1,3

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.

R. C. Read, personal communication.

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).

Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

N. J. A. Sloane, Table of n, a(n) for n = 1..100

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

E. Lucas, Th\'{e}orie des Nombres. Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

If n>1 then a(n) = 1/8 * (F(n) + C(n) + 2 * R(n) + 2 * D(n)), where F(n) = A000142(n) [all solutions, i.e. factorials], C(n) = A037223(n) [central symmetric solutions], R(n) = A037224(n) [rotationally symmetric solutions], and D(n) = A000085(n) [symmetric solutions by reflection at a diagonal] - Matthias Engelhardt (Matthias.R.Engelhardt(AT)web.de), Apr 05 2000

For asymptotics see the Robinson paper.

For n=4 the 7 solutions may be taken to be 1234,1243,1324,1423,1432,2143,2413.

Maple programs for A000142, A037223, A122670, A001813, A000085, A000898, A000407, A000902, A000900, A000901, A000899, A000903

P:=n->n!; # Gives A000142

G:=proc(n) local k; k:=floor(n/2); k!*2^k; end; # Gives A037223, A000165

R:=proc(n) local m; if n mod 4 = 2 or n mod 4 = 3 then RETURN(0); fi; m:=floor(n/4); (2*m)!/m!; end; # Gives A122670, A001813

unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085

B:=proc(n) option remember; if n <= 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(B(n-1)); fi; 2*B(n-2) + (n-2)*B(n-4); end; # Gives A000898 (doubled up)

rho:=n->R(n)/2; # Gives A000407, aerated

beta:=n->B(n)/2; # Gives A000902, doubled up

delta:=n->(D(n)-B(n))/2; # Gives A000900

unprotect(gamma); gamma:=n-> if n <= 1 then RETURN(0) else (G(n)-B(n)-R(n))/4; fi; # Gives A000901, doubled up

alpha:=n->P(n)/8-G(n)/8+B(n)/4-D(n)/4; # Gives A000899

unprotect(sigma); sigma:=n-> if n <= 1 then RETURN(1); else P(n)/8+G(n)/8+R(n)/4+D(n)/4; fi; #Gives A000903

Cf. A000142, A037223, A037224, A000085, A005635.

Adjacent sequences: A000900 A000901 A000902 this_sequence A000904 A000905 A000906

Sequence in context: A073344 A038119 A006986 this_sequence A049021 A002494 A032264

nonn,nice

njas

More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jul 13 2003