Difference between revisions of "Arrangement"
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An arrangement can be regarded as a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342011.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342012.png" /> and taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342013.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342015.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342016.png" /> are usually called cells (or urns), while those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342017.png" /> are called particles (or balls); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342018.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342020.png" />, respectively, belong to the same class if and only if there is a permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342023.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342024.png" />. In this case, the number of such classes, or, as it is called, the number of arrangements filling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342025.png" /> different cells by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342026.png" /> indistinguishable particles, is the number of combinations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342027.png" /> out of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342028.png" /> elements, with repetition allowed (cf. [[Combination|Combination]]). | An arrangement can be regarded as a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342011.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342012.png" /> and taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342013.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342015.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342016.png" /> are usually called cells (or urns), while those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342017.png" /> are called particles (or balls); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342018.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342020.png" />, respectively, belong to the same class if and only if there is a permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342023.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342024.png" />. In this case, the number of such classes, or, as it is called, the number of arrangements filling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342025.png" /> different cells by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342026.png" /> indistinguishable particles, is the number of combinations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342027.png" /> out of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342028.png" /> elements, with repetition allowed (cf. [[Combination|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342030.png" /> respectively, belong to the same class if there exists a permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342032.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342034.png" />. In this case the number of arrangements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342035.png" /> different particles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342036.png" /> indistinguishable cells, that is, the number of classes, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342038.png" /> | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342030.png" /> respectively, belong to the same class if there exists a permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342032.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342034.png" />. In this case the number of arrangements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342035.png" /> different particles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342036.png" /> indistinguishable cells, that is, the number of classes, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342038.png" /> are the [[Stirling numbers]] of the second kind: |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342039.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013420/a01342039.png" /></td> </tr></table> | ||
Revision as of 19:19, 21 November 2014
with repetitions of 
out of 
elements
A finite sequence 
of elements from some set 
. If all the term of 
are distinct, then 
is called an arrangement without repetitions. The number of arrangements of 
out of 
elements with repetitions is 
, and that without repetitions in 
.
An arrangement can be regarded as a function 
given on 
and taking values in 
: 
, 
. The elements of 
are usually called cells (or urns), while those of 
are called particles (or balls); 
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 
and 
, respectively, belong to the same class if and only if there is a permutation 
of 
such that 
for all 
. In this case, the number of such classes, or, as it is called, the number of arrangements filling 
different cells by 
indistinguishable particles, is the number of combinations of 
out of 
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 
and 
respectively, belong to the same class if there exists a permutation 
of 
for which 
for all 
. In this case the number of arrangements of 
different particles in 
indistinguishable cells, that is, the number of classes, is 
, where 
are the Stirling numbers of the second kind:
 |
 |
If one does not distinguish either particles or cells, then an arrangement of 
identical particles in 
identical cells is obtained; the number of such arrangements is 
, where 
is the number of additive partitions of 
into 
natural numbers. One can also consider other partitions of arrangements into classes, for example when the above-mentioned permutations 
and 
are taken from subgroups of the symmetric groups of degrees 
and 
, 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" .
References
| [1] | A.V. Sachkov, "Combinatorial methods in discrete mathematics" , Moscow (1977) (In Russian) |
| [2] | J. Riordan, "An introduction to combinational analysis" , Wiley (1958) |
Comments
References
| [a1] | L. Comtet, "Advanced combinatorics" , Reidel (1977) (Translated from French) |
How to Cite This Entry:
Arrangement. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arrangement&oldid=34716
Arrangement. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arrangement&oldid=34716
This article was adapted from an original article by V.M. Mikheev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article