The insertion encoding of Cayley permutations

TL;DR

This work extends the insertion encoding framework to Cayley permutations via vertical (leftmost maxima) and horizontal (rightmost values) constructions, and provides a complete classification of which Cayley permutation classes admit regular insertion encodings. It develops a tilings-based DFA approach to compute rational generating functions and implements it to enumerate and analyze classes, including solving Cerbai’s hare pop-stack sortable problem. Key results include a non-ambiguous CFG for Av(211,312) with a closed-form generating function, and a proof that a finitely based class has a regular encoding iff it is slot-bounded, characterized by nine vertical juxtapositions and four horizontal juxtapositions. The methodology yields practical, automatable enumeration and generating-function computation for a broad spectrum of Cayley permutation classes, with an accompanying open-source implementation.

Abstract

We introduce the vertical and horizontal insertion encodings for Cayley permutations which naturally generalise the insertion encoding for permutations. In both cases, we fully classify the Cayley permutation classes for which these languages are regular, and provide an algorithm for computing the rational generating functions. We use our algorithm to solve an open problem of Cerbai by enumerating the hare pop-stack sortable Cayley permutations.

Figures (11)

• Figure 1: The plot of the Cayley permutation $21321$.
• Figure 2: The four ways to insert a number $n$ into a slot.
• Figure 3: Examples of vertical juxtapositions.
• Figure 4: $(35125224, ((0,1), (0,1), (0,0), (0,0), (1,1), (1,0), (1,0), (1,1)))$.
• Figure 5: An example tiling showing obstructions in red and requirements in blue.

Theorems & Definitions (18)

• Proposition 3.1
• proof
• Proposition 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof