

A322859


The number of permutations p of {1,...,n} such that the numbers 2p(1)1,...,2p(n)n are all distinct.


0



1, 1, 2, 4, 14, 52, 256, 1396, 9064, 62420, 500000, 4250180, 40738880, 410140060, 4572668112, 53214384548, 676739353112
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OFFSET

0,3


COMMENTS

If p=(i,j) is a transposition on letters 1,...,n with 1 <= i < j <= n, then the numbers 2p(1)1, ..., 2p(n)n are all distinct if and only if either j >= 2i or j > (i+n)/2. It follows that the number b(n) of such permutations equals A000212(n)=floor(n^2/3).


LINKS



FORMULA

Conjecture: n! ~ n^(1+o(1))*a(n).
Conjecture: (n2)a(n1) <= a(n) <= (n1)a(n1).
Conjecture: The polynomial a(1)+a(2)x+...+a(n)x^(n1) is irreducible for all n. Indeed, it seems that the polynomials are irreducible for any permutation of coefficients except for n=7 where the exceptional permutations are (1,7,3,5,4,6) and (1,3,4,6,2).


EXAMPLE

For n=4, the a(4)=14 permutations are (), (2,4), (2,3,4), (1,4), (1,4,3,2), (1,4,2,3), (1,4)(2,3), (1,2,4,3), (1,2)(3,4), (1,2,3,4), (1,3), (1,3,2), (1,3)(2,4), (1,3,2,4).


PROG

(GAP) Number(Filtered(SymmetricGroup(n), p>Number(Unique(List([1..n], i>2*i^pi)))=n));


CROSSREFS

A099152 counts the permutations of {1,...,n} such that the numbers p(i)i (or p(i)+i) are distinct for i=1,...,n.


KEYWORD

nonn,hard,more


AUTHOR



EXTENSIONS



STATUS

approved



