Universal updates of Dyck-nest signatures

TL;DR

The anchored Dyck words of length \(n=2k+1\) (obtained by prefixing a 0-bit to each Dyck word of length \(2k\)) represent in \(O_k\) the cycles of an \(n\)- (resp.

Abstract

Let $0<k\in\mathbb{Z}$. The anchored Dyck words of length $n=2k+1$ (obtained by prefixing a 0-bit to each Dyck word of length $2k$ and used to reinterpret the Hamilton cycles in the odd graph $O_k$ and the middle-levels graph $M_k$ found by Mütze et al.) represent in $O_k$ (resp., $M_k$) the cycles of an $n$- (resp., $2n$-) 2-factor and its cyclic (resp., dihedral) vertex classes, and are equivalent to Dyck-nest signatures. A sequence is obtained by updating these signatures according to the depth-first order of a tree of restricted growth strings (RGS's), reducing the RGS-generation of Dyck words by collapsing to a single update the time-consuming $i$-nested castling used to reach each non-root Dyck word or Dyck nest. This update is universal, for it does not depend on $k$.

Figures (5)

• Figure 1: List of $k$-germs $\alpha$, $n$-nests $F(\alpha)$, signatures and update entries, for $k=2,3,4,5$.
• Figure 2: List of $k$-germs $\alpha$, $n$-nests $F(\alpha)$, signatures and update entries, for $k=6$.
• Figure 3: Extension of Table \ref{['tab2']} and partial view of $\Delta'$, for $k=2,3,4,5,6,7$
• Figure 4: Members of $\Phi_1$, for $k=2,3,4,5,6,7$
• Figure 5: Information for $\Phi_2,\Phi_3,\Phi_4,\Phi_5$

Theorems & Definitions (40)

• Lemma 1
• proof
• Theorem 2
• proof
• Example 3
• Example 4
• Theorem 5
• proof
• Corollary 6
• Claim 7