On Inequalities between Subword Histories

By taking out letters from a word we get a subword. Both continuous sub-words (also called factors or simply subwords) and scattered subwords were extensively studied. In [4] the authors introduced Parikh matrices, structures that contain more information about the words than Parikh vectors, which tell us only the number of di $ffere\tau 1T$ letters building the word. $I_{l1}[5]$ the notion of subword histories appeared and has been developed into a powerful tool in the investigation of relations between certain scattered subwords of a given word. In particular, several characteristic equalities and inequalities regarding sums of subword occurrences were presented, perhaps most notably the Cauchy inequality for words [5]. The decidability of equalities between subword histories was settled with a positive answer. This paper tries to answer the question about the decidability of inequalities between subword histories and succeeds in giving partial results, that is, certain cases where the inequalities hold and an algorithm to decide whether a subword inequality belongs to one of these particular casas. By alphaber we mean a