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(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.
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 ...
By converting a number less than n! to factorial representation, one obtains a sequence of n digits that can be converted to a permutation of n elements in a straightforward way, either using them as Lehmer code or as inversion table [1] representation; in the former case the resulting map from integers to permutations of n elements lists them ...
The HP-20S (F1890A) is an algebraic programmable scientific calculator produced by Hewlett-Packard from 1987 to 2000.. A member of HP's Pioneer series, the 20S was a low cost model targeted at students, using the same hardware as the HP-10B business calculator.
TI SR-50A, a 1975 calculator with a factorial key (third row, center right) The factorial function is a common feature in scientific calculators. [73] It is also included in scientific programming libraries such as the Python mathematical functions module [74] and the Boost C++ library. [75]
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 ...
The two copies of k must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is k − 1, with n positions into which the adjacent pair of k values may be placed. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction. [1]
Comparison of Stirling's approximation with the factorial. 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 .