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The sum of the numbers in the factorial number system representation gives the number of inversions of the permutation, and the parity of that sum gives the signature of the permutation. Moreover, the positions of the zeroes in the inversion table give the values of left-to-right maxima of the permutation (in the example 6, 8, 9) while the ...
(n factorial) is the number of n-permutations; !n (n subfactorial) is the number of derangements – n-permutations where all of the n elements change their initial places. In combinatorial mathematics , a derangement is a permutation of the elements of a set in which no element appears in its original position.
In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base , although factorials do not function as base , but as place value of digits.
The word "factorial" (originally French: factorielle) was first used in 1800 by Louis François Antoine Arbogast, [18] in the first work on Faà di Bruno's formula, [19] but referring to a more general concept of products of arithmetic progressions. The "factors" that this name refers to are the terms of the product formula for the factorial. [20]
The last formula can be understood directly, by considering the n! permutations of all the elements of S. Each such permutation gives a k -combination by selecting its first k elements. There are many duplicate selections: any combined permutation of the first k elements among each other, and of the final ( n − k ) elements among each other ...
Enumerations of specific permutation classes; Factorial. Falling factorial; Permutation matrix. Generalized permutation matrix; Inversion (discrete mathematics) Major index; Ménage problem; Permutation graph; Permutation pattern; Permutation polynomial; Permutohedron; Rencontres numbers; Robinson–Schensted correspondence; Sum of permutations ...
[10] [12] [13] The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are instead given by the telephone numbers, which may be expressed as a summation involving double factorials. [14] Stirling permutations, permutations of the multiset of numbers 1, 1, 2, 2, ..., k, k in which each pair of equal numbers ...
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n {\displaystyle n} .