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

Author

Manfred Scheucher, Jan 03 2018

Keywords

• nonn

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

Crossrefs

Cf. A000085, A001813, A006882, A135401.

Programs

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 *)

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))

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.)