Arrangement

with repetitions of $ n $ out of $ m $ elements

A finite sequence $ \overline{a}\; = ( a _ {i _ {1} } \dots a _ {i _ {n} } ) $ of elements from some set $ A = \{ a _ {1} \dots a _ {m} \} $. If all the term of $ \overline{a}\; $ are distinct, then $ \overline{a}\; $ is called an arrangement without repetitions. The number of arrangements of $ n $ out of $ m $ elements with repetitions is $ m ^ {n} $, and that without repetitions in $ (m) _ {n} = m ( m - 1 ) \dots (m-n-1) $.

An arrangement can be regarded as a function $ \phi $ given on $ Z _ {n} = \{ 1 \dots n \} $ and taking values in $ A $: $ \phi ( k ) = a _ {i _ {k} } $, $ k = 1 \dots n $. The elements of $ A $ are usually called cells (or urns), while those of $ Z _ {n} $ are called particles (or balls); $ \phi $ defines the filling of the various cells by the various particles. If one speaks of indistinguishable particles or cells, it is understood that classes of arrangements are being considered. Thus, if all the particles are indistinguishable, then two arrangements defined by functions $ \phi $ and $ \psi $, respectively, belong to the same class if and only if there is a permutation $ \sigma $ of $ Z _ {n} $ such that $ \phi ( \sigma (k) ) = \psi (k) $ for all $ k \in Z _ {n} $. In this case, the number of such classes, or, as it is called, the number of arrangements filling $ m $ different cells by $ n $ indistinguishable particles, is the number of combinations of $ n $ out of $ m $ elements, with repetition allowed (cf. Combination).

If one says that all the cells are indistinguishable, then one means that the arrangements are put into the classes in such a way that two arrangements, defined by functions $ \phi $ and $ \psi $ respectively, belong to the same class if there exists a permutation $ \tau $ of $ A $ for which $ \tau ( \phi (k) ) = \psi (k) $ for all $ k \in Z _ {n} $. In this case the number of arrangements of $ n $ different particles in $ m $ indistinguishable cells, that is, the number of classes, is $ \sum _ {k=1} ^ {m} S ( n , k ) $, where $ S ( n , k ) $ are the Stirling numbers of the second kind:

$$ S ( n , k ) = \ S ( n -1 , k - 1 ) + k S ( n - 1 , k ) ,\ \ S ( 0 , 0 ) = 1 , $$

$$ S ( n , k ) = 0 \ \textrm{ for } k > n \textrm{ and } k = 0 , n > 0 . $$

If one does not distinguish either particles or cells, then an arrangement of $ n $ identical particles in $ m $ identical cells is obtained; the number of such arrangements is $ \sum _ {k=1} ^ {n} p _ {n} (k) $, where $ p _ {n} (k) $ is the number of additive partitions of $ n $ into $ k $ natural numbers. One can also consider other partitions of arrangements into classes, for example when the above-mentioned permutations $ \sigma $ and $ \tau $ are taken from subgroups of the symmetric groups of degrees $ n $ and $ m $, respectively (this and other generalizations can be found in [1], [2]). Synonyms of "arrangement" are the terms "n-permutationn-permutation" , and "ordered n-sampleordered n-sample of a population" .

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