A polynomial counting some packings in $\mathbb Z/N\mathbb Z$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:25:21Z http://mathoverflow.net/feeds/question/47413 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47413/a-polynomial-counting-some-packings-in-mathbb-z-n-mathbb-z A polynomial counting some packings in $\mathbb Z/N\mathbb Z$ Roland Bacher 2010-11-26T09:00:58Z 2010-11-26T09:00:58Z <p>Given two integers $n$ and $N$ such that $N>{n+1\choose 2}$, we denote by $\alpha_n(N)$ the number of elements $(x_1,\dots,x_n)$ in $(\mathbb Z/N\mathbb Z)^n$ such that the $2n$ elements $x_1,x_1+1,x_2,x_2+2,x_3,x_3+3,\dots,x_n,x_n+n$ of $\mathbb Z/N\mathbb Z$ are all distinct. The number $\alpha_n(N)$ is then a polynomial function of the form $N^n-2nN^{n-1}+\dots$. </p> <p>Is there a nice formula for the polynomial $\alpha_n(N)$?</p> <p>Remark: The analogous problem such that $x_1,x_1+1,x_2,x_2+2,x_3,x_3+4,x_4,x_4+8,\dots, x_n,x_n+2^{n-1}$ (and many similar variations and generalizations) is much easier and I have a solution (described in <a href="http://fr.arxiv.org/abs/1011.0975" rel="nofollow">http://fr.arxiv.org/abs/1011.0975</a>).</p>