Nearly k-Universal Words - Investigating a Part of Simon’s Congruence

Abstract

AbstractDetermining 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., \(\mathtt {oath}\) is a scattered factor of \(\mathtt {logarithm}\) but \(\mathtt {tail}\) is not. Following the idea of scattered factor k-universality (also known as k-richness), we investigate nearly k-universality, i.e., words where exactly one scattered factor of length k is absent. We present a full characterisation as well as the index of the congruence in this special case and the shortlex normal form for each such class. Moreover, we extend the definition to m-nearly k-universality (exactly m scattered factors of length k are absent), show some results for \(m>1\), and give a full combinatorial characterisation of m-nearly k-universal words which are additionally \((k-1)\)-universal.