8.4 Permutations and related structures

Permutations and cycles are key concepts in labelled structures. They help us understand how elements can be rearranged and form patterns within sets. This knowledge is crucial for analyzing complex combinatorial problems and developing efficient algorithms.

Exponential generating functions (EGFs) are powerful tools for studying permutations. They allow us to represent and manipulate permutation-related structures mathematically, making it easier to solve counting problems and derive important combinatorial identities.

Permutations and Cycles

Understanding Permutations and Their Structure

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• Permutations rearrange elements of a set in a specific order
• Number of permutations for n distinct elements equals n!
• Represent permutations using two-line notation or one-line notation
• Compose permutations by applying them sequentially
• Inverse permutation reverses the effect of the original permutation

Analyzing Cycles and Special Permutations

• Cycles form closed loops within permutations
• Decompose permutations into disjoint cycles
• Cycle notation provides a compact representation of permutations
• Derangements permute elements without fixed points
• Calculate number of derangements using inclusion-exclusion principle
• Involutions are permutations that are their own inverse ($\sigma^2 = id$)
• Classify involutions based on their cycle structure (fixed points and 2-cycles)

Counting Permutations

Factorial-based Counting Techniques

• Factorials calculate the number of ways to arrange n distinct objects
• Compute n! as the product of all positive integers up to n
• Use factorials in permutation formulas ($P(n,r) = \frac{n!}{(n-r)!}$)
• Apply factorial properties in combinatorial proofs

Advanced Counting Methods and Statistics

• Stirling numbers of the first kind count permutations with a specific number of cycles
• Denote Stirling numbers of the first kind as $\stirling{n}{k}$ or s(n,k)
• Calculate Stirling numbers recursively or using generating functions
• Analyze permutation statistics (inversions, descents, cycles)
• Compute expected values and variances of permutation statistics

Generating Functions

Exponential Generating Functions for Permutations

• Define exponential generating function (EGF) for permutations as $P(z) = \sum_{n \geq 0} n! \frac{z^n}{n!} = \frac{1}{1-z}$
• Derive EGF for permutations from the series expansion of $\frac{1}{1-z}$
• Use EGF to analyze properties of permutations and related structures
• Apply EGF in solving combinatorial problems involving permutations
• Combine EGFs to study more complex permutation-related structures