[Paper Review] Central binomial coefficients also count (2431,4231,1432,4132)-avoiders

Marie-Louise Bruner|arXiv (Cornell University)|May 19, 2015

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper proves that permutations avoiding the four patterns 2431, 4231, 1432, and 4132 are enumerated by the central binomial coefficients $\binom{2(n-1)}{n-1}$. Using a bijective encoding of permutations into code words—comprising integers and markers B/E representing insertion positions—the authors establish a direct correspondence between these code words and lattice paths, yielding a combinatorial proof of the enumeration formula via a constructive bijection.

ABSTRACT

This short paper is concerned with the enumeration of permutations avoiding the following four patterns: $2431$, $4231$, $1432$ and $4132$. Using a bijective construction, we prove that these permutations are counted by the central binomial coefficients.

Motivation & Objective

• To determine the exact enumeration of permutations avoiding the four specific 4-patterns: 2431, 4231, 1432, and 4132.
• To establish a bijective connection between such permutations and a combinatorial class of code words.
• To show that the number of such permutations of length $n$ is given by the central binomial coefficient $\binom{2(n-1)}{n-1}$.
• To provide a constructive, bijective proof that avoids generating functions or rewriting rules, focusing on structural insertion constraints.

Proposed method

• Define a code word system over the alphabet $\{2,3,\ldots\} \cup \{B,E\}$, where $B$ and $E$ denote beginning and end markers, and integers represent insertion positions.
• Establish constraints on code words: non-B/E symbols must be non-decreasing and not adjacent to other non-B/E symbols.
• Construct a bijection between permutations in the class $\mathcal{S}_n$ and code words of length $n-1$ by tracking the insertion position of the largest element $n$ in recursive constructions.
• Map non-initial segments of code words to Dyck-like lattice paths to enable enumeration via combinatorial path counting.
• Prove that the number of valid code words of length $n-1$ equals $\binom{2(n-1)}{n-1}$ using a generalized summation identity and induction.
• Verify the formula by setting $m = n$ in the identity $\sum_{j=0}^{n-1} 2^{m-j} \binom{m+j-1}{j} (m-j) = n \cdot 2^{m-n+1} \binom{m+n-1}{n}$, yielding the central binomial coefficient.

Experimental results

Research questions

• RQ1 What is the exact enumeration of permutations avoiding the four 4-patterns: 2431, 4231, 1432, and 4132?
• RQ2 Can a bijective correspondence be established between such permutations and a combinatorial class of words with a known enumeration?
• RQ3 Is the number of such permutations of length $n$ equal to the central binomial coefficient $\binom{2(n-1)}{n-1}$?
• RQ4 Can this enumeration be proven combinatorially, without relying on generating functions or rewriting rules?
• RQ5 Are there direct bijections to other objects counted by central binomial coefficients, such as granny walks or Dyck paths?

Key findings

• The number of permutations of length $n$ avoiding the four patterns 2431, 4231, 1432, and 4132 is exactly $\binom{2(n-1)}{n-1}$, the central binomial coefficient.
• A bijective encoding maps each such permutation to a unique code word of length $n-1$ over the alphabet $\{2,3,\ldots\} \cup \{B,E\}$, satisfying specific structural constraints.
• The set of valid code words is enumerated by the same formula as the number of lattice paths with certain step types, establishing a combinatorial link.
• The proof uses a generalized summation identity that, when specialized with $m = n$, yields the central binomial coefficient $\binom{2n}{n}$.
• The authors identify twelve potential Wilf-equivalence classes of 4-pattern-avoiding permutations that may also be enumerated by the central binomial coefficients, with one already proven via a Catalan-based insertion argument.
• The paper leaves open the challenge of constructing a direct bijection from these permutations to standard combinatorial objects like granny walks or Dyck paths.