Nathan Fox

A Graph Polynomial Approach to Primitivity

Recently, Tittmann et al. introduced the subgraph component polynomial and showed that its power for distinguishing graphs is quite different from the power of other graph polynomials that appear in the literature such as the matching polynomial, the Tutte polynomial, the characteristic polynomial, the chromatic polynomial, etc. The subgraph component polynomial enumerates vertex induced subgraphs in a given undirected graph with respect to the number of components. We show the use of the subgraph component polynomial to count the number of primitive partial words of a given length over an alphabet of a fixed size, which leads to a method for enumerating such partial words.

On the Asymptotic Abelian Complexity of Morphic Words

The subword complexity of an infinite word counts the number of subwords of a given length, while the abelian complexity counts this number up to letter permutation. Although a lot of research has been done on the subword complexity of morphic words, i.e., words obtained as fixed points of iterated morphisms, little is known on their abelian complexity. In this paper, we undertake the classification of the asymptotic growths of the abelian complexities of fixed points of binary morphisms. Some general results we obtain stem from the concept of factorization of morphisms. We give an algorithm that yields all canonical factorizations of a given morphism, describe how to use it to check quickly whether a binary morphism is Sturmian, discuss how to fully factorize the Parry morphisms, and finally derive a complete classification of the abelian complexities of fixed points of uniform binary morphisms.

Linear recurrent subsequences of generalized meta-Fibonacci sequences

In a recent paper, Frank Ruskey asked whether every linear recurrent sequence can occur in some solution of a meta-Fibonacci sequence. In this paper, we consider the natural generalization of meta-Fibonacci recurrences to more than two terms. In this context, we show, using an explicit construction, that any sequence satisfying a linear recurrence with positive coefficients occurs as an evenly-spaced subsequence in some generalized meta-Fibonacci sequence.

Abelian-primitive partial words☆

Abstract In this paper we count the number of abelian-primitive partial words of a given length over a given alphabet size, which are partial words that are not abelian powers. Partial words are sequences that may have undefined positions called holes. This combinatorial problem was considered recently for full words (those without holes). It turns out that, even for the full word case, it is a nontrivial problem as opposed to the counting of the number of primitive full words, well-known to be easily derived using the Mobius function.

Discovering Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation

Abstract The Hofstadter Q -sequence, with its simple definition, has defied all attempts at analyzing its behavior. Defined by a simple nested recurrence and an initial condition, the sequence looks approximately linear, though with a lot of noise. But, it is unknown whether the sequence is even infinite. In the years since Hofstadter published his sequence, various people have found variants with predictable behavior. Commonly, the variant sequences eventually satisfy linear recurrences. Proofs of such behaviors are inductive and highly automatable. This suggests that a search for more sequences like these may be fruitful. In this paper, we develop an algorithm to search for these sequences. Using this algorithm, we determine that such sequences come in infinite families that are themselves plentiful. In fact, there are hundreds of easy to describe families based on the Hofstadter Q -recurrence alone.