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A multiset may be formally defined as an ordered pair (A, m) where A is the underlying set of the multiset, formed from its distinct elements, and : + is a function from A to the set of positive integers, giving the multiplicity – that is, the number of occurrences – of the element a in the multiset as the number m(a).
The Stirling numbers of the second kind, written (,) or {} or with other notations, count the number of ways to partition a set of labelled objects into nonempty unlabelled subsets.
In combinatorial mathematics, a Stirling permutation of order k is a permutation of the multiset 1, 1, 2, 2, ..., k, k (with two copies of each value from 1 to k) with the additional property that, for each value i appearing in the permutation, any values between the two copies of i are larger than i. For instance, the 15 Stirling permutations ...
If M is a finite multiset, then a multiset permutation is an ordered arrangement of elements of M in which each element appears a number of times equal exactly to its multiplicity in M. An anagram of a word having some repeated letters is an example of a multiset permutation.
The permutations of the multiset {,,,, …,,} which have the property that for each k, all the numbers appearing between the two occurrences of k in the permutation are greater than k are counted by the double factorial number ()!!.
The formula counting all functions N → X is not useful here, because the number of them grouped together by permutations of N varies from one function to another. Rather, as explained under combinations , the number of n -multicombinations from a set with x elements can be seen to be the same as the number of n -combinations from a set with x ...
In computer science, bogosort [1] [2] (also known as permutation sort and stupid sort [3]) is a sorting algorithm based on the generate and test paradigm. The function successively generates permutations of its input until it finds one that is sorted. It is not considered useful for sorting, but may be used for educational purposes, to contrast ...
In a 1977 review of permutation-generating algorithms, Robert Sedgewick concluded that it was at that time the most effective algorithm for generating permutations by computer. [2] The sequence of permutations of n objects generated by Heap's algorithm is the beginning of the sequence of permutations of n+1 objects.