bifix codes and sturmian words we prove new results concerning the relation between bifix codes episturmian words and subgroups of free groups we study bifix codes in factorial sets of words we generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set f of words f maximal bifix codes in the case of bifix codes contained in sturmian sets of words we obtain several new results let f be a sturmian set of words defined as the set of factors of a strict episturmian word our results express the fact that an f maximal bifix code of degree d behaves just as the set of words of f of length d an f maximal bifix code of degree d in a sturmian set of words on an alphabet with k letters has k 1 d 1 elements this generalizes the fact that a sturmian set contains k 1 d 1 words of length d moreover given an infinite word x if there is a finite maximal bifix code x of degree d such that x has at most d factors of length d in x then x is ultimately periodic our main result states that any f maximal bifix code of degree d on the alphabet a is the basis of a subgroup of index d of the free group on a

linear transformations that are tridiagonal with respect to both eigenbases of a leonard pair let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let x denote the set of linear transformations x v v such that the matrix representing x with respect to the basis i is tridiagonal and the matrix representing x with respect to the basis ii is tridiagonal we show that x is spanned by i a a aa a a and these elements form a basis for x provided the dimension of v is at least 3

on brlek reutenauer conjecture brlek and reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2d u email protected n 0 t u n in which d u denotes the defect of u and t u n denotes c u n 1 c u n 2 p u n 1 p u n where c u and p u are the factor and palindromic complexity of u respectively brlek and reutenauer verified their conjecture for periodic infinite words using their result we prove the conjecture for uniformly recurrent words moreover we summarize results and some open problems related to defects which may be useful for the proof of the brlek reutenauer conjecture in full generality

on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

palindromic length of words and morphisms in class p abstract we study the palindromic length of factors of infinite words fixed by morphisms of the so called class p introduced by hof knill and simon we show that it grows at most logarithmically with the length of the factor for the fibonacci word and the thue morse word we provide explicit bounds on their rate of growth we also construct an infinite word rich in palindromes for which the palindromic length grows as n

burrows wheeler transform and palindromic richness the investigation of the extremal case of the burrows wheeler transform leads to study the words w over an ordered alphabet a a 1 a 2 a k with a 1

matrix units associated with the split basis of a leonard pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v it is known that there exists a basis for v with respect to which the matrix representing a is lower bidiagonal and the matrix representing a is upper bidiagonal in this paper we give some formulae involving the matrix units associated with this basis

characteristic morphisms of generalized episturmian words in a recent paper with l q zamboni the authors introduced the class of q episturmian words an infinite word over a is standard q episturmian where q is an involutory antimorphism of a if its set of factors is closed under q and its left special factors are prefixes when q is the reversal operator one obtains the usual standard episturmian words in this paper we introduce and study q characteristic morphisms that is morphisms which map standard episturmian words into standard q episturmian words they are a natural extension of standard episturmian morphisms the main result of the paper is a characterization of these morphisms when they are injective in order to prove this result we also introduce and study a class of biprefix codes which are overlap free i e any two code words do not overlap properly and normal i e no proper suffix prefix of any code word is left right special in the code a further result is that any standard q episturmian word is a morphic image by an injective q characteristic morphism of a standard episturmian word

sharp tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfies the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 a conjecture of tatsuro ito and the second author states that if k is algebraically closed then a a is sharp in order to better understand and eventually prove the conjecture in this paper we begin a systematic study of the sharp tridiagonal pairs our results are summarized as follows assuming a a is sharp and using the data a vi i 0d a vi i 0d we define a finite sequence of scalars called the parameter array we display some equations that show the geometric significance of the parameter array we show how the parameter array is affected if is replaced by a vi i 0d a vi i 0d or a vd i i 0d a vi i 0d or a vi i 0d a vd i i 0d we prove that if the isomorphism class of is determined by the parameter array then there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v

a classification of sharp tridiagonal pairs let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide the pair a a is called sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp in this paper we classify up to isomorphism the sharp tridiagonal pairs as a corollary we classify up to isomorphism the tridiagonal pairs over an algebraically closed field we obtain these classifications by proving the conjecture

factors of characteristic words location and decompositions let a be an irrational number with 0 a 1 and let f a be the characteristic word of a the location of a factor w of f a is defined to be the set of all positions in f a at which w occurs in this paper an explicit formula of the location of each factor w of f a is obtained the decompositions red a w and sod a w of f a associated with w are established the former one involves all return words of w while the latter one involves all occurrences of w and the corresponding separate factors and overlap factors these results generalize the work by the present authors 2005 z x wen and z y wen 1994 g melancon 1999 w t cao and z y wen 2003 i m araujo and v bruyere 2005 and a glen 2006

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

balanced leonard pairs let k denote a field and let v denote a vector space over k with finite positive dimension by a leonard pair on v we mean an ordered pair of linear transformations a v v and a v v that satisfy the following two conditions i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal let v0 v1 vd respectively v0 v1 vd denote a basis for v that satisfies i respectively ii for 0 i d let ai denote the coefficient of vi when we write avi as a linear combination of v0 v1 vd and let ai denote the coefficient of vi when we write a vi as a linear combination of v0 v1 vd in this paper we show a0 ad if and only if a0 ad moreover we show that for d 1 the following are equivalent i a0 ad and a1 ad 1 ii a0 ad and a1 ad 1 iii ai ad i and ai ad i for 0 i d these give a proof of a conjecture by the second author we say a a is balanced whenever ai ad i and ai ad i for 0 i d we say a a is essentially bipartite respectively essentially dual bipartite whenever ai respectively ai is independent of i for 0 i d observe that if a a is essentially bipartite or dual bipartite then a a is balanced for d 2 we show that if a a is balanced then a a is essentially bipartite or dual bipartite

lattices and multi dimensional words in the present paper we develop a formalism to generate multi dimensional words using lattices which generalizes the construction of real numbers one dimensional words from a sequence of partial quotients using continued fractions the construction was introduced in a special case by simpson and tijdeman in order to derive a multi dimensional generalization of the theorem of fine and wilf we show that the produced multi dimensional words are intrinsically connected with k dimensional sturmian words

episturmian words and episturmian morphisms infinite episturmian words are a generalization of sturmian words which includes the arnoux rauzy sequences we continue their study and that of episturmian morphisms begun previously in relation with the action of the shift operator palindromic and periodic factors of these words are described we consider in particular the case where these words are generated by morphisms and introduce then a notion of intercept generalizing that of sturmian words finally we prove that the frequencies of the factors in a strong sense do exist for all episturmian words

replacing model categories with simplicial ones in this paper we show that model categories of a very broad class can be replaced up to quillen equivalence by simplicial model categories

balance properties of multi dimensional words a word u is called 1 balanced if for any two factors v and w of u of equal length we have 1 v i w i1 for each letter i where v i denotes the number of occurrences of i in the factor v the aim of this paper is to extend the notion of balance to multi dimensional words we first characterize all 1 balanced words on zn in particular we prove they are fully periodic for n 1 we then give a quantitative measure of non balancedness for some words on z2 with irrational density including two dimensional sturmian words

standard sturmiam morphisms abstract we give a characterization of morphisms which preserve finite and infinite standard sturmian words the class of such morphisms coincides with the monoid d e of the endomorphisms of a where a a b generated by the two elementary morphisms e which interchanges the letter a with b and d which is the fibonacci morphism defined as d a ab d b a some new properties of these morphisms are shown in particular we prove the following result let x n n 0 be the sequence of sets inductively defined as x 0 e x 1 a x n 1 ax n for n 1 where e denotes the empty word and is the operator of left palindrome closure associating to each word w the shortest word having w as suffix for all n 0 one has x n 1 e g d x n a where denotes the identity map from this we derive a new characterization of the set per of all words w having two periods p and q which are coprimes and such that w p q 2 indeed one has that per aa xcup n 0 e l od x n a

towards a classification of the tridiagonal pairs let k denote a field and let v denote a vector space over k with finite positive dimension let end v denote the k algebra consisting of all k linear transformations from v to v we consider a pair a a end v that satisfy i iv below i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v let e0 denote the element of end v such that e0 i v0 0 and e0 vi 0 for 1 i d let d resp d denote the k subalgebra of end v generated by a resp a in this paper we prove that the span of e0 dd de0 equals the span of e0 de0 de0 and that the elements of e0 de0 mutually commute we relate these results to some conjectures of tatsuro ito and the second author that are expected to play a role in the classification of tridiagonal pairs

the determinant of aa a a for a leonard pair a a abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in this paper we investigate the commutator aa a a our results are as follows abbreviate d dim v 1 and first assume d is odd we show aa a a is invertible and display several attractive formulae for the determinant next assume d is even we show that the null space of aa a a has dimension 1 we display a nonzero vector in this null space we express this vector as a sum of eigenvectors for a and as a sum of eigenvectors for a

a new characteristic property of rich words originally introduced and studied by the third and fourth authors together with j justin and s widmer 2008 rich words constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes several characterizations of rich words have already been established a particularly nice characteristic property is that all complete returns to palindromes are palindromes in this note we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix

extensions of rich words a word w is rich if it has w 1 many distinct palindromic factors including the empty word this article contains several results about rich words particularly related to extending them a word w can be eventually extended richly in n ways if there exists a finite word u and n distinct letters a alph w such that wua is rich we will prove that every non unary rich word can be eventually extended richly in at least two different ways but not always in three or more ways we will also prove that every rich word can be extended to both periodic and aperiodic infinite rich words the defect of a finite word w is defined by d w w 1 pal w this concept has been studied in various papers here we will define a new concept infinite defect for a finite word w the definition is d w min d z zis an infinite word which has factorw we will show that the infinite defect of a finite word is always finite and give some upper bounds for it the difference between defect and infinite defect is also investigated we will also give an upper and a lower bound for the number of rich words a new class of words two dimensional rich words is also introduced

tridiagonal pairs and the q tetrahedron algebra abstract the q tetrahedron algebra q was recently introduced and has been studied in connection with tridiagonal pairs in this paper we further develop this connection let k denote an algebraically closed field and let q denote a nonzero scalar in k that is not a root of unity let v denote a vector space over k with finite positive dimension and let a a denote a tridiagonal pair on v let i i 0 d resp i i 0 d denote a standard ordering of the eigenvalues of a resp a ito and terwilliger have shown that when i q 2 i d and i q d 2 i 0 i d there exists an irreducible q module structure on v such that the q generators x 01 x 23 act as a a respectively in this paper we examine the case in which there exists a nonzero scalar c in k such that i q 2 i d and i q 2 i d cq d 2 i for 0 i d in this case we associate to a a a polynomial p in one variable and prove the following theorem as our main result theorem the following are equivalent 1 i there exists a q module structure on v such that x 01 acts as a and x 30 cx 23 acts as a where x 01 x 30 x 23 are standard generators for q 2 ii p q 2 d 2 q q 1 2 0 suppose i and ii hold then the q module structure on v is unique and irreducible

sturmian words structure combinatorics and their arithmetics abstract we prove some new results concerning the structure the combinatorics and the arithmetics of the set per of all the words w having two periods p and q p q which are coprimes and such that w p q 2 a basic theorem relating per with the set of finite standard sturmian words was proved in de luca and mignosi 1994 the main result of this paper is the following simple inductive definition of per the empty word belongs to per if w is an already constructed word of per then also aw and bw belong to per where denotes the operator of palindrome left closure i e it associates to each word u the smallest palindrome word u having u as a suffix we show that by this result one can construct in a simple way all finite and infinite standard sturmian words we prove also that up to the automorphism which interchanges the letter a with the letter b any element of per can be codified by the irreducible fraction p q this allows us to construct for any n 0 a natural bijection that we call farey correspondence of the set of the farey series of order n 1 and the set of special elements of length n of the set st of all finite sturmian words finally we introduce the concepts of farey tree and farey monoid this latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q

rauzy substitutions and multi dimensional sturmian words recently berthe and the author developed a method to construct a multi dimensional sturmian word as a limit of a sequence of words on lattice domains in the present paper the reach of the approach is studied in the special case of rauzy substitutions

leonard pairs having zero diagonal td td form abstract fix an algebraically closed field f and an integer n 1 let mat n f denote the f algebra consisting of the n n matrices that have all entries in f we consider a pair of diagonalizable matrices in mat n f each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one such a pair is called a leonard pair in mat n f in the present paper we find all leonard pairs a a in mat n f such that each of a and a is irreducible tridiagonal with all diagonal entries 0 this solves a problem given by paul terwilliger

powers in a class of a strict standard episturmian words this paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic sturmian words in particular we explicitly determine all integer powers occurring in such infinite words extending recent results of damanik and lenz d damanik d lenz powers in sturmian sequences european j combin 24 2003 377 390 doi 10 1016 s0195 6698 03 00026 x who studied powers in sturmian words the key tools in our analysis are canonical decompositions and a generalization of singular words which were originally defined for the ubiquitous fibonacci word our main results are demonstrated via some examples including the k bonacci word a generalization of the fibonacci word to a k letter alphabet k 2

on the least number of palindromes contained in an infinite word we investigate the least number of palindromic factors in an infinite word we first consider general alphabets and give answers to this problem for periodic and non periodic words closed or not under reversal of factors we then investigate the same problem when the alphabet has size two

balancing and clustering of words in the burrows wheeler transform compression algorithms based on burrows wheeler transform bwt take advantage of the fact that the word output of bwt shows a local similarity and then turns out to be highly compressible the aim of the present paper is to study such clustering effect by using notions and methods from combinatorics on words the notion of balance of a word plays a central role in our investigation empirical observations suggest that balance is actually the combinatorial property of input word that ensure optimal bwt compression moreover it is reasonable to assume that the more balanced the input word is the more local similarity we have after bwt and therefore the better the compression is this hypothesis is here corroborated by experiments on real text by using local entropy as a measure of the degree of balance of a word in the setting of combinatorics on words a sound confirmation of previous hypothesis is given by a result of mantaci et al 2003 27 which states that in the case of a binary alphabet there is an equivalence between circularly balanced words words having a clusterized bwt and the conjugates of standard words in the case of alphabets of size greater than two there is no more equivalence the last section of the present paper is devoted to investigate the relationships between these notions and other related ones as for instance palindromic richness in the case of a general alphabet

the structure of a tridiagonal pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of k linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide in this paper we show that the following i iv hold provided that k is algebraically closed i each of v0 v0 vd vd has dimension 1 ii there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for all u v v iii there exists a unique anti automorphism of end v that fixes each of a a iv the pair a a is determined up to isomorphism by the data i i 0d i i 0d i i 0d where i resp i is the eigenvalue of a resp a on vi resp vi and i i 0d is the split sequence of a a corresponding to i i 0d and i i 0d

on k abelian palindromes abstract a word is called a palindrome if it is equal to its reversal in the paper we consider a k abelian modification of this notion two words are called k abelian equivalent if they contain the same number of occurrences of each factor of length at most k we say that a word is a k abelian palindrome if it is k abelian equivalent to its reversal a question we deal with is the following how many distinct palindromes can a word contain it is well known that a word of length n can contain at most n 1 distinct palindromes as its factors such words are called rich on the other hand there exist infinite words containing only finitely many distinct palindromes as their factors such words are called poor we show that in the k abelian case there exist infinite words containing finitely many distinct k abelian palindromic factors for rich words we show that there exist finite words of length n containing n 2 distinct k abelian palindromes as their factors

on an extension of the notion of reedy category we extend the classical notion of a reedy category so as to allow non trivial automorphisms our extension includes many important examples occurring in topology such as segal s category or the total category of a crossed simplicial group such as connes cyclic category for any generalized reedy category mathbb r and any cofibrantly generated model category mathcal e the functor category mathcal e mathbb r is shown to carry a canonical model structure of reedy type

tridiagonal pairs of krawtchouk type let f denote an algebraically closed field with characteristic 0 and let v denote a vector space over f with finite positive dimension let a a denote a tridiagonal pair on v with diameter d we say that a a has krawtchouk type whenever the sequence d 2i i 0d is a standard ordering of the eigenvalues of a and a standard ordering of the eigenvalues of a assume a a has krawtchouk type we show that there exists a nondegenerate symmetric bilinear form on v such that au v u av and a u v u a v for u v v we show that the following tridiagonal pairs are isomorphic i a a ii a a iii a a iv a a we give a number of related results and conjectures

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

more properties of the fibonacci word on an infinite alphabet abstract recently the fibonacci word w on an infinite alphabet was introduced by zhang et al 2017 13 as a fixed point of the morphism 2 i 2 i 2 i 1 2 i 1 2 i 2 over all i n in this paper we investigate the occurrence of squares palindromes and lyndon factors in this infinite word

the switching element for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let v i i 0 d resp w i i 0 d denote a basis for v referred to in i resp ii we show that there exists a unique linear transformation s v v that sends v 0 to a scalar multiple of v d fixes w 0 and sends w i to a scalar multiple of w i for 1 i d we call s the switching element we describe s from many points of view

on the number of episturmian palindromes episturmian words are a suitable generalization to arbitrary alphabets of sturmian words in this paper we are interested in the problem of enumerating the palindromes in all episturmian words over a k letter alphabet a k we give a formula for the map g k giving for any n the number of all palindromes of length n in all episturmian words over a k this formula extends to k 2 a similar result obtained for k 2 by the second and third authors in 2006 the map g k is expressed in terms of the map p k counting for each n the palindromic prefixes of all standard episturmian words epicentral words for any n 0 p 2 n f n 2 where f is the totient euler function the map p k plays an essential role also in the enumeration formula for the map l k counting for each n the finite episturmian words over a k similarly to euler s function the behavior of p k is quite irregular the first values of p k and of the related maps g k and l k for 3 k 6 have been calculated and reported in the paper some properties of p k are shown in particular broad upper and lower bounds for p k as well as for m 0 np k m and g k are determined finally some conjectures concerning the map p k are formulated

transition maps between the 24 bases for a leonard pair abstract let v denote a vector space with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below 1 i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal 2 ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v in an earlier paper we described 24 special bases for v one feature of these bases is that with respect to each of them the matrices that represent a and a are i diagonal and irreducible tridiagonal or ii irreducible tridiagonal and diagonal or iii lower bidiagonal and upper bidiagonal or iv upper bidiagonal and lower bidiagonal for each ordered pair of bases among the 24 there exists a unique linear transformation from v to v that sends the first basis to the second basis we call this the transition map in this paper we find each transition map explicitly as a polynomial in a a

the end parameters of a leonard pair abstract fix an algebraically closed field f and an integer d 3 let v be a vector space over f with dimension d 1 a leonard pair on v is a pair of diagonalizable linear transformations a v v and a v v each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one there is an object related to a leonard pair called a leonard system it is known that a leonard system is determined up to isomorphism by a sequence of scalars i i 0 d i i 0 d i i 1 d i i 1 d called its parameter array the scalars i i 0 d resp i i 0 d are mutually distinct and the expressions i 2 i 1 i 1 i i 2 i 1 i 1 i are equal and independent of i for 2 i d 1 write this common value as 1 in the present paper we consider the end parameters 0 d 0 d 1 d 1 d of the parameter array we show that a leonard system is determined up to isomorphism by the end parameters and we display a relation between the end parameters and using this relation we show that there are up to isomorphism at most d 1 2 leonard systems that have specified end parameters the upper bound d 1 2 is best possible

on z factorization and c factorization of standard episturmian words abstract ziv lempel and crochemore factorization are two kinds of factorizations of words related to text processing in this paper we find these factorizations for standard epiesturmian words thus the previously known c factorization of characteristic sturmian words is provided as a special case moreover the two factorizations are compared

inequalities characterizing standard sturmian and episturmian words considering the smallest and the greatest factors with respect to the lexicographic order we associate to each infinite word r two other infinite words min r and max r in this paper we prove that the inequalities as min s max s bs characterize standard sturmian words proper ones and periodic ones and that the condition for any x a and lexicographic order satisfying x min a one has xs min s characterizes standard episturmian words

rich sturmian and trapezoidal words in this paper we explore various interconnections between rich words sturmian words and trapezoidal words rich words first introduced by the second and third authors together with j justin and s widmer constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors every finite sturmian word is rich but not conversely trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite sturmian words unfortunately this property does not characterize finite sturmian words in this note we show that the only trapezoidal palindromes are sturmian more generally we show that sturmian palindromes can be characterized either in terms of their subword complexity the trapezoidal property or in terms of their palindromic complexity we also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity

on different generalizations of episturmian words in this paper we study some classes of infinite words generalizing episturmian words and analyse the relations occurring among such classes in each case the reversal operator r is replaced by an arbitrary involutory antimorphism q of the free monoid a in particular we define the class of q words with seed whose standard elements q standard words with seed are constructed by an iterative q palindrome closure process starting from a finite word u 0 called the seed when the seed is empty one obtains q words episturmian words are exactly the r words one of the main theorems of the paper characterizes q words with seed as infinite words closed under q and having at most one left special factor of each length n n where n is some nonnegative integer depending on the word when n 0 we call such words q episturmian further results on the structure of q episturmian words are proved in particular some relationships between q words with or without seed and q episturmian words are shown

on a generalization of christoffel words epichristoffel words sturmian sequences are well known as the ones having minimal complexity over a 2 letter alphabet they are also the balanced sequences over a 2 letter alphabet and the sequences describing discrete lines they are famous and have been extensively studied since the 18th century one of the extensions of these sequences over a k letter alphabet with k 3 is the episturmian sequences which generalizes a construction of sturmian sequences using the palindromic closure operation there exists a finite version of the sturmian sequences called the christoffel words they have been known since the works of christoffel and have interested many mathematicians in this paper we introduce a generalization of christoffel words for an alphabet with 3 letters or more using the episturmian morphisms we call them the epichristoffel words we define this new class of finite words and show how some of the properties of the christoffel words can be generalized naturally or not for this class

tridiagonal pairs and the μ conjecture let f denote a field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering vi i 0d of the eigenspaces of a such that a vi vi 1 vi vi 1 for 0 i d where v 1 0 and vd 1 0 iii there exists an ordering vi i 0 of the eigenspaces of a such that avi vi 1 vi vi 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of vi vd i vi vd i coincide we say the pair a a is sharp whenever dimv0 1 it is known that if f is algebraically closed then a a is sharp a conjectured classification of the sharp tridiagonal pairs was recently introduced by t ito and the second author we present a result which supports the conjecture given scalars i i 0d i i 0d in f that satisfy the known constraints on the eigenvalues of a tridiagonal pair we define an f algebra t by generators and relations we consider the f algebra e0 te0 for a certain idempotent e0 t let f x1 xd denote the polynomial algebra over f involving d mutually commuting indeterminates we display a surjective f algebra homomorphism f x1 xd e0 te0 we conjecture that is an isomorphism we show that this conjecture implies the classification conjecture and that the conjecture holds for d 5

leonard pairs having specified end entries fix an algebraically closed field f and an integer d 3 let v be a vector space over f with dimension d 1 a leonard pair on v is an ordered pair of diagonalizable linear transformations a v v and a v v each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one let vi i 0d resp vi i 0d be such an eigenbasis for a resp a for 0 i d define a linear transformation ei v v such that eivi vi and eivj 0 if j i 0 j d define ei v v in a similar way the sequence a ei i 0d a ei i 0d is called a leonard system on v with diameter d with respect to the basis vi i 0d let i i 0d resp ai i 0d be the diagonal entries of the matrix representing a resp a with respect to the basis vi i 0d let i i 0d resp ai i 0d be the diagonal entries of the matrix representing a resp a it is known that i i 0d resp i i 0d are mutually distinct and the expressions i 1 i 2 i i 1 i 1 i 2 i i 1 are equal and independent of i for 1 i d 2 write this common value as 1 in the present paper we consider the end entries 0 d 0 d a0 ad a0 ad we prove that a leonard system with diameter d is determined up to isomorphism by its end entries and if and only if either i 2 and qd 1 1 where q q 1 or ii 2 and char f 2

on christoffel and standard words and their derivatives we introduce and study natural derivatives for christoffel and standard words as well as for characteristic sturmian words these derivatives which are defined as inverse images under suitable morphisms preserve the aforementioned classes of words in the case of christoffel words the morphisms involved map a to a k 1 b resp a b k and b to a k b resp a b k 1 for a suitable k 0 as long as derivatives are not just a single letter higher order derivatives are naturally obtained we define the depth of a christoffel or of a standard word as the smallest order for which the derivative is a single letter we give several combinatorial and arithmetic descriptions of the depth and tight lower and upper bounds for it

tridiagonal pairs of q racah type abstract let f denote an algebraically closed field and let v denote a vector space over f with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that a w w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d for 0 i d let i resp i denote the eigenvalue of a resp a associated with v i resp v i the pair a a is said to have q racah type whenever i a b q 2 i d c q d 2 i and i a b q 2 i d c q d 2 i for 0 i d where q a b c a b c are scalars in f with q b c b c nonzero and q 2 1 1 this type is the most general one we classify up to isomorphism the tridiagonal pairs over f that have q racah type our proof involves the representation theory of the quantum affine algebra u q sl 2

q inverting pairs of linear transformations and the q tetrahedron algebra abstract as part of our study of the q tetrahedron algebra q we introduce the notion of a q inverting pair roughly speaking this is a pair of invertible semisimple linear transformations on a finite dimensional vector space each of which acts on the eigenspaces of the other according to a certain rule our main result is a bijection between the following two sets i the isomorphism classes of finite dimensional irreducible q modules of type 1 ii the isomorphism classes of q inverting pairs

palindromes in circular words there is a very short and beautiful proof that the number of distinct non empty palindromes in a word of length n is at most n in this paper we show with a very complicated proof that the number of distinct non empty palindromes with length at most n in a circular word of length n is less than 5n 3 for n divisible by 3 we present circular words of length n containing 5n 3 2 distinct palindromes so the bound is almost sharp the paper finishes with some open problems

a generalized palindromization map in free monoids the palindromization map j in a free monoid a was introduced in 1997 by the first author in the case of a binary alphabet a and later extended by other authors to arbitrary alphabets acting on infinite words j generates the class of standard episturmian words including standard arnoux rauzy words in this paper we generalize the palindromization map starting with a given code x over a the new map j x maps x to the set pal of palindromes of a in this way some properties of j are lost and some are saved in a weak form when x has a finite deciphering delay one can extend j x to x w generating a class of infinite words much wider than standard episturmian words for a finite and maximal code x over a we give a suitable generalization of standard arnoux rauzy words called x ar words we prove that any x ar word is a morphic image of a standard arnoux rauzy word and we determine some suitable linear lower and upper bounds to its factor complexity for any code x we say that j x is conservative when j x x x we study conservative maps j x and conditions on x assuring that j x is conservative we also investigate the special case of morphic conservative maps j x i e maps such that email protected email protected email protected email protected email protected for an injective morphism f finally we generalize j x by replacing palindromic closure with q palindromic closure where q is any involutory antimorphism of a this yields an extension of the class of q standard words introduced by the authors in 2006

palindromes and sturmian words abstract an infinite word x over the alphabet a is sturmian if and only if g x n n 1 for any integer n where g x n is the number of distinct words of length n occurring in x a palindrome is a word that can be read indistinctly from left to right or from right to left we prove that x is sturmian if and only if h x n 1 n mod 2 for any integer n where h x n is the number of palindromes of length n occurring in x an infinite word x over the alphabet a is generated by a morphism f if there exists a letter c a such that lim n f n c x we prove the existence of a morphism that generates the palindromes of any infinite sturmian word generated by a morphism

palindrome complexity we study the palindrome complexity of infinite sequences on finite alphabets i e the number of palindromic factors blocks of given length occurring in a given sequence we survey the known results and obtain new results for some sequences in particular for rote sequences and for fixed points of primitive morphisms of constant length belonging to class p of hof knill simon we also give an upper bound for the palindrome complexity of a sequence in terms of its block complexity

on theta palindromic richness in this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism q it generalizes the notion of a palindrome into a q palindrome a word invariant under q for languages closed under q we give the relation email protected complexity and factor complexity we generalize the notion of richness email protected and we prove analogous characterizations of words that are q rich especially in the case of set of factors invariant under q a criterion for q richness of q episturmian words is given together with other examples of q rich words

some trace formulae involving the split sequences of a leonard pair let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy i and ii below i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal we call such a pair a leonard pair on v let diag 0 1 d denote the diagonal matrix referred to in ii above and let diag 0 1 d denote the diagonal matrix referred to in i above it is known that there exists a basis u0 u1 ud for v and there exist scalars 1 2 d in k such that aui iui ui 1 0 i d 1 aud dud a ui iui 1 i ui 1 i d a u0 0 u0 the sequence 1 2 d is called the first split sequence of the leonard pair it is known that there exists a basis v0 v1 vd for v and there exist scalars 1 2 d in k such that avi d ivi vi 1 0 i d 1 avd 0vd a vi ivi 1 i vi 1 i d a v0 0 v0 the sequence 1 2 d is called the second split sequence of the leonard pair we display some attractive formulae for the first and second split sequence that involve the trace function

palindromic richness for languages invariant under more symmetries for a given finite group g consisting of morphisms and antimorphisms of a free monoid a we study infinite words with language closed under the group g we focus on the notion of g richness which describes words rich in generalized palindromic factors i e in factors w satisfying q w w for some antimorphism email protected g we give several equivalent descriptions which are generalizations of known characterizations of rich words in the terms of classical palindromes and show two examples of g rich words

conjugacy and episturmian morphisms episturmian morphisms generalize sturmian morphisms here we study some intrinsic properties of these morphisms invertibility presentation cancellativity unitarity characterization by conjugacy most of them are generalizations of known properties of sturmian morphisms but we present also some results on episturmian morphisms that have not already been stated in the particular case of sturmian morphisms characterization of the episturmian morphisms that preserve palindromes new algorithms to compute conjugates we also study the conjugation of morphisms in the general case and show that the monoid of invertible morphisms on an alphabet containing at least three letters is not finitely generated

sturmian words and the stern sequence central standard and christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart to characteristic sturmian words a natural arithmetization of the theory is obtained by representing central and christoffel words by irreducible fractions labeling respectively two binary trees the raney or calkin wilf tree and the stern brocot tree the sequence of denominators of the fractions in the raney tree is the famous stern diatomic numerical sequence an interpretation of the terms s n of stern s sequence as lengths of christoffel words when n is odd and as minimal periods of central words when n is even allows one to interpret several results on christoffel and central words in terms of stern s sequence and conversely to obtain new insight into the combinatorics of christoffel and central words one of our main results is a non commutative version of the alternating bit sets theorem by calkin and wilf we also study the length distribution of christoffel words corresponding to nodes of equal height in the tree obtaining some interesting bounds and inequalities

proof of the brlek reutenauer conjecture brlek and reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2d u email protected n 0 t u n in which d u denotes the defect of u and t u n denotes c u n 1 c u n 2 p u n 1 p u n where c u and p u are the factor and palindromic complexity of u respectively this conjecture was verified for periodic words by brlek and reutenauer themselves using their results for periodic words we have recently proved the conjecture for uniformly recurrent words in the present article we prove the conjecture in its general version by a new method without exploiting the result for periodic words

left fibrations and homotopy colimits for a small category mathbf a we prove that the homotopy colimit functor from the category of simplicial diagrams on mathbf a to the category of simplicial sets over the nerve of mathbf a establishes a left quillen equivalence between the projective or reedy model structure on the former category and the covariant model structure on the latter we compare this equivalence to a quillen equivalence in the opposite direction previously established by lurie from our results we deduce that a categorical equivalence of simplicial sets induces a quillen equivalence on the corresponding over categories equipped with the covariant model structures also we show that a version of quillen s theorem a for infty categories easily follows

on graphs of central episturmian words episturmian sequences are a natural extension of sturmian sequences to the case of finite alphabets of arbitrary cardinality in this paper we are interested in central episturmian words or simply epicentral words i e the palindromic prefixes of standard episturmian sequences an epicentral word admits a variety of faithful representations including as a directive word as a certain type of period vector as a parikh vector as a certain type of fine and wilf extremal word as a suitable modular matrix and as a labeled graph various interconnections between the different representations of an epicentral word are analyzed in particular we investigate the structure of the graphs of epicentral words proving some curious and surprising properties

a condition of boshernitzan and uniform convergence in the multiplicative ergodic theorem this paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts if such a subshift satisfies a certain condition originally introduced by boshernitzan every locally constant sl 2 r valued cocycle is uniform as a consequence the corresponding schrodinger operators exhibit cantor spectrum of lebesgue measure zero an investigation of boshernitzan s condition then shows that these results cover all earlier results of this type and moreover provide various new ones in particular boshernitzan s condition is shown to hold for almost all circle maps and almost all arnoux rauzy subshifts

palindrome pattern matching a palindrome is a string that reads the same forward and backward for a string x let pals x be the set of all maximal palindromes of x where each maximal palindrome in pals x is encoded by a pair c r of its center c and its radius r given a text t of length n and a pattern p of length m the palindrome pattern matching problem is to compute all positions i of t such that pals p pals t i i m 1 we present linear time algorithms to solve this problem

sturmian words and a criterium by michaux villemaire michaux and villemaire s proof of cobham s theorem relies on the characterization of ultimately periodic words by means of the behaviour of certain repetitions in the word namely they consider the length of the smallest shift between repetitions of a given length and the first position at which that smallest shift is observed in this paper we study those properties for characteristic sturmian words in particular we answer a question posed by michaux and villemaire in that context

palindromic language of thin discrete planes we work on the rveills hyperplane p v 0 with normal vectorvrd shift 0 and thicknessr such a hyperplane is connected as soon as is greater than some value v 0 called the connecting thickness of v with null shift in the case where v satisfies the so called kraaikamp and meester criterion at the connecting thickness the hyperplane has very specific properties first of all the adjacency graph of the voxels forms a tree this tree appeared in many works both in discrete geometry and in discrete dynamical systems in addition it is well known that for a finite coding of length n of discrete lines the number of palindromes in the language is exactly n 1 we extend this notion of language to labeled trees and we compute the number of distinct palindromes in fact for our voxel adjacency trees with n letters we show that the number of palindromes in the language is also n 1 this result establishes a first link between combinatorics on words palindromic languages voxel adjacency trees and connecting thickness of rveills hyperplanes it also provides a better understanding of the combinatorial structure of discrete planes