aspects of this question: the number of distinct quartics in an n×n 2D string is 𝒪(n²log n) and they can be computed in the worst-case optimal 𝒪(n²log n) time. As expected, our solution heavily exploits the periodic structure implied by occurrences of quartics. However, the two-dimensional nature of the problem introduces some technical challenges. Somewhat surprisingly, we overcome the final challenge for the combinatorial bound using a result of Marcus and Tardos [JCTA 2004] for permutation avoidance on matrices.

Given two indeterminate equal-length strings p and t with a set of characters per position in both strings, we obtain a determinate string p_w from p and a determinate string t_w from t by choosing one character per position. Then, we say that p and t match when p_w and t_w match for some choice of the characters. While the most standard notion of a match for determinate strings is that they are simply identical, in certain applications it is more appropriate to use other definitions, with the prime examples being parameterized matching, order-preserving matching, and the recently introduced Cartesian tree matching. We provide a systematic study of the complexity of string matching for indeterminate equal-length strings, for different notions of matching. We use n to denote the length of both strings, and r to be an upper-bound on the number of uncertain characters per position. First, we provide the first polynomial time algorithm for the Cartesian tree version that runs in deterministic 𝒪(nlog² n) and expected 𝒪(nlog nlog log n) time using 𝒪(nlog n) space, for constant r. Second, we establish NP-hardness of the order-preserving version for r=2, thus solving a question explicitly stated by Henriques et al. [CPM 2018], who showed hardness for r=3. Third, we establish NP-hardness of the parameterized version for r=2. As both parameterized and order-preserving indeterminate matching reduce to the standard determinate matching for r=1, this provides a complete classification for these three variants.