Rich, Sturmian, and trapezoidal words

In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced in arXiv:0801.1656 by the second and third authors together with J. Justin and S. Widmer, constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors. Every finite Sturmian word is rich, but not conversely. Trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite Sturmian words. Unfortunately this property does not characterize finite Sturmian words. In this note we show that the only trapezoidal palindromes are Sturmian. More generally we show that Sturmian palindromes can be characterized either in terms of their subword complexity (the trapezoidal property) or in terms of their palindromic complexity. We also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity.

💡 Research Summary

The paper investigates the interrelations among three important families of words in combinatorics on words: rich words, Sturmian words, and trapezoidal words. A finite word W is called rich if it contains exactly |W| + 1 distinct palindromic factors (including the empty word). Droubay, Justin and Pirillo proved that any word can have at most this many palindromes, and all Sturmian words are known to be rich, though the converse does not hold.

A trapezoidal word is defined via its factor complexity C_W(n). For a finite word W, C_W(n) first increases by one for a block of length r, stays constant for a block of length s, then decreases by one for another block of length r, producing a “trapezoid” shape when plotted. Equivalently, |W| = R_W + K_W where R_W is the smallest length without a right‑special factor and K_W is the length of the shortest unrepeated suffix. Trapezoidal words were introduced to describe the complexity profile of finite Sturmian words, but they do not characterize Sturmianity (e.g., “aaab” is trapezoidal but not Sturmian).

The authors first prove that every trapezoidal word is rich. The proof proceeds by induction on length. Assuming a minimal counter‑example, they locate a non‑palindromic complete return to a palindrome aU a, deduce structural constraints on the word, and eventually show that the word must be Sturmian, contradicting the assumption that it is not rich. This yields the inclusion “trapezoidal ⊂ rich”, while noting that the converse fails (e.g., “aabbaa” is rich but not trapezoidal).

The core contributions are two characterizations expressed through the palindromic complexity P_W(n) (the number of distinct palindromic factors of length n) and the subword complexity C_W(n) (the number of distinct factors of length n).

1. Rich palindromes: Theorem 1 states that for a finite word W the following are equivalent:

   • (A) W is a rich palindrome.
   • (B) For every n with 0 ≤ n ≤ |W|,
     P_W(n) + P_W(n + 1) = C_W(n + 1) − C_W(n) + 2. The direction (B)⇒(A) is proved by evaluating the equality at n = |W|, which forces P_W(|W|)=1, i.e., W itself is a palindrome, and then summing the equalities to obtain the total number of palindromes S = |W| + 1. Conversely, assuming W is a rich palindrome, the authors inductively remove the outer letters, compare the complexities of the shortened word V and W, and handle the first new factor U that appears in W but not in V. Careful analysis shows that U must be both a prefix and a suffix (hence a palindrome) and that the equality (B) propagates to all n.

2. Sturmian palindromes: Theorem 2 gives three equivalent conditions for a word W of length N:

   • (A′) W is a Sturmian palindrome.
   • (B′) For every n, 0 ≤ n ≤ N, P_W(n) + P_W(N − n) = 2.
   • (C′) W is a trapezoidal palindrome. The equivalence (A′)⇔(C′) follows from known results that every finite Sturmian word is trapezoidal and from an inductive argument showing that a trapezoidal palindrome must be Sturmian; the argument uses the fact that trapezoidal words are binary, that they are rich (hence have a unique longest repeated suffix), and that the minimal period satisfies π_W = R_W + 1, which by a known proposition forces Sturmianity. The equivalence (A′)⇔(B′) rests on the classic Sturmian palindromic complexity: for Sturmian words P_W(n) is 1 for even n and 2 for odd n, which directly yields (B′). Conversely, assuming (B′), the authors show that P_W(1) cannot be 0; if it is 1 the word is constant (hence Sturmian), and if it is 2 the word is binary. They then prove that any violation of the balance property would create two distinct palindromes of the same length, contradicting (B′). Hence the word must be balanced and therefore Sturmian.

The paper also discusses the θ‑palindrome interpretation of condition (B′) and provides an alternative short proof (suggested by A. De Luca) that a trapezoidal palindrome must be Sturmian by examining its longest proper palindromic suffix and relating the longest repeated suffix K_W to the minimal period.

In summary, the authors achieve three main results:

• Every trapezoidal word is rich, but the converse does not hold.
• Rich palindromes are exactly those finite words whose palindromic and subword complexities satisfy the linear relation of Theorem 1.
• Sturmian palindromes are precisely the trapezoidal palindromes, and they are also characterized by the symmetric palindromic‑complexity condition of Theorem 2.

These findings unify three previously separate notions—richness, Sturmianity, and trapezoidal complexity—through precise combinatorial identities, deepening our understanding of how palindromic structure interacts with factor complexity in finite and infinite words.