A322859 The number of permutations p of {1,...,n} such that the numbers 2p(1)-1,...,2p(n)-n are all distinct.

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

Offset: 0

Author

M. Farrokhi D. G., Dec 29 2018

Keywords

• nonn
• hard
• more

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).

Examples

			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).
		

Crossrefs

Cf. A000212.

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

Programs

GAP

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

Formula

Conjecture: n! ~ n^(1+o(1))*a(n).

Conjecture: (n-2)a(n-1) <= a(n) <= (n-1)a(n-1).

Conjecture: The polynomial a(1)+a(2)x+...+a(n)x^(n-1) 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).

Extensions

a(15)-a(16) from Bert Dobbelaere, Sep 18 2019