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).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

Sequence in context: A073344 A038119 A006986 this_sequence A049021 A002494 A032264

Adjacent sequences: A000900 A000901 A000902 this_sequence A000904 A000905 A000906

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

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