Subword balance, position indices and power sums

AbstractIn this paper, we investigate various ways of characterizing words, mainly over a binary alphabet, using information about the positions of occurrences of letters in words. We introduce two new measures associated with words, the position index and sum of position indices. We establish some characterizations, connections with Parikh matrices, and connections with power sums. One particular emphasis concerns the effect of morphisms and iterated morphisms on words

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced

Polynomials, Primes and the PTE Problem

This dissertation considers three different topics. In the first part of the dissertation, we use Newton Polygons to show that for the arithmetic functions g(n) = n t , where t ≥ 1 is an integer, the polynomials defined with initial condition P g 0 (X) = 1 and recursion P g n (X) = X n Xn k=1 g(k)P g n−k (X) are X/ (n!) times an irreducible polynomial. In the second part of the dissertation, we show that, for 3 ≤ n ≤ 8, there are infinitely many 2-adic integer solutions to the Prouhet-Tarry-Escott (PTE) problem, that are not rational integer solutions. In particular, we look at the 2-adic valuation of a certain constant associated with the PTE problem and for the case n = 8 there exist solutions whose valuation is strictly less than any known rational integer solution. In the third part of the dissertation, we obtain a number of results pertaining to polynomials f(x) with non-negative integer coefficients that take on a prime value at x = b, where b ≥ 2 is an integer. In particular, we give an explicit bound M1(b) such that if the coefficients of f(x) are each ≤ M1(b), then f(x) is irreducible. We also show that there are similarly explicit bounds M2(b), M3(b) and M4(b), for b sufficiently large (made explicit), that can be placed on the coefficients of f(x) such that if f(x) is reducible then it must be divisible by at least one of the shifted cyclotomic polynomials Φ3(x − b), Φ4(x − b) or Φ6(x − b)