The insertion encoding of restricted growth functions

TL;DR

This work classifies when insertion encodings of restricted growth functions (RGFs) yield regular languages, by adapting horizontal and vertical encodings from Cayley permutations and employing slot-boundedness and grid-class techniques. It develops precise criteria: horizontal regularity corresponds to containment in slot-bounded families $\\mathcal{SB}_H(k)$, while vertical regularity corresponds to avoidance of nine griddable grid classes via $\\mathcal{SB}_V(k)$, with canonical bases specified for non-slot-bounded cases. The paper also extends the framework to RGFs arising from matchings, deriving reduced alphabets and corresponding regularity conditions. Finally, it provides practical enumeration tools, reports extensive size-3 pattern class statistics, and points to future work on context-free language questions and broader grid-class analyses. These results connect pattern avoidance for RGFs with formal language properties, enabling automatic generation of generating functions and systematic exploration of RGF classes.

Abstract

We adapt the vertical and horizontal insertion encodings of Cayley permutations to enumerate restricted growth functions, which are in bijection with unordered set partitions. For both insertion encodings, we fully classify the classes for which these languages are regular. For the horizontal insertion encoding, we also prove that the conditions to be regular are the same for restricted growth functions of matchings.

Figures (7)

• Figure 1: The plot of the Cayley permutation $1 2 1 3 2 1 4$ with points from an occurrence of $2213$ circled.
• Figure 2: $14242534$ is a horizontal juxtaposition of the Cayley permutations $1323$ and $1423$.
• Figure 3: $51521443$ is a vertical juxtaposition of the Cayley permutations $1213$ and $2211$.
• Figure 4: The four ways to insert a value $n$ into a slot.
• Figure 5: The Cayley permutation $135641742$ with the points in $\pi[[3, 7] \times [4]]$ shown in a dashed rectangle.

Theorems & Definitions (21)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5
• Corollary 2.6