patternsnumber

Patternsnumber is a numerical attribute used in combinatorics on words to quantify the diversity of pattern structures within a finite word. For a word w over a finite alphabet, patternsnumber(w) is defined as the number of distinct pattern templates that occur among all substrings of w. A pattern template is the equivalence class of a substring under pattern isomorphism: two substrings are equivalent if one can be obtained from the other by renaming the letters via a bijection that preserves the relative positions of equal letters. For example, in the word aba, the substrings are a, b, ab, ba, and aba, with patterns 1, 1, 12, 12, and 121 respectively. The distinct patterns are {1, 12, 121}, so patternsnumber(aba) = 3.

Computation of patternsnumber typically proceeds by enumerating substrings and replacing each substring with its canonical pattern form, then counting distinct forms. A naive approach runs in roughly cubic time in the length of the word, since there are O(n^2) substrings and each pattern normalization can take O(n) time. Practical methods use optimizations based on suffix trees or automata to reduce constant factors, though the theoretical complexity remains dependent on word length and alphabet size. In practice, patternsnumber serves as a proxy for repetitiveness and structural variety in strings, rather than as a single fixed invariant.

Properties of patternsnumber include invariance under renaming of the alphabet, and the general bound 1 ≤ patternsnumber(w) ≤ n(n+1)/2 for a word of length n, with many strings yielding far smaller values due to repeated pattern types. Related concepts include pattern complexity and the study of equivalence classes of substrings in combinatorics on words.