Abstract. The binomial coefficient of two words u and v is the number of times v occurs as a subsequence of u. Based on this classical notion, we introduce the m-binomial equivalence of two words refining the abelian equivalence. The m-binomial complexity of an infinite word x maps an integer n to the number of m-binomial equivalence classes of factors of length n occurring in x. We study the first properties of m-binomial equivalence. We compute the m-binomial complexity of the Sturmian words and of the Thue-Morse word. We also mention the possible avoidance of 2-binomial squares.

A graph <i>G</i>=(<i>V,E</i>) is representable if there exists a word <i>W</i> over the alphabet <i>V</i> such that letters <i>x</i> and <i>y</i> alternate in <i>W</i> if and only if (<i>x</i> ,<i>y</i>) is in <i>E</i> for each <i>x</i> not equal to <i>y</i> . The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view. In this paper, we prove that for <i>n</i> greater than 3, the line graph of an <i>n</i>-wheel is non-representable. This not only provides a new construction of non-repre- sentable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for <i>n</i> greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph <i>G</i> which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of <i>G k</i>-times is guaranteed to be non-representable for <i>k</i> greater than 3

An infinite permutation α is a linear ordering of N. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we define maximal pattern complexity p * α (n) for infinite permutations and show that this complexity function is ultimately constant if and only if the permutation is ultimately periodic; otherwise its maximal pattern complexity is at least n, and the value p * α (n) ≡ n is reached exactly on the family of permutations constructed by Sturmian words.