Sets of period sets for words of length n.

We consider finite words of length n. Each word has a set of periods, but many words can have the same set of periods. For a definition of a period of a word, see [1]. A set of periods is a subset of the set {0, 1, ..., n-1}, but not all subsets of {0, 1, ..., n-1} are period sets. The set denoted Gamma(n) contains all possible period sets corresponding to at least one word of length n. For a definition of Gamma(n), see references [1] or [2]. For more details see reference [4].

The series of text files provide the list of period set, one per line, for Gamma(n), for n = 1, 2, ..., 100.Each line contains a list of integers sorted by increasing value: this list constitutes one period set. The  separator symbol is a space. Hence, the number of non-empty lines in a file gives the cardinality of Gamma(n).The period set are sorted by their basic period.  For a definition of the notion of basic period, see [1] or [2].

The files for n=61, ..., 100, where computed with the incremental algorithm described in [5].

The sequence of the cardinalities of the set Gamma(n), is also called, the Number of distinct autocorrelations of binary words of length n, and corresponds to the sequence A005434 in the Encyclopedia of Integer Sequences (EOIS) link [3].

The files have generic name formatted as follows: gamma.n.bpswhere n is the word length, for n = 1, 2, ..., 100.

References:

1. Eric Rivals, Sven Rahmann.   Combinatorics of Periods in Strings.   Proc. 28th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science vol. 2076, p. 615-26., P. Orejas, P. G. Spirakis, J. van Leuween editors, Springer Verlag, Berlin, 2001.   doi: https://doi.org/10.1007/3-540-48224-5_512. Eric Rivals, Sven Rahmann.   Combinatorics of Periods in Strings.   Journal of Combinatorial Theory - Series A, 104(1), p. 95-113, October 2003   doi: https://doi.org/10.1016/s0097-3165(03)00123-73. Entry A005434 from The On-Line Encyclopedia of Integer Sequences.   URL: https://oeis.org/A0054344. Autocorrelation of Strings. A comment on entries A005434 and A045690 of the Encyclopedia of Integer Sequences.   URL: https://www.lirmm.fr/~rivals/RESEARCH/PERIOD/ 5. Eric Rivals. Incremental computation of the set of period sets. arXiv:2410.12077,  2024. https://doi.org/10.48550/arXiv.2410.12077