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

0,3

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

Table of n, a(n) for n=0..16.

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).

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^p-i)))=n));

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.

Sequence in context: A129876 A038055 A006385 * A183949 A131180 A047990

Adjacent sequences: A322856 A322857 A322858 * A322860 A322861 A322862

Keyword

nonn,hard,more

Author

M. Farrokhi D. G., Dec 29 2018

Extensions

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

Status

approved