Two bijections for sets of words with forbidden factors

TL;DR

This paper establishes explicit bijections for two specific sets of words avoiding certain patterns, solving open problems and connecting these sets to known sequences in combinatorics.

Contribution

It provides explicit bijections for two open problems related to pattern-avoiding words, linking them to well-known integer sequences.

Findings

Solved two open problems on bijections for pattern-avoiding words

Connected pattern-avoiding words to sequences A007070 and A048739

Enhanced understanding of combinatorial structures in pattern avoidance

Abstract

In a recent paper by Kitaev and Remmel, several formulas for the number of words of length n avoiding some generalized patterns were established. Each time the obtained function of n had been found in Sloane's Encyclopedia as the number of some other objects, but the bijections between the two sets did not follow from the proof. Kitaev and Remmel stated four open problems on finding respective bijections. Here we solve two of them, concerning sequences A007070 and A048739.