When arrow patterns meet classical patterns

Abstract: Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures. In this paper, we resolve all three conjectures. First, we enumerate all six subclasses of permutations that simultaneously avoid a classical pattern of length 3 and a fixed arrow pattern of length 3, thereby confirming the first two conjectures. Second, we settle the third conjecture (which involves a different arrow pattern) by providing two independent proofs. These proofs rely on a restriction of Biane's bijection to non-nesting involutions and Krattenthaler's bijection from 321-avoiding permutations to Dyck paths, respectively.

• The paper’s main contribution is the explicit enumeration of permutations avoiding a length-3 classical and the arrow pattern (12;1→3), resolving conjectures by Archer and Laudone.
• It employs two bijective proofs—one via non-nesting involutions and another using Dyck paths—to derive formulas connecting Catalan, Fibonacci, and Motzkin numbers.
• The study further introduces bivariate generating functions and refined statistics that link permutation structures with classical combinatorial sequences.

Enumeration of Permutations Avoiding Arrow and Classical Patterns

Introduction and Motivation

The paper "When arrow patterns meet classical patterns" (2607.04094) provides a comprehensive study on the avoidance of arrow patterns in conjunction with classical permutation patterns, resolving conjectures previously proposed by Archer and Laudone. Arrow patterns, introduced by Berman and Tenner, serve to bridge the one-line and cycle notation representations of permutations, capturing structure sensitive simultaneously to several permutation statistics. Understanding their interplay with classical patterns is critical for advances in enumerative combinatorics, with ramifications in the study of permutation statistics and bijective combinatorics.

Definitions and Preliminaries

Arrow patterns generalize classical pattern avoidance by encoding additional constraints using the inverse of a variant of Foata's fundamental bijection (denoted $\Phi$). Specifically, an arrow pattern $\alpha = (\nu; H)$ involves an order pattern $\nu$ and a set of arrows $H$, interpreted as constraints on the images of certain indices of the permutation under $\Phi^{-1}$. The principal arrow patterns analyzed are $(12;1\to 3)$ and $(12;1\to 2)$, denoted as $\alpha$ and $\beta$, respectively.

Given classical pattern $\pi$ and arrow pattern $\alpha = (\nu; H)$0, $\alpha = (\nu; H)$1 denotes the set of $\alpha = (\nu; H)$2-permutations avoiding both. The enumeration of $\alpha = (\nu; H)$3 is the primary objective, with a systematic determination for all length-3 classical patterns $\alpha = (\nu; H)$4, and arrow patterns $\alpha = (\nu; H)$5 or $\alpha = (\nu; H)$6.

Main Results: Exact Enumerations

The central contribution is the exhaustive enumeration of permutations simultaneously avoiding any classical pattern of length 3 and the fixed arrow pattern $\alpha = (\nu; H)$7. The enumerations confirm and generalize three conjectures by Archer and Laudone. Explicitly:

• For $\alpha = (\nu; H)$8: $\alpha = (\nu; H)$9,
• For $\nu$0 or $\nu$1: $\nu$2 (Catalan numbers),
• For $\nu$3, $\nu$4, or $\nu$5: $\nu$6 (odd-indexed Fibonacci numbers).

The case analysis is structured on a recursive characterization of $\nu$7-avoiding permutations, revealing how the arrow pattern constraint intersects the standard inclusion patterns of classical avoidance. The proofs leverage refined combinatorial arguments and recurrence relations, such as $\nu$8 for the $\nu$9 case.

For the third conjecture, concerning $H$0 and $H$1, the paper establishes $H$2 (Motzkin numbers), confirming the conjecture via two independent bijections.

Bijective Proofs and Combinatorial Structures

Two distinct bijective proofs underpin the Motzkin enumeration for $H$3- and $H$4-avoiding permutations:

1. Involution-Based Bijection

Permutations in $H$5 correspond via $H$6 to non-nesting involutions (i.e., $H$7-avoiding involutions) of size $H$8. There exists a canonical bijection (essentially a restriction of Biane's bijection) between Motzkin paths of length $H$9 and such involutions, where the number of up-steps in the path correlates with the number of 2-cycles.

2. Dyck Path-Based Statistic

A second approach deploys Krattenthaler's bijection between $\Phi^{-1}$0-avoiding permutations and Dyck paths, mapping $\Phi^{-1}$1-occurrences to a local path statistic: the number of triple-up-steps (UUU). The number of Dyck paths without any UUU subword is also $\Phi^{-1}$2. This approach establishes a direct statistic-preserving correspondence, elucidating the structural implications of the avoidance constraint.

Refined Enumeration and Generating Functions

The paper deepens the enumeration by introducing bivariate generating functions capturing refined statistics, such as the number of occurrences of the arrow pattern or descents. The functional equations and closed forms, such as for joint avoidance of $\Phi^{-1}$3 or $\Phi^{-1}$4 with $\Phi^{-1}$5, yield the odd-indexed Fibonacci numbers via algebraic generating functions.

The study further analyzes the tail statistic and its $\Phi^{-1}$6-enumeration over $\Phi^{-1}$7- and $\Phi^{-1}$8-avoiding permutations, establishing an explicit relation with differences of consecutive odd-indexed Fibonacci numbers, and connecting them to classical summation identities.

Implications and Future Directions

This work closes several open conjectures regarding joint avoidance of arrow and classical patterns, providing explicit enumerative solutions and bijective constructions. The methods, particularly the bijective frameworks intertwining Motzkin paths, involutions, and Dyck paths, offer templates for further explorations in pattern avoidance, notably in the generalization to larger patterns and more intricate arrow configurations.

Additionally, the combinatorial tools and the connections to permutation statistics such as descents, cycles, and fixed points, open possibilities for statistical enumeration and Wilf-equivalence classifications within the combined arrow-classical pattern framework.

The direct correspondence between arrow pattern occurrence and structural subwords in Dyck/Motzkin paths suggests potential for further development of the combinatorics of permutation statistics under mixed avoidance constraints. Applications might extend beyond enumerative combinatorics, including the analysis of sorting algorithms and the structure of restricted permutations in algebraic combinatorics and representation theory.

Conclusion

The paper establishes a comprehensive classification and explicit enumeration for all pairs of length-3 classical patterns and the principal arrow pattern $\Phi^{-1}$9, resolving prior conjectures with rigorous combinatorial arguments and bijections. The introduction of two distinct bijective strategies for the Motzkin correspondence further enriches the theoretical toolkit for researchers interested in advanced pattern avoidance phenomena. These results form a foundational component for further development in structural and enumerative combinatorics involving generalized pattern types.