A052655 a(2) = 6, otherwise a(n) = n*n!.

0, 1, 6, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

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

Examples

			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.
		

Links

• Richard A. Brualdi, John L. Goldwasser, T. S. Michael, Maximum permanents of matrices of zeros and ones, J. Combin. Theory Ser. A 47 (1988), 207-245.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 602

Crossrefs

Cf. A000255. A089480 gives occurrence counts for permanents of non-singular (0, 1)-matrices, A051752 number of (0, 1)-matrices with maximal determinant A003432.

Essentially the same as A001563.

Programs

Maple

spec := [S,{S=Prod(Z,Union(Z,Prod(Sequence(Z),Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

Join[{0,1,6},Table[n*n!,{n,3,20}]] (* Harvey P. Dale, Apr 20 2012 *)

Formula

E.g.f.: x*(-2*x^2+x^3+x+1)/(-1+x)^2.