A072948 Number of permutations p of {1,2,3,...,2n} such that Sum_{k=1..2n} abs(k-p(k)) = 2n.

1, 1, 7, 46, 327, 2350, 17222, 127508, 952299, 7159090, 54107670, 410729140, 3129241874, 23914923644, 183254996828, 1407497158968, 10832287881639, 83516348514010, 644935028526278, 4987483388201684, 38619491922881310, 299390833303838980, 2323441087636417604

Offset: 0

Author

Benoit Cloitre, Aug 20 2002

Keywords

• nonn

Comments

This is impossible if the number of symbols is odd.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..300 (terms 0..71 from Alois P. Heinz)
• Mathieu Guay-Paquey and T. Kyle Petersen, The generating function for total displacement, arXiv:1404.4674 [math.CO], 2014.

Crossrefs

Cf. A072949, A062869, A357329.

Programs

Mathematica

f[n_] := If[n == 1, 1, Floor[n/2] t^Floor[(n - 1)/2] z];
F[t_, z_] = ContinuedFractionK[f[i], 1, {i, 1, 8}];
a[n_] := a[n] = SeriesCoefficient[F[t, z], {z, 0, 2 n}, {t, 0, n}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 15}] (* Jean-François Alcover, Feb 25 2019 *)

PARI

a(n)=sum(k=1,n!,if(sum(i=1,n,abs(i-component(numtoperm(n,k),i)))-n,0,1))

Formula

a(n) = A062869(2n,n).

Extensions

a(5) from Michel ten Voorde Jun 13 2003

a(6) from Ryan Propper, Mar 26 2007

a(7)-a(8) from Sean A. Irvine, Sep 22 2009

a(9)-a(12) from Robert Gerbicz, Nov 27 2010

a(13)-a(16) from Alois P. Heinz, May 02 2014 using formula given by Guay-Paquey and Petersen

a(17)-a(22) from Alois P. Heinz, Oct 01 2022