A007016 Number of permutations of length n with 1 fixed and 1 reflected point. (Formerly M4491)

0, 1, 0, 0, 8, 20, 96, 656, 5568, 48912, 494080, 5383552, 65097600, 840566080, 11833898496, 176621049600, 2838024476672, 48060623405312, 868000333234176, 16441638519762944, 329723762151352320, 6907027877807330304

Offset

0,5

Comments

Number of distinct solutions to the order n checkerboard problem, including symmetrical solutions: place n pieces on an n X n board so there is exactly one piece in each row, column and main diagonal. Compare A064280.

Number of magic permutation matrices of order n. - Chai Wah Wu, Jan 15 2019

Upper bound for the number of diagonal transversals in a Latin square: A287647(n) <= A287648(n) <= a(n). - Eduard I. Vatutin, Jan 02 2020

References

Simpson, Todd; Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Jean-François Alcover, Table of n, a(n) for n = 0..100

F. Rakotondrajao, Magic squares, rook polynomials and permutations, Séminaire Lotharingien de Combinatoire, vol. 54A, article B54Ac, 2006.

T. Simpson, Letter to N. J. A. Sloane, Mar. 1992

T. Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy)

E. Vatutin, Upper bound for the number of diagonal transversals in a Latin square (in Russian).

Formula

a(2*m) = m*(x(2*m) - (2*m-3)*x(2*m-1)), a(2*m+1) = (2*m+1)*x(2*m) + 3*m*x(2*m-1) - 2*m*(m-1)*x(2*m-2), where x(n) = A003471(n).

Mathematica

x[n_] := x[n] = Integrate[If[EvenQ[n], (x^2 - 4*x + 2)^(n/2), (x - 1)*(x^2 - 4*x + 2)^((n - 1)/2)]/E^x, {x, 0, Infinity}];
a[n_ /; EvenQ[n]] := With[{m = n/2}, m*(x[2*m] - (2*m - 3)*x[2*m - 1])];
a[n_ /; OddQ[n]] := With[{m = (n - 1)/2}, (2*m + 1)*x[2*m] + 3*m*x[2*m - 1] - 2*m*(m - 1)*x[2*m - 2]];
Table[a[n], {n, 0, 21}] // Quiet (* Jean-François Alcover, Jun 29 2018 *)

Prog

(PARI)
a(n) = {my(v = vector(n)); \\ v is A003471
for(n=4, length(v), v[n] = (n-1)*v[n-1] + 2*if(n%2==1, (n-1)*v[n-2], (n-2) * if(n==4, 1, v[n-4])));
if(n<4, [1, 0, 0][n], if(n%2==0, n*(v[n] - (n-3)*v[n-1]), 2*n*v[n-1] + 3*(n-1)*v[n-2] - (n-1)*(n-3)*v[n-3])/2)} \\ Andrew Howroyd, Sep 12 2017

Crossrefs

Cf. A003471, A064280.

Sequence in context: A066011 A333156 A365606 * A129550 A215181 A014584

Adjacent sequences: A007013 A007014 A007015 * A007017 A007018 A007019

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved