Related papers: Counting the occurrences of generalized patterns i…
A word is level if each letter appears in it the same number of times, plus or minus 1. We give a complete characterization of the lengths for which level ternary circular square-free words exist. Key words: combinatorics on words, circular…
A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A…
We study the properties of the sequence of words $(B_i)$, where $B_1 = 101$ and $B_{i+1} = B_i C_i$ for $i \geq 1$, where $C_i$ is $B_i$ with the first $i$ symbols removed, and the infinite binary sequence ${\bf b} = 10101101011011101…
We develop the technique of reduced word manipulation to give a range of results concerning reduced words and permutations more generally. We prove a broad connection between pattern containment and reduced words, which specializes to our…
We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on…
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…
Prompted models have demonstrated impressive few-shot learning abilities. Repeated interactions at test-time with a single model, or the composition of multiple models together, further expands capabilities. These compositions are…
Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of…
In the unidimensional unfolding model, given m objects in general position there arise 1+m(m-1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. By changing these m objects, we can generate various…
We describe new, simple, recursive methods of construction for orientable sequences over an arbitrary finite alphabet, i.e. periodic sequences in which any sub-sequence of n consecutive elements occurs at most once in a period in either…
Recently, Bill Chen, together with his disciples Alvin Dai and Robin Zhou, discovered, and very elegantly proved, an algebraic equation satisfied by the generating function enumerating 123-avoiding words with two occurrences of each of 1,…
A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. In a recent paper, tight general bounds on the block entropy of patterns of…
We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all…
A graph $G = (\{1, 2, \ldots, n\}, E)$ is $12$-representable if there is a word $w$ over $\{1, 2, \ldots, n\}$ such that two vertices $i$ and $j$ with $i < j$ are adjacent if and only if every $j$ occurs before every $i$ in $w$. These…
In this paper, we prove results on enumerations of sets of Rota-Baxter words in a finite number of generators and a finite number of unary operators. Rota-Baxter words are words formed by concatenating generators and images of words under…
In this note we present a characterisation of all unary and binary patterns that do not only contain variables, but also reversals of their instances. These types of variables were studied recently in either more general or particular…
We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit…
A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern $p$ is unavoidable if, over every finite alphabet, every sufficiently long word encounters $p$. A theorem by…
In a language corpus, the probability that a word occurs $n$ times is often proportional to $1/n^2$. Assigning rank, $s$, to words according to their abundance, $\log s$ vs $\log n$ typically has a slope of minus one. That simple Zipf's law…
We count the number of distinct (scattered) subwords occurring in the base-b expansion of the non-negative integers. More precisely, we consider the sequence $(S_b(n))_{n\ge 0}$ counting the number of positive entries on each row of a…