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In the same way that the double factorial generalizes the notion of the single factorial, the following definition of the integer-valued multiple factorial functions (multifactorials), or α-factorial functions, extends the notion of the double factorial function for positive integers : ! = {()!
The main application of the combinatorial number system is that it allows rapid computation of the k-combination that is at a given position in the lexicographic ordering, without having to explicitly list the k-combinations preceding it; this allows for instance random generation of k-combinations of a given set.
(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 factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value is to be multiplied by (i − 1)!
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.
The Bell numbers come up in a card shuffling problem mentioned in the addendum to Gardner 1978. If a deck of n cards is shuffled by repeatedly removing the top card and reinserting it anywhere in the deck (including its original position at the top of the deck), with exactly n repetitions of this operation, then there are n n different shuffles ...
The latter is far from optimal, but the former, which changes only one variable at a time, is worse. See also the factorial experimental design methods pioneered by Sir Ronald A. Fisher. Reasons for disfavoring OFAT include: OFAT requires more runs for the same precision in effect estimation; OFAT cannot estimate interactions
These symbols are collectively called factorial powers. [2] The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (), where n is a non-negative integer. It may represent either the rising or the falling