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  2. Lehmer code - Wikipedia

    en.wikipedia.org/wiki/Lehmer_code

    The usual way to prove that there are n! different permutations of n objects is to observe that the first object can be chosen in n different ways, the next object in n − 1 different ways (because choosing the same number as the first is forbidden), the next in n − 2 different ways (because there are now 2 forbidden values), and so forth.

  3. Inversion (discrete mathematics) - Wikipedia

    en.wikipedia.org/wiki/Inversion_(discrete...

    Finite permutations that have only 0s and 1s in their inversion vectors: A059590 (their inversion sets: A211364) Number of permutations of n elements with k inversions; Mahonian numbers: A008302 (their row maxima; Kendall-Mann numbers: A000140) Number of connected labeled graphs with n edges and n nodes: A057500

  4. Enumerative combinatorics - Wikipedia

    en.wikipedia.org/wiki/Enumerative_combinatorics

    3 out of 4638576 [1] or out of 580717, [2] if rotations and reflections are not counted as distinct, Hamiltonian cycles on a square grid graph 8х8. Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.

  5. Cycles and fixed points - Wikipedia

    en.wikipedia.org/wiki/Cycles_and_fixed_points

    Thus, π = (1 2 4 3)(6 8), would be an appropriate way to express this permutation. There are different ways to write a permutation as a list of its cycles, but the number of cycles and their contents are given by the partition of S into orbits, and these are therefore the same for all such expressions.

  6. Inclusion–exclusion principle - Wikipedia

    en.wikipedia.org/wiki/Inclusion–exclusion...

    In the given example, there are 12 = 2(3!) permutations with property P 1, 6 = 3! permutations with property P 2 and no permutations have properties P 3 or P 4 as there are no restrictions for these two elements. The number of permutations satisfying the restrictions is thus: 4! − (12 + 6 + 0 + 0) + (4) = 24 − 18 + 4 = 10.

  7. Steinhaus–Johnson–Trotter algorithm - Wikipedia

    en.wikipedia.org/wiki/Steinhaus–Johnson...

    The ! permutations of the numbers from 1 to may be placed in one-to-one correspondence with the ! numbers from 0 to ! by pairing each permutation with the sequence of numbers that count the number of positions in the permutation that are to the right of value and that contain a value less than (that is, the number of inversions for which is the ...

  8. Comparison sort - Wikipedia

    en.wikipedia.org/wiki/Comparison_sort

    Sorting a set of unlabelled weights by weight using only a balance scale requires a comparison sort algorithm. A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation (often a "less than or equal to" operator or a three-way comparison) that determines which of two elements should occur first in the final sorted list.

  9. Combinatorics - Wikipedia

    en.wikipedia.org/wiki/Combinatorics

    Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.