Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems

We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite words occurring in this paper belong to the language of a Parry numeration system satisfying the Bertrand property, i.e., we can add or remove trailing zeroes to valid representations. It is a folklore fact that the Sierpi\'{n}ski gasket is the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from the classical Pascal triangle modulo $2$. In a similar way, we describe and study the subset of $[0, 1] \times [0, 1]$ associated with the latter generalization of the Pascal triangle modulo a prime number.Comment: 30 pages; 32 figure

Avoiding 5/4-powers on the alphabet of nonnegative integers

We identify the structure of the lexicographically least word avoiding 5/4-powers on the alphabet of nonnegative integers. Specifically, we show that this word has the form $p \tau(\varphi(z) \varphi^2(z) \cdots)$ where $p, z$ are finite words, $\varphi$ is a 6-uniform morphism, and $\tau$ is a coding. This description yields a recurrence for the $i$th letter, which we use to prove that the sequence of letters is 6-regular with rank 188. More generally, we prove $k$-regularity for a sequence satisfying a recurrence of the same type.Comment: 35 pages, 3 figure

Automatic sequences: from rational bases to trees

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider these built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with $r$-block substitutions where $r$ morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence.Comment: 25 pages, 15 figure

Pascal triangles and Sierpiński gasket extended to binomial coefficients of words

The binomial coefficient (u,v) of two finite words u and v (on a finite alphabet) is the number of times the word v appears inside the word u as a subsequence (or, as a "scattered" subword). For instance, (abbabab,ab)=4. This concept naturally extends the classical binomial coefficients of integers, and has been widely studied for about thirty years (see, for instance, Simon and Sakarovitch). In this talk, I present the research lead from October 2015 on an extension of the Pascal triangles to base-2 expansions of integers. In a first part, I define two new objects that both generalize the classical Pascal triangle and the Sierpinski gasket. In a second part, I define a new sequence extracted from the Pascal triangle in base 2 and study its regularity. In a third part, I exhibit an exact formula for the behavior of the summatory function of the latter sequence

Some generalizations of the Pascal triangle: base 2 and beyond

The binomial coefficient (u,v) of two finite words u and v (on a finite alphabet) is the number of times the word v appears inside the word u as a subsequence (or, as a "scattered" subword). For instance, (abbabab,ab)=4. This concept naturally extends the classical binomial coefficients of integers, and has been widely studied for about thirty years (see, for instance, Simon and Sakarovitch). In this talk, I present the research lead from October 2015: I give the main ideas that lead to an extension of the Pascal triangles to base-2 expansions of integers. After that, I extend the work to any Parry-Bertrand numeration system including the Fibonacci numeration system

Generalized Pascal triangles to binomial coefficients of finite words

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we show the existence of a subset of [0, 1]×[0, 1] associated with this extended Pascal triangle modulo a prime p

Generalized Pascal triangles for binomial coefficients of words: a short introduction

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p

Triangles de Pascal généralisés et coefficients binomiaux de mots

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p. From the extended Pascal triangle obtained when p is equal to 2, we derive a sequence of which we study the regularity and the asymptotic behavior of the summatory function. Inspired from this regularity, we extend our results to another famous numeration systems, namely the Zeckendorff numeration system

Automatic sequences in rational base numeration systems (and even more)

The nth term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of n in a suitable numeration system. Here, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider these built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with r-block substitutions where r morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence