Pattern Avoiding Permutations Enumerated by Inversions

TL;DR

The paper investigates enumerating indecomposable permutations by their inversion count, focusing on pattern avoidance for patterns of length at most $3$ and establishing rich bijections to classical combinatorial objects. By leveraging inversion tables, decomposition into indecomposable components, and pattern symmetries, it derives exact counts and generating functions for single patterns (e.g., $132$ ↔ partitions, $231$ ↔ fountains, $321$ ↔ parallelogram polyominoes) and for all pairs of patterns, linking to almost triangular, Gorenstein, and distinct-part partitions among others. It further classifies many multi-pattern sets, revealing that higher-pattern avoidance often aligns with well-structured partition families and known combinatorial sequences, sometimes via explicit recurrences or closed-form generating functions. The work provides novel bijections and structural insights that may inform growth-rate bounds and cross-objects mappings within pattern-avoiding permutation theory. $\$

Abstract

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion and Schmidt, we investigate all combinations of permutation patterns of length at most 3.

Figures (6)

• Figure 1: Example of bijective map in Theorem \ref{['ik231']}.
• Figure 2: Example of parallelogram polyomino.
• Figure 3: Example of bijective map in Theorem \ref{['parallelo']}.
• Figure 4: Example of decomposition into smaller fountains.
• Figure 5: Hasse/Ferrers diagram of a Gorenstein partition.

Theorems & Definitions (46)

• Lemma 1: akc, Lemma 8
• Lemma 2: Lemma 10, akc
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Lemma 6
• proof