Double Derangement Permutations
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Abstract: Let n be a positive integer. A permutation a of the symmetric group 
of permutations of 
is called a derangement if 
for each 
. Suppose that x and y are two arbitrary permutations of 
. We say that a
permutation a is a double
derangement with respect to x and y if 
and 
for each 
. In this paper, we give an explicit formula for 
, the number of double
derangements with respect to x and y.
Let 
and let 
and 
be two subsets of 
with 
and 
. Suppose that 
denotes the number of derangements x such that 
. As the main result,
we show that if 
and z is a permutation such that 
for 
and 
for 
, then 
where 
.
of permutations of is called a derangement if for each . Suppose that x and y are two arbitrary permutations of . We say that a
permutation a is a double
derangement with respect to x and y if and for each . In this paper, we give an explicit formula for , the number of double
derangements with respect to x and y.
Let and let and be two subsets of with and . Suppose that denotes the number of derangements x such that . As the main result,
we show that if and z is a permutation such that for and for , then where .
Cite this paper:
Daneshmand, P. , Mirzavaziri, K. and Mirzavaziri, M. (2016) Double Derangement Permutations. Open Journal of Discrete Mathematics, 6, 99-104. doi: 10.4236/ojdm.2016.62010.
References
[1] Graham, R.L., Knuth, D.E. and Patashnik, O. (1988) Concrete Mathematics. Addison-Wesley, Reading.
[2] Pitman, J. (1997) Some Probabilistic Aspects of Set Partitions. American Mathematical Monthly, 104, 201-209.
http://dx.doi.org/10.2307/2974785
[3] Knopfmacher, A., Mansour, T. and Wagner, S. (2010) Records in Set Partitions. The Electronic Journal of Combinatorics, 17, R109.