Difference between revisions of "Derangement"
From Encyclopedia of Mathematics
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| − | A permutation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524301.png" /> elements in which the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524302.png" /> cannot occupy the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524303.png" />-th position, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524304.png" />. The problem of calculating the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524305.png" /> of derangements is known as the "[[Montmort matching problem]]" or "problème des rencontres" (cf. [[Classical combinatorial problems]]. The following formula holds: | + | A permutation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524301.png" /> elements in which the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524302.png" /> cannot occupy the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524303.png" />-th position, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524304.png" />. The problem of calculating the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052430/i0524305.png" /> of derangements is known as the "[[Montmort matching problem]]" or "problème des rencontres" (cf. [[Classical combinatorial problems]]). The following formula holds: |
<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/i/i052/i052430/i0524306.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/i/i052/i052430/i0524306.png" /></td> </tr></table> | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H.J. Ryser, "Combinatorial mathematics" , ''Carus Math. Monogr.'' , '''14''' , Wiley & Math. Assoc. Amer. (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Riordan, "An introduction to combinatorial mathematics" , Wiley (1958)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> H.J. Ryser, "Combinatorial mathematics" , ''Carus Math. Monogr.'' , '''14''' , Wiley & Math. Assoc. Amer. (1963)</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> J. Riordan, "An introduction to combinatorial mathematics" , Wiley (1958)</TD></TR> | ||
| + | </table> | ||
Revision as of 07:28, 2 December 2016
derangement
A permutation of 
elements in which the element 
cannot occupy the 
-th position, 
. The problem of calculating the number 
of derangements is known as the "Montmort matching problem" or "problème des rencontres" (cf. Classical combinatorial problems). The following formula holds:
 |
Derangements are a special case of permutations satisfying a specific restriction on the position of the permuted elements. For example, the "problème des ménages" consists in calculating the number 
of permutations conflicting with the two permutations 
and 
. (Two permutations of 
elements are called conflicting if the 
-th element occupies different positions in each of them for all 
). The number 
is given by the formula:
 |
The number 
of Latin squares (cf. Latin square) of size 
for 
can be calculated in terms of 
and 
by the formulas
 |
 |
References
| [1] | H.J. Ryser, "Combinatorial mathematics" , Carus Math. Monogr. , 14 , Wiley & Math. Assoc. Amer. (1963) |
| [2] | J. Riordan, "An introduction to combinatorial mathematics" , Wiley (1958) |
How to Cite This Entry:
Derangement. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derangement&oldid=39882
Derangement. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derangement&oldid=39882
This article was adapted from an original article by V.M. Mikheev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article