On universal partial words for word-patterns and set partitions

Chen, Herman Z. Q. and Kitaev, Sergey (2020) On universal partial words for word-patterns and set partitions. RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), 54. 5. ISSN 1290-385X (https://doi.org/10.1051/ita/2020004)

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Universal words are words containing exactly once each element from a given set of combinatorial structures admiting encoding by words. Universal partial words (u-p-words) contain, in addition to the letters from the alphabet in question, any number of occurrences of a special ``joker'' symbol. We initiate the study of u-p-words for word-patterns (essentially, surjective functions) and (2-)set partitions by proving a number of existence/non-existence results and thus extending the results in the literature on u-p-words and u-p-cycles for words and permutations. We apply methods of graph theory and combinatorics on words to obtain our results.