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%I #18 Dec 01 2018 08:20:25
%S 0,1,6,18,96,600,4320,35280,322560,3265920,36288000,439084800,
%T 5748019200,80951270400,1220496076800,19615115520000,334764638208000,
%U 6046686277632000,115242726703104000,2311256907767808000
%N a(2) = 6, otherwise a(n) = n*n!.
%C a(n) = number of real non-singular (0,1)-matrices of order n having maximal permanent = A000255(n). Proof: [W. Edwin Clark and Richard Brualdi] The maximum permanent is per A where A has all 1's except for n-1 0's on the main diagonal. By Corollary 4.4 in the Brualdi et al. reference for n >= 4 any n X n (0,1)-matrix B with per B = per A can be obtained from A by permuting rows and columns. Since there are n ways to place the single 1 on the main diagonal and then n! ways to permute the distinct rows, a(n) = n*n! if n >=4. Direct computation shows this also holds for n = 1 and 3. - _W. Edwin Clark_, Nov 15 2003
%H Richard A. Brualdi, John L. Goldwasser, T. S. Michael, <a href="http://dx.doi.org/10.1016/0097-3165(88)90019-2">Maximum permanents of matrices of zeros and ones</a>, J. Combin. Theory Ser. A 47 (1988), 207-245.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=602">Encyclopedia of Combinatorial Structures 602</a>
%F E.g.f.: x*(-2*x^2+x^3+x+1)/(-1+x)^2.
%e a(2)=6 because there are 6 (0,1)-matrices with nonzero determinant having permanent=1. See example in A089482. The (0,1)-matrix with maximal permanent=2 ((1,1),(1,1)) has det=0.
%p spec := [S,{S=Prod(Z,Union(Z,Prod(Sequence(Z),Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t Join[{0,1,6},Table[n*n!,{n,3,20}]] (* _Harvey P. Dale_, Apr 20 2012 *)
%Y Cf. A000255. A089480 gives occurrence counts for permanents of non-singular (0, 1)-matrices, A051752 number of (0, 1)-matrices with maximal determinant A003432.
%Y Essentially the same as A001563.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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