# Block patterns in generalized Euler Permutations

## TL;DR

This paper extends the cluster method to analyze the distribution of block patterns in generalized Euler permutations, broadening the scope of pattern analysis in restricted combinatorial objects.

## Contribution

It introduces the generalized cluster method, an extension of Goulden and Jackson's approach, for studying pattern distributions in various restricted combinatorial structures.

## Key findings

- Developed the generalized cluster method for restricted objects
- Applied the method to block patterns in generalized Euler permutations
- Enhanced understanding of pattern distributions in combinatorial classes

## Abstract

Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural classes of combinatorial objects which start with either permutations or words and add additional restrictions. These include up-down permutations, generalized Euler permutations, words with no consecutive repeated letters, Young tableaux, and non-backtracking random walks. We develop an extension of the cluster method which we call the {\em generalized cluster method} to study the distribution of certain consecutive patterns in such restricted combinatorial objects. In this paper, we focus on block patterns in generalized Euler permutations.