We introduced the notation of a set of prohibitions and give definitions of a complete set and a crucial word with respect to a given set of prohibitions. We consider 3 particular sets which appear in different areas of mathematics and for each of them examine the length of a crucial word. One of these sets is proved to be incomplete. The problem of determining lengths of words that are free from a set of prohibitions is shown to be NP-complete, although the related problem of whether or not a given set of prohibitions is complete is known to be effectively solvable.
Recently, Grytczuk, Kordulewski, and Niewiadomski defined an extremal word over an alphabet $\mathbb{A}$ to be a word with the property that inserting any letter from $\mathbb{A}$ at any position in the word yields a given pattern. In this paper, we determine the number of extremal $XY_1XY_2X\dots XY_tX$-avoiding words on a $k$-letter alphabet. We also derive a lower bound on the shortest possible length of an extremal square-free word on a $k$-letter alphabet that grows exponentially in $k$.
A word is \emph{square-free} if it does not contain non-empty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over $3$-letter alphabet. We consider a new type of square-free words. A square-free word is \emph{extremal} if it cannot be extended to a new square-free word by inserting a single letter on arbitrary position. We prove that there exist infinitely many extremal words over $3$-letter alphabet. Some parts of our construction relies on computer verifications. We also pose some related open problems.
A word is square-free if it does not contain any square (a word of the form $XX$), and is extremal square-free if it cannot be extended to a new square-free word by inserting a single letter at any position. Grytczuk, Kordulewski, and Niewiadomski proved that there exist infinitely many ternary extremal square-free words. We establish that there are no extremal square-free words over any alphabet of size at least 17.
An overlap-free (or $β$-free) word $w$ over a fixed alphabet $Σ$ is extremal if every word obtained from $w$ by inserting a single letter from $Σ$ at any position contains an overlap (or a factor of exponent at least $β$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every extended real number $β$ such that $2^+\leqβ\leq 8/3$, we show that there are arbitrarily long extremal $β$-free binary words.
A square-free word $w$ over a fixed alphabet $Σ$ is extremal if every word obtained from $w$ by inserting a single letter from $Σ$ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal square-free word, and demonstrated that there are arbitrarily long extremal square-free ternary words. We find all lengths which admit an extremal square-free ternary word. In particular, we show that there is an extremal square-free ternary word of every sufficiently large length. We also solve the analogous problem for circular words.
We prove that the Fibonacci word $f$ satisfies among all characteristic Sturmian words, three interesting extremal properties. The first concerns the length and the second the minimal period of its palindromic prefixes. Each of these two properties characterizes $f$ up to a renaming of its letters. A third property concerns the number of occurrences of the letter $b$ in its palindromic prefixes. It characterizes uniquely $f$ among all characteristic Sturmian words having the prefix $abaa$.
Given an infinite word X over an alphabet A a letter b occurring in X, and a total order σon A, we call the smallest word with respect to σstarting with b in the shift orbit closure of X an extremal word of X. In this paper we consider the extremal words of morphic words. If X = g(f^ω(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantees that all extremal words are morphic. This happens, in particular, when X is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the Period-doubling word and the…
One of the prominent areas of research in graph theory is the degree-diameter problem, in which we seek to determine how many vertices a graph may have when constrained to a given degree and diameter. Different variants of this problem are obtained by considering further restrictions on the graph, such as whether it is directed, undirected, mixed, vertex-transitive etc. The currently best known extremal constructions are the Gómez graphs, both when considered as directed and undirected. As the Gómez graphs are similar to the Faber-Moore-Chen graphs, we give a natural generalisation of their de…
A word is "crucial" with respect to a given set of "prohibited words" (or simply "prohibitions") if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A "minimal crucial word" is a crucial word of the shortest length. A word W contains an "abelian k-th power" if W has a factor of the form X_1X_2...X_k where X_i is a permutation of X_1 for 2<= i <= k. When k=2 or 3, one deals with "abelian squares" and "abelian cubes", respectively. In 2004 (arXiv:math/0205217), Evdokimov and Kitaev showed that a minimal crucial word o…
Let $[n]=\{1, \ldots, n\}$. A sequence $u=a_1a_2\dots a_l$ over $[n]$ is called $k$-sparse if $a_i = a_j$, $i > j$ implies $i-j\geq k$. In other words, every consecutive subsequence of $u$ of length at most $k$ does not have letters in common. Let $u,v$ be two sequences. We say that $u$ is $v$-free, if $u$ does not contain a subsequence isomorphic to $v$. Suppose there are only $k$ letters appearing in $v$. The extremal function Ex$(v,n)$ is defined as the maximum length of all the $v$-free and $k$-sparse sequences. In this paper, we study a generalization of the extremal function Ex$(v,n)$.
We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincaré series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are expressed in terms of Lyndon words in an appropriate language, thereby relating counting of BPS states to the …
Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in distribution of real numbers modulo 1 via combinatorics on words, we survey some combinatorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some of which have been rediscovered several times.
In a 1983 paper, G. Ramharter asks what are the extremal arrangements for the cyclic analogues of the regular and semi-regular continuants first introduced by T.S. Motzkin and E.G. Straus in 1956. In this paper we answer this question by showing that for each set $A$ consisting of positive integers $1<a_1<a_2<\cdots <a_k$ and a $k$-term partition $P: n_1+n_2 + \cdots + n_k=n$, there exists a unique (up to reversal) cyclic word $x$ which maximizes (resp. minimizes) the regular cyclic continuant $K^{\circlearrowright}(\cdot)$ amongst all cyclic words over $A$ with Parikh vector $(n_1,n_2,\ldots,…
Abelian maximal pattern complexity and extremal words
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematics and Information Science, Shandong Technology and Business University（山东工商学院数学与信息科学学院）
Let us recall the well-known Shirshov's Height Theorem. "Let A be a finitely generated algebra of degree d. Then there exists a finite set Y which is the subset of A that A has and an integer h' = h(A) such that A has Shirshov's height h' over set Y. For Y we may take the set of words of length <d. Such Y is called a Shirshov's base of algebra A." Shirshov's original proof was purely combinatorical, but did not provide a reasonable upper estimate for the height. Kolotov obtained an estimate for h(A) < s^(s^m) (m = deg(A), and s is the number of generators) in 1981. I. Zelmanov asked for an exp…
A finite word $w\in\{0,1\}^*$ is balanced if for every equal-length factors $u$ and $v$ of every cyclic shift of $w$ we have $||u|_1-|v|_1| <= 1$. This new class of finite words were defined in [JZ]. In [J], there was proved several results considering finite balanced words and majorization. One of the main results was that the base-2 orbit of the balanced word is the least element in the set of orbits with respect to partial sum. It was also proved that the product of the elements in the base-2 orbit of a word is maximized precisely when the word is balanced. It turns out that the words $0^{q…
In this paper we study the following problem. Let $A$ be a fixed graph, and let $\hom(G,A)$ denote the number of homomorphisms from a graph $G$ to $A$. Furthermore, let $v(G)$ denote the number of vertices of $G$, and let $\mathcal{G}_d$ denote the family of $d$--regular graphs. The general problem studied in this paper is to determine $$\inf_{G\in \mathcal{G}_d}\hom(G,A)^{1/v(G)}.$$ It turns out that in many instances the infimum is not achieved by a finite graph, but a sequence of graphs with girth (i. e., length of the shortest cycle) tending to infinity. In other words, the optimization pr…
This paper analyses stable commutator length in groups Z^r * Z^s. We bound scl from above in terms of the reduced wordlength (sharply in the limit) and from below in terms of the answer to an associated subset-sum type problem. Combining both estimates, we prove that, as n tends to infinity, words of reduced length n generically have scl arbitrarily close to n/4 - 1. We then show that, unless P=NP, there is no polynomial time algorithm to compute scl of efficiently encoded words in F2. All these results are obtained by exploiting the fundamental connection between scl and the geometry of certa…
Turán's famous tetrahedron problem is to compute the Turán density of the tetrahedron $K_4^3$. This is equivalent to determining the maximum $\ell_1$-norm of the codegree vector of a $K_4^3$-free $n$-vertex $3$-uniform hypergraph. We introduce a new way for measuring extremality of hypergraphs and determine asymptotically the extremal function of the tetrahedron in our notion. The codegree squared sum, $\text{co}_2(G)$, of a $3$-uniform hypergraph $G$ is the sum of codegrees squared $d(x,y)^2$ over all pairs of vertices $xy$, or in other words, the square of the $\ell_2$-norm of the codegree v…
On the Number of Rational Power Factors in a Finite Word
Hangzhou International Innovation Institute of Beihang University（北京航空航天大学杭州创新研究院）；LMIB-School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）
Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a $d$-regular graph on $n$ vertices is at most $(2^{d+1}-1)^{n/2d}$ by the Kahn-Zhao theorem. Relaxing the regularity constraint to a minimum degree condition, Galvin conjectured that, for $n\geq 2d$, the number of independent sets in a graph with $δ(G)\geq d$ is at most that in $K_{d,n-d}$. In this paper, we give a lower bound on the number of independent sets in a $d$-regular graph mirroring the upper bound in the Kahn-Zhao theorem. The main r…
