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The Narayana Morphism and Related Words

Jeffrey Shallit

TL;DR

This work analyzes the Narayana morphism $\\nu$ and its fixed point ${\\mathbf n}$ through automata theory and Walnut, enabling first-order proofs of intricate properties. It develops the Narayana numeration system, derives a closed form for the Narayana numbers, and establishes a broad automata-based framework to study ${\\mathbf n}$, including appearance, critical exponent, subword complexity, and abelian properties. The paper proves the Kimberling–Moses conjecture and Cloître-type inequalities, connects Narayana representations to the $3$-Zeckendorf array and additive number theory, and explores related sequences such as the Allouche–Johnson word and Hofstadter-type constructions. It also sketches generalizations to a family of words ${\\mathbf x}_k$, highlighting connections to classical morphisms and proposing several open conjectures for future work.

Abstract

The Narayana morphism $ν$ maps $0 \rightarrow 01$, $1 \rightarrow 2$, $2 \rightarrow 0$ and has a fixed point $\mathbf{n} = n_0 n_1 n_2 \cdots = {\tt 0120010120120}\cdots$. In this paper we study the properties of this word and related words using automata theory.

The Narayana Morphism and Related Words

TL;DR

This work analyzes the Narayana morphism and its fixed point through automata theory and Walnut, enabling first-order proofs of intricate properties. It develops the Narayana numeration system, derives a closed form for the Narayana numbers, and establishes a broad automata-based framework to study , including appearance, critical exponent, subword complexity, and abelian properties. The paper proves the Kimberling–Moses conjecture and Cloître-type inequalities, connects Narayana representations to the -Zeckendorf array and additive number theory, and explores related sequences such as the Allouche–Johnson word and Hofstadter-type constructions. It also sketches generalizations to a family of words , highlighting connections to classical morphisms and proposing several open conjectures for future work.

Abstract

The Narayana morphism maps , , and has a fixed point . In this paper we study the properties of this word and related words using automata theory.

Paper Structure

This paper contains 19 sections, 34 theorems, 43 equations, 9 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The Narayana numbers and the Narayana numeration system
  3. Closed form and estimates for the Narayana numbers
  4. Finite automata and Walnut
  5. Properties of the sequence $\bf n$
  6. Appearance
  7. Critical exponent
  8. Subword complexity
  9. Palindromes
  10. Right-special factors
  11. Positions of symbols
  12. The $3$-Zeckendorf array
  13. Some additive number theory
  14. The Kimberling-Moses conjecture
  15. Abelian properties
  16. ...and 4 more sections

Key Result

Theorem 1

Every positive integer $m$ has a unique representation $e_1 \cdots e_t$, provided the leading digit $e_1$ is nonzero, and there are no occurrences of the block $11$ or $101$ in $e_1 \cdots e_{t}$.

Figures (9)

  • Figure 2: Incrementer automaton for Narayana representations.
  • Figure 3: Narayana automaton for maximum of period.
  • Figure 4: Membership automaton for the set $P_1 + P_1$.
  • Figure 5: Narayana automaton computing $a(i)$.
  • Figure 6: Narayana automaton computing $b(i)$.
  • ...and 4 more figures

Theorems & Definitions (77)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 3
  • proof
  • Corollary 4
  • Lemma 5
  • proof
  • Remark 6
  • ...and 67 more