A002777 -id:A002777 - 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

A003471 Number of permutations with no hits on 2 main diagonals.

Original entry on oeis.org

1, 0, 0, 0, 4, 16, 80, 672, 4752, 48768, 440192, 5377280, 59245120, 839996160, 10930514688, 176547098112, 2649865335040, 48047352500224, 817154768973824, 16438490531536896, 312426715251262464, 6906073926286725120, 145060238642780180480, 3495192502897779875840

Offset: 0

N. J. A. Sloane

Permanent of the binary matrix with an entry equal to 0 iff the entry is on the main diagonal or the main antidiagonal. - Simone Severini, Oct 14 2004
From Toby Gottfried, Dec 05 2008: (Start)
Suppose you have a group of married couples (plus perhaps one other person).
You wish to organize a gift exchange so that:
- each person gives and receives one gift.
- no one gives himself a gift.
- no one gives his/her spouse a gift.
Then the sequence gives the number of ways that this can be done. (End)

			G.f. = 1 + 4*x^4 + 16*x^5 + 80*x^6 + 672*x^7 + 4752*x^8 + ... - _Michael Somos_, Jun 17 2023

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 187.
Todd Simpson, 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).

Robert Israel, Table of n, a(n) for n = 0..200
S. Even and J. Gillis, Derangements and Laguerre polynomials, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 79, Issue 1, January 1976, pp. 135-143.
S. Hertzsprung, Løsning og Udvidelse af Opgave 402, Tidsskrift for Math., 4 (1879), 134-140.
Mathematics Stack Exchange, Derivation of integral formula for even n and for odd n.
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]. See Vol. 3 page 468. There may have been some confusion here of this sequence with A002777.
John Riordan and N. J. A. Sloane, Correspondence, 1974
Todd Simpson, Letter to N. J. A. Sloane, Mar. 1992
Todd Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy)

Cf. A000166, A002777, A225740.

Column k=0 of A335872.

a:= proc(n) option remember; `if`(n<5, [1, 0$3, 4][n+1],
      (n-1)*a(n-1)+2*`if`(n::even, (n-2)*a(n-4), (n-1)*a(n-2)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 27 2020

a[n_] := Integrate[m = Mod[n, 2]; k = (n-m)/2; (x^2-4*x+2)^k*(x-1)^m*Exp[-x], {x, 0, Infinity}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Sep 09 2013, after Felix A. Pahl *)
nmax=20; b=ConstantArray[0,nmax+1]; b[[1]]=1; b[[2]]=0; b[[3]]=0; b[[4]]=0; b[[5]]=4; Do[b[[n+1]] = (n-1)*b[[n]] + If[EvenQ[n],2*(n-2)*b[[n-3]],2*(n-1)*b[[n-1]]],{n,5,nmax}]; b  (* Vaclav Kotesovec, Mar 07 2014 *)
a[ n_] = If[n<4, Boole[n==0], With[{m =2-Mod[n, 2]}, a[n-1]*(n-1) + 2*(n-m)*a[n-2*m]]]; (* Michael Somos, Jun 17 2023 *)

{a(n) = if(n<4, n==0, my(m = 2-n%2); a(n-1)*(n-1) + 2*(n-m)*a(n-2*m))}; /* Michael Somos, Jun 17 2023 */

a(n) = (n-1)*a(n-1) + 2*(n-d)*a(n-e), where (d, e) = (2, 4) if n even, (1, 2) if n odd.
a(n) = Integral_{ x = 0..oo} (x^2-4*x+2)^k * (x-1)^m * exp(-x) dx, where n=2*k+m, m=n mod 2. - Felix A. Pahl, Dec 27 2011
Recurrence: (n-3)*(3*n^3 - 36*n^2 + 137*n - 162)*a(n) = (n-5)*(3*n^3 - 27*n^2 + 71*n - 50)*a(n-1) + (n-2)*(3*n^5 - 45*n^4 + 248*n^3 - 606*n^2 + 608*n - 156)*a(n-2) - 2*(n-3)*(3*n^3 - 28*n^2 + 87*n - 94)*a(n-3) + 2*(3*n^5 - 51*n^4 + 334*n^3 - 1060*n^2 + 1650*n - 1028)*a(n-4) - 4*(n-4)*(n^2 + n - 14)*a(n-5) - 4*(n-5)*(n-4)*(n-2)*(3*n^3 - 27*n^2 + 74*n - 58)*a(n-6). - Vaclav Kotesovec, Mar 07 2014
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Mar 07 2014

More terms from Larry Reeves (larryr(AT)acm.org), Sep 24 2001

cp's OEIS Frontend

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

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