Combinatorics of Palindromes

We investigate the structure and reconstruction complexity of Manacher arrays. First, we establish a combinatorial lower bound, proving that the number of rooted tandem repeat trees with $n+1$ genes exceeds the number of distinct Manacher arrays of length $n$. Second, we introduce a graph-theoretic framework that associates a graph to each Manacher array, where every proper vertex coloring yields a string consistent with the array. Finally, we analyze a reconstruction algorithm by I et al. (SPIRE 2010), showing that it simultaneously achieves a globally minimal alphabet size, uses at most $\log_2(n{-}1) + 2$ distinct symbols, and can be adapted to produce reconstructions over arbitrary alphabets when possible. Our results also resolve an open problem posed by the original authors. Together, these findings advance the combinatorial understanding of Manacher arrays and open new directions for string reconstruction under structural constraints.