Twelvefold Way
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High Quality Content by WIKIPEDIA articles! In combinatorics, the twelvefold way is a name given to 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 idea of the classification is due to Gian-Carlo Rota, and the name was suggested by Joel Spencer.Formally, the last three cases mean that the problem is taken to be counting the orbits of the natural action of respectively the symmetric group of N, the symmetric group o...
High Quality Content by WIKIPEDIA articles! In combinatorics, the twelvefold way is a name given to 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 idea of the classification is due to Gian-Carlo Rota, and the name was suggested by Joel Spencer.Formally, the last three cases mean that the problem is taken to be counting the orbits of the natural action of respectively the symmetric group of N, the symmetric group of X, and of the product of the two groups, on the appropriate sets of functions. These criteria can be paired in 3 × 4 = 12 ways. These 12 types of problems do not involve the same difficulties, and there is not one systematic method for solving them.
