Permutation
A permutation is a bijective rearrangement of the elements of a finite set, equivalently an ordering in which each element appears exactly once.
Statistics Every Programmer Needs
A permutation without replacement is an ordered arrangement in which each item may be selected at most once.
- How it is used
- Counts possible outcomes when a probability problem selects items in a meaningful order.
- How deep it goes
- Defines order and replacement, derives the factorial count through shrinking choices, and computes an example in Python.
- Examples
- Uses briefcase-lock sequences and a five-item, three-position calculation yielding 60 arrangements.
- What you can do after
- Apply it via Python’s math.perm to count ordered selections without replacement and check whether repetition is allowed.
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Statistics Slam Dunk
A permutation without replacement is an ordered arrangement in which a selected element cannot be selected again.
- How it is used
- Frames NBA rest analysis by determining whether prior-days-off groupings require permutations or combinations.
- How deep it goes
- Defines ordering and replacement, distinguishes formulas conceptually, and computes or prints arrangements without developing formal derivations.
- Examples
- Uses NBA lineups, standings, lottery selections, coach play calls, and home-away rest pairings.
- What you can do after
- Apply it via gtools to count and print ordered arrangements for an NBA data analysis.
Optimization Algorithms: AI techniques for design, planning, and control problems
A permutation is an ordered arrangement of elements in which each element appears exactly once.
- How it is used
- Frames permutations as discrete decision structures for combinatorial optimization and route-ordering problems.
- How deep it goes
- Defines order and repetition, then works through permutation encoding, tour-cost calculation, and optimizer parameters.
- Examples
- Uses fitness-plan selections, city tours, distance matrices, Python functions, and a twenty-city PSO route.
- What you can do after
- Apply it via PySwarms by encoding candidate orders, evaluating tour distances, and converting the best position into a route.
Fabulous Adventures in Data Structures and Algorithms
A permutation is an ordering of all elements of a sequence or set.
- How it is used
- Uses reorderings to generate test cases, shuffle sequences, and study finite combinatorial structures.
- How deep it goes
- Defines permutations, derives n! growth, compares generation strategies, and demonstrates direct application while deferring full algorithm details.
- Examples
- Uses integer sequences, operation-order test cases, card shuffling, and a generic animal-list reordering.
- What you can do after
- Apply via a C# extension method, generate test orderings, and choose an appropriate permutation-generation strategy.
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Classic Computer Science Problems in Swift: Essential techniques for practicing programmers
A permutation is an ordering of the elements of a collection in which each element appears exactly once.
- How it is used
- Uses exhaustive orderings to evaluate candidate routes in the traveling salesman problem.
- How deep it goes
- Explains recursive swapping, base cases, in-place mutation, and backtracking, without developing permutation theory or performance analysis.
- Examples
- Demonstrates permutations with three letters, Vermont cities, distance tables, and closed traveling-salesman routes.
- What you can do after
- Build all permutations in Swift and apply them to exhaustive route evaluation or constrained word generation.
Advanced Algorithms and Data Structures
A permutation is an arrangement of the elements of a sequence in which each element occurs exactly once.
- How it is used
- Models candidate routes for traveling-salesperson optimization and encodes solutions for simulated annealing and genetic algorithms.
- How deep it goes
- Defines the representation, cost calculation, symmetry pruning, initialization, and transition operators, but not permutation theory beyond optimization.
- Examples
- Uses six-city and ten-vertex tours, explicit city sequences, cost landscapes, random initialization, swaps, and random-permutation transitions.
- What you can do after
- Build candidate TSP solutions, calculate closed-tour costs, prune equivalent arrangements, and select search transitions.
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Real-World Cryptography
A permutation is a reversible one-to-one mapping of a set of elements onto itself.
- How it is used
- Clarifies reversible mappings underlying SHA-3 and keyed block-cipher encryption.
- How deep it goes
- Defines one-to-one reversibility and connects it to SHA-3 and AES, but stops short of designing the underlying permutations.
- Examples
- Uses four shapes, all possible 8-bit strings, and 128-bit AES plaintext-ciphertext mappings.
- What you can do after
- Understand how reversible mappings support hashing and encryption constructions.
Quantum Programming in Depth
A permutation is an arrangement containing each member of a finite set exactly once.
- How it is used
- Reduces the candidate space for a Grover-based N queens solver by excluding repeated column indices.
- How deep it goes
- Contrasts N^N with N!, quantifies the N = 4 improvement, and notes the harder state preparation without deriving its circuit.
- Examples
- Uses column-index encodings for N queens and compares the four-queen spaces of 256 unrestricted inputs and 24 permutations.
- What you can do after
- Evaluate/understand how permutation-based encodings reduce Grover iterations while complicating equal-superposition preparation.
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Get Programming: Learn to code with Python
A permutation is an arrangement of selected items in which order distinguishes the arrangement.
- How it is used
- Uses permutations to reason about differently ordered lists and possible arrangements of Scrabble tiles.
- How deep it goes
- Defines permutations with list examples and exercises, but stops short of implementing the permutation-checking function.
- Examples
- Uses integer lists with reordered and repeated values, plus Scrabble tiles whose letters could form valid words.
- What you can do after
- Apply the concept by checking whether two lists contain identical elements with identical counts despite different ordering.
Read how Get Programming: Learn to code with Python explains permutation →
Related concepts
- combinationcontrasts with · A combination is a selection of elements in which the order of selection does not matter.
- permutation with replacementincludes · A permutation with replacement is an ordered selection in which each item remains available for subsequent selection.
- sequenceuses · A sequence is an ordered collection of elements whose positions determine their order of traversal or processing.
- shuffleproduced by · Shuffling is the random rearrangement of elements in a collection, sequence, or dataset.
- search spacepart of · A search space is the set of possible states, choices, configurations, or candidate solutions that may be considered when exploring or solving a problem.
- keccak-fincludes · keccak-f is the fixed-size permutation used as the internal transformation in SHA-3.
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