Avoiding abelian squares in partial words

AbstractErdős raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes. Several of our constructions are based on iterating morphisms. In the case of one hole, we prove that the minimal alphabet size is four, while in the case of more than one hole, we prove that it is five. We also investigate the number of partial words of length n with a fixed number of holes over a five-letter alphabet that avoid abelian squares and show that this number grows exponentially with n

Higher Lie characters and cyclic descent extension on conjugacy classes

A cyclic descent extension of the classical notion of descent set, for permutations as well as standard Young tableaux, was introduced and studied in recent years. The main result of this paper is a full characterization of conjugacy classes in the symmetric group, which carry a cyclic descent extension. Letting $S_n$ be the symmetric group on $n$ letters and \C_\mu\subset S_n be a conjugacy class of cycle type $\mu$, it is shown that the descent map on \C_\mu has a cyclic extension if and only if $\mu$ is not of the form $(r^s)$ for some square-free $r$. The proof involves a detailed study of hook constituents in higher Lie characters.Comment: 27 page

A powerful abelian square-free substitution over 4 letters

AbstractIn 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet. An abelian square is a non-empty word uv, where u and v are permutations (anagrams) of each other. The case of the four letter alphabet Σ4={a,b,c,d} turned out to be the most challenging and remained open until 1992 when the author presented an abelian square-free (a-2-free) endomorphism g85 of Σ4∗. The size of this g85, i.e., |g85(abcd)|, is equal to 4×85 (uniform modulus). Until recently, all known methods for constructing arbitrarily long a-2-free words on Σ4 have been based on the structure of g85 and on the endomorphism g98 of Σ4∗ found in 2002.In this paper, a great many new a-2-free endomorphisms of Σ4∗ are reported. The sizes of these endomorphisms range from 4×102 to 4×115. Importantly, twelve of the new a-2-free endomorphisms, of modulus m=109, can be used to construct an a-2-free (commutatively functional) substitution σ109 of Σ4∗ with 12 image words for each letter.The properties of σ109 lead to a considerable improvement for the lower bound of the exponential growth of cn, i.e., of the number of a-2-free words over 4 letters of length n. It is obtained that cn>β−50βn with β=121/m≃1.02306. Originally, in 1998, Carpi established the exponential growth of cn by showing that cn>β−tβn with β=219/t=219/(853−85)≃1.000021, where t=853−85 is the modulus of the substitution that he constructs starting from g85

A bijection to count (1-23-4)-avoiding permutations

A permutation is (1-23-4)-avoiding if it contains no four entries, increasing left to right, with the middle two adjacent in the permutation. Here we give a 2-variable recurrence for the number of such permutations, improving on the previously known 4-variable recurrence. At the heart of the proof is a bijection from (1-23-4)-avoiding permutations to increasing ordered trees whose leaves, taken in preorder, are also increasing.Comment: latex, 16 page