A297708 Number of permutations p of [n] such that p(p(i)) = i for all i or p(n+1-p(i)) = n+1-i for all i.

1, 1, 2, 6, 14, 46, 132, 444, 1452, 5164, 18680, 71080, 278920, 1135624, 4774448, 20692560, 92381072, 423566224, 1994458656, 9619233888, 47516407008, 239904464608, 1237764055616, 6515682543040, 34984350444736, 191360856810688, 1065970229647232, 6041353305197184

Offset

0,3

Comments

a(n) is likewise the number of ways to place n points on an n X n grid with pairwise distinct abscissa, pairwise distinct ordinate, and mirror symmetry using one or the other of the diagonals of the grid as axis of symmetry. See also A000085 and A135401. For rotational symmetry see A001813 and A006882.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..800 (terms n = 0..99 from Manfred Scheucher)

Manfred Scheucher, Python program for enumeration

Formula

a(n) = 2*A000085(n) - A135401(n). (Proof: A000085 counts the permutations satisfying the first condition as well as the permutations satisfying the second condition. A135401 counts the permutations satisfying both conditions.)

Maple

a:= proc(n) option remember; `if`(n<7, [1$2, 2, 6, 14, 46, 132][n+1],
      ((-25*n+149)*a(n-1)+(2*(10*n^2-7*n-106))*a(n-2)+
       (45*n^2-268*n+298)*a(n-3)-(2*(10*n^2-7*n-61))*a(n-4)
       -(65*n^2-367*n+522)*a(n-5)-(2*(10*n^3-67*n^2+96*n+1))*a(n-6)
       -(45*n-113)*(n-4)*(n-6)*a(n-7))/(20*n-79))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 07 2018

Mathematica

a[n_] := 2*Sum[2^k*BellB[k, 1/2]*StirlingS1[n, k], {k, 0, n}] - Sum[2^k*BellB[k]*StirlingS1[Floor[n/2], k], {k, 0, Floor[n/2]}];
Array[a, 30, 0] (* Jean-François Alcover, May 29 2019 *)

Prog

(SageMath)
def a135401(n): return sum( binomial(floor(n/2), 2*k)*binomial(2*k, k)*factorial(k)*2^(floor(n/2)-2*k) for k in range(1+floor(n/4)))
def a85(n): return sum( factorial(n) / (factorial(n-2*k) * 2^k * factorial(k)) for k in range(1+floor(n/2)))
def a297708(n): return 2*a85(n) - a135401(n)
for n in range(100): print(n, a297708(n))

Crossrefs

Cf. A000085, A001813, A006882, A135401.

Sequence in context: A366223 A122109 A133155 * A284701 A011455 A188491

Adjacent sequences: A297705 A297706 A297707 * A297709 A297710 A297711

Keyword

nonn

Author

Manfred Scheucher, Jan 03 2018

Status

approved