max permanent of nonsingular nxn (0,1)-matrices

[Jaap Spies]

I wrote:
> 
> Let a(n) be the permanent of the (0,1)-matrix of order nxn with n-1
> diagonal 0s only.
> Let b(n) be the permanent of the (0,1)-matrix of order nxn with n
> diagonal 0s only.
> 
> Then by expanding along the first row
> 
> b(n) =  0.b(n-1) + (n-1) a(n-1) = (n-1) a(n-1)
> a(n) =  1.b(n-1) + (n-1) a(n-1)
> 
> So a(n) = (n-2) a(n-2) + (n-1) a(n-1) and we get
> with a(1)=1 and a(2)=1
> the sequence 1, 1, 3, 11, 53, etc.
> 
> Which is in fact A000255.
> 

In fact there is this formula: b(n) = n! * sum((-1)^k / k!, k=0..n)
See Brualdi, Ryser: Combinatorial Matrix Theory, 7.2 p. 202.
And also A000166.
So
	a(n-1) = b(n)/(n-1) = (n-1)! * sum((-1)^k / k!,k=0..n)
etcetera.
  This or a similar result can be obtained by applying theorem 7.2.1
of Brualdi & Ryser to the matrix belonging to a(n)

a(n)=sum((-1)^k * binomial(n-1,k) * (n-k)!, k=0..n-1);

Jaap Spies