CYCLIC PARTIAL WORDS

Partial words are linear words with holes. Cyclic words are derived from linear words by linking its first letter after the last one. Both partial words and cyclic words have wide applications in DNA sequencing. In this paper we introduce cyclic partial words and discuss their periodicity and certain properties. We also establish representation of a cyclic partial word using trees.

Słowa miesiąca roku 2018 w polach leksykalno-semantycznych.

Words of the month in 2018 in lexical and semantic fi elds Summary The aim of this study is to present the results of a lexical and semantic analysis of the so-called words of the month in 2018, that is words (or their combinations) with high frequency in the Polish daily press, published and commented by a group of linguists from the Institute of Polish Language, University of Warsaw, on www.slowanaczasie.uw.edu.pl over the past year. Apart from the analysis of the thematic structure of the excerpted lexemes and their assignment to contextually determined word fi elds, the examination covers the total number of occurrences of the units in the monthly list of words of the month. The outcome of the linguistic analysis not only permitted identifi cation of the most topical issues for Poles in a given year, but also enabled verifi cation of the conclusions drawn by various researchers of cyclic words to date. They have found, among others, a considerable share of borrowings from foreign languages, neologisms and neosemantisms, pointed to a seasonal motivation of the units, and argued that the greatest share in the set belonged to names related to politics, economics, and broadly understood social issues.

Staircase words and Chebyshev polynomials

A word ? = ?1... ?n over the alphabet [k]={1,2,...,k} is said to be a staircase if there are no two adjacent letters with difference greater than 1. A word ? is said to be staircase-cyclic if it is a staircase word and in addition satisfies |?n??1|?1. We find the explicit generating functions for the number of staircase words and staircase-cyclic words in [k]n, in terms of Chebyshev polynomials of the second kind. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate staircase necklaces, which are staircase-cyclic words that are not equivalent up to rotation.