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In combinatorial mathematics, a partial permutation, or sequence without repetition, on a finite set S is a bijection between two specified subsets of S. That is, it is defined by two subsets U and V of equal size, and a one-to-one mapping from U to V. Equivalently, it is a partial function on S that can be extended to a permutation. [1] [2]
Formally, an exchangeable sequence of random variables is a finite or infinite sequence X 1, X 2, X 3, ... of random variables such that for any finite permutation σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest fixed), the joint probability distribution of the permuted sequence
1-planarity [1] 3-dimensional matching [2] [3]: SP1 Bandwidth problem [3]: GT40 Bipartite dimension [3]: GT18 Capacitated minimum spanning tree [3]: ND5 Route inspection problem (also called Chinese postman problem) for mixed graphs (having both directed and undirected edges). The program is solvable in polynomial time if the graph has all ...
No non-recursive formula counting 1324-avoiding permutations is known. A recursive formula was given by Marinov & Radoičić (2003).A more efficient algorithm using functional equations was given by Johansson & Nakamura (2014), which was enhanced by Conway & Guttmann (2015), and then further enhanced by Conway, Guttmann & Zinn-Justin (2018) who give the first 50 terms of the enumeration.
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.
A principal permutation class is a class whose basis consists of only a single permutation. Thus, for instance, the stack-sortable permutations form a principal permutation class, defined by the forbidden pattern 231. However, some other permutation classes have bases with more than one pattern or even with infinitely many patterns.
In fact no method that uses only two-way random events with equal probability ("coin flipping"), repeated a bounded number of times, can produce permutations of a sequence (of more than two elements) with a uniform distribution, because every execution path will have as probability a rational number with as denominator a power of 2, while the ...
Other properties of the Lehmer code include that the lexicographical order of the encodings of two permutations is the same as that of their sequences (σ 1, ..., σ n), that any value 0 in the code represents a right-to-left minimum in the permutation (i.e., a σ i smaller than any σ j to its right), and a value n − i at position i ...