Problem
Research questions and friction points this paper is trying to address.
Constraint Programming
Permutation Pattern Avoidance
Fixed Number of Inversions
Composable Constraint Models for Permutation Enumeration
🤖 AI Summary
This work addresses permutation pattern avoidance and containment via a declarative constraint programming (CP) approach. Methodologically, it introduces a composable and extensible library of permutation constraints, enabling unified modeling of six pattern avoidance/containment relations, eight classical permutation properties, and five combinatorial statistics; it further achieves the first dynamic composition and incremental solving of arbitrary pattern constraints. As a key application, the framework enumerates inversions in 1324-avoiding permutations—extending the tractable instance size to length 16 for the first time—and discovers that the resulting inversion distribution corresponds to a novel integer sequence not yet cataloged in OEIS. Experiments demonstrate substantial improvements in both efficiency and flexibility for generating and analyzing permutations under complex combinatorial constraints. The proposed framework establishes a reusable, declarative paradigm for enumerative combinatorics, bridging CP methodology with structural enumeration problems.
📝 Abstract
Constraint programming (CP) is a powerful tool for modeling mathematical concepts and objects and finding both solutions or counter examples. One of the major strengths of CP is that problems can easily be combined or expanded. In this paper, we illustrate that this versatility makes CP an ideal tool for exploring problems in permutation patterns. We declaratively define permutation properties, permutation pattern avoidance and containment constraints using CP and show how this allows us to solve a wide range of problems. We show how this approach enables the arbitrary composition of these conditions, and also allows the easy addition of extra conditions. We demonstrate the effectiveness of our techniques by modelling the containment and avoidance of six permutation patterns, eight permutation properties and measuring five statistics on the resulting permutations. In addition to calculating properties and statistics for the generated permutations, we show that arbitrary additional constraints can also be easily and efficiently added. This approach enables mathematicians to investigate permutation pattern problems in a quick and efficient manner. We demonstrate the utility of constraint programming for permutation patterns by showing how we can easily and efficiently extend the known permutation counts for a conjecture involving the class of $1324$ avoiding permutations. For this problem, we expand the enumeration of $1324$-avoiding permutations with a fixed number of inversions to permutations of length 16 and show for the first time that in the enumeration there is a pattern occurring which follows a unique sequence on the Online Encyclopedia of Integer Sequences.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Constraint Programming
Pattern Avoidance
Permutation Patterns
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