Avoiding 2-letter signed patterns

TL;DR

This paper classifies the sizes of signed permutation sets avoiding certain 2-letter signed patterns, expressing these sizes through well-known combinatorial sequences like binomial coefficients, Catalan numbers, and Fibonacci numbers.

Contribution

It provides a complete enumeration of signed permutations avoiding 2-letter patterns, linking these counts to classical combinatorial sequences.

Findings

Cardinalities expressed via binomial coefficients

Cardinalities expressed via Catalan numbers

Cardinalities expressed via Fibonacci numbers

Abstract

Let B_n be the hyperoctahedral group; that is, the set of all signed permutations on n letters, and let B_n(T) be the set of all signed permutations in B_n which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets B_n(T) where $T \subseteq B_2$. This allow us to express these cardinalities via inverse of binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.