Nearly k-universal words – Investigating a part of Simon's congruence

Journal article, 2023

Determining the index of Simon's congruence is a long outstanding open problem. Two words u and v are called Simon congruent if they have the same set of scattered factors (also known as subwords or subsequences), which are parts of the word in the correct order but not necessarily consecutive, e.g., oath is a scattered factor of logarithm but tail is not. Following the idea of scattered factor k-universality (also known as k-richness), we investigate m-nearly k-universality, i.e., words where exactly m scattered factors of length k are absent. We present full characterisations as well as the indexes of the congruence for very small and very large m. Moreover, we give a full combinatorial characterisation of m-nearly k-universal words which are additionally (k−1)-universal.