A064280 -id:A064280 - cp's OEIS frontend

A007016 Number of permutations of length n with 1 fixed and 1 reflected point.

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

Offset: 0

N. J. A. Sloane

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

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).

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).

Cf. A003471, A064280.

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 *)

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

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).

Conjecture D-finite with recurrence (365968635435167109808*n^2 -5566069866485493251505*n +20525522573033552369132)*a(n) +(-1215369044326430542311*n^2 +19103429957352794982854*n -73690801030090785944295)*a(n-1) +(-365968635435167109808*n^4 +6663975772790994580929*n^3 -35836353442786038818589*n^2 +34878550744402035813586*n +124043542472821007763204)*a(n-2) +(483431773456096322695*n^4 -10754417727097457203127*n^3 +85154149458907095778621*n^2 -277683967994722584206067*n +286254870342835757751852)*a(n-3) +2*(-393241909113483884738*n^4 +9142334951839265043383*n^3 -78427160779754271402777*n^2 +309283968160862567580813*n -465057422344277141977923)*a(n-4) +2*(-745044547502580209919*n^4 +21471238686323774026196*n^3 -222067832543690193789255*n^2 +944698954932049830084232*n -1372732531859619119793978)*a(n-5) +4*(365968635435167109808*n^4 -5227374504728642916627*n^3 +19793104565012302929789*n^2 +391834816007939927082*n -57365695502678698166146)*a(n-6) +4*(-483431773456096322695*n^4 +7592214312314395379733*n^3 -45284933032689911393913*n^2 +117535885088909103449165*n -84799883220517633629252)*a(n-7) +8*(n-7)*(393241909113483884738*n^3 -4789400677912625536335*n^2 +17834478528905815208536*n -23668675533486426523455)*a(n-8) +8*(n-7)*(n-8)*(745044547502580209919*n^2 -6086915962816073505121*n +12854159797389104313178)*a(n-9)=0. - R. J. Mathar, Feb 27 2025

A292080 Number of nonequivalent ways to place n non-attacking rooks on an n X n board with no rook on 2 main diagonals up to rotations and reflections of the board.

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 14, 84, 630, 6096, 55336, 672160, 7409300, 104999520, 1366363752, 22068387264, 331233939624, 6005919062528, 102144359744192, 2054811316442112, 39053339674065360, 863259240785840640, 18132529836143846560, 436899062862222484480

Offset: 0

Andrew Howroyd, Sep 12 2017

nonn

For odd n, there are no symmetrical configurations of non-attacking rooks without a rook in the main diagonal, so a(2n+1) = A003471(2n+1) / 8. For even n, configurations with rotational and diagonal symmetry are possible.

			Case n=4: The 2 nonequivalent solutions are:
   _ x _ _     _ x _ _
   x _ _ _     _ _ _ x
   _ _ _ x     x _ _ _
   _ _ x _     _ _ x _
Case n=5: The 2 nonequivalent solutions are:
   _ x _ _ _   _ x _ _ _
   x _ _ _ _   _ _ _ _ x
   _ _ _ x _   x _ _ _ _
   _ _ _ _ x   _ _ x _ _
   _ _ x _ _   _ _ _ x _

Andrew Howroyd, Table of n, a(n) for n = 0..100
Andrew Howroyd, Nonequivalent and Symmetric Solutions

Cf. A000166, A000903, A003471, A037224, A053871, A064280.

sf[n_] := n! * SeriesCoefficient[Exp[-x ] / (1 - x), {x, 0, n}];
F[n_] := (Clear[v]; v[_] = 0; For[m = 4, m <= n, m++, v[m] = (m - 1)*v[m - 1] + 2*If[OddQ[m], (m - 1)*v[m - 2], (m - 2)*If[m == 4, 1, v[m - 4]]]]; v[n]);
d[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*(2k)!/(2^k*k!), {k, 0, n}];
R[n_] := If[OddQ[n], 0, (n - 1)!*2/(n/2 - 1)!];
a[0] = 1; a[n_] := (F[n] + If[OddQ[n], 0, m = n/2; 2^m * sf[m] + 2*R[m] + 2*d[m]])/8;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 28 2017, after Andrew Howroyd *)

\\ here sf is A000166, F is A003471, D is A053871, R(n) is A037224(2n).
sf(n) = {n! * polcoeff( exp(-x + x * O(x^n)) / (1 - x), n)}
F(n) = {my(v = vector(n)); 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]))); v[n]}
D(n) = {sum(k=0, n, (-1)^(n-k) * binomial(n,k) * (2*k)!/(2^k*k!))}
R(n) = {if(n%2==1, 0, (n-1)!*2/(n/2-1)!)}
a(n) = {(F(n) + if(n%2==1, 0, my(m=n/2); 2^m * sf(m) + 2*R(m) + 2*D(m)))/8}

a(2n+1) = A003471(2n+1) / 8, a(2n) = (A003471(2n) + 2^n * A000166(n) + 2*A037224(2*n) + 2*A053871(n)) / 8.

cp's OEIS Frontend

A007016 Number of permutations of length n with 1 fixed and 1 reflected point.

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