A generalized pseudostandard prefix with no string attractor of size four

A string attractor of a finite word is a set of positions such that every nonempty factor has an occurrence containing one of them. Binary generalized pseudostandard sequences are built by iterating palindromic and antipalindromic closures along a directive bi-sequence $(\Delta,\Theta)$, where $\Delta$ lists the letters appended and $\Theta$ the closures taken, $R$ for palindromic and $E$ for antipalindromic. Dvořáková and Hendrychová (2023) determined the minimal string attractors of the pseudopalindromic prefixes of standard complementary-symmetric Rote sequences, gave attractors of at most three positions for the antipalindromic prefixes of pseudostandard sequences, and conjectured that the pseudopalindromic prefixes of every binary generalized pseudostandard sequence have a minimal string attractor of size at most four. We show that this bound fails at length 36: for $\Delta$ beginning $00101$ and $\Theta$ beginning $EEEER$, the fifth word of the recurrence, the palindromic prefix $010110010101100101101001101010011010$, has smallest attractor size five. The lower bound is an elementary argument with seven factors; the upper bound is an explicit five-position attractor. Both bounds are also confirmed by computer. A second example has length 43.

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