Search: id:A000903
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%I A000903 M1761 N0698
%S A000903 1,1,2,7,23,115,694,5282,46066,456454,4999004,59916028,778525516,
%T A000903 10897964660,163461964024,2615361578344,44460982752488,800296985768776,
%U A000903 15205638776753680,304112757426239984,6386367801916347184
%N A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n 
               board.
%D A000903 L. C. Larson, The number of essentially different nonattacking rook arrangements, 
               J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
%D A000903 R. C. Read, personal communication.
%D A000903 R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial 
               Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%D A000903 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000903 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000903 Z. Stankova and J. West, A new class of Wilf-equivalent permutations, 
               J. Algeb. Combin., 15 (2002), 271-290.
%H A000903 N. J. A. Sloane, <a href="b000903.txt">Table of n, a(n) for n = 1..100</
               a>
%H A000903 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000903 E. Lucas, <a href="http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&O=N029021">
               Th\'{e}orie des Nombres</a>. Gauthier-Villars, Paris, 1891, Vol. 
               1, p. 222.
%H A000903 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RookNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000903 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RooksProblem.html">Link to a section of The World of Mathematics.</
               a>
%F A000903 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
%F A000903 For asymptotics see the Robinson paper.
%e A000903 For n=4 the 7 solutions may be taken to be 1234,1243,1324,1423,1432,2143,
               2413.
%p A000903 Maple programs for A000142, A037223, A122670, A001813, A000085, A000898, 
               A000407, A000902, A000900, A000901, A000899, A000903
%p A000903 P:=n->n!; # Gives A000142
%p A000903 G:=proc(n) local k; k:=floor(n/2); k!*2^k; end; # Gives A037223, A000165
%p A000903 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
%p A000903 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
%p A000903 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)
%p A000903 rho:=n->R(n)/2; # Gives A000407, aerated
%p A000903 beta:=n->B(n)/2; # Gives A000902, doubled up
%p A000903 delta:=n->(D(n)-B(n))/2; # Gives A000900
%p A000903 unprotect(gamma); gamma:=n-> if n <= 1 then RETURN(0) else (G(n)-B(n)-R(n))/
               4; fi; # Gives A000901, doubled up
%p A000903 alpha:=n->P(n)/8-G(n)/8+B(n)/4-D(n)/4; # Gives A000899
%p A000903 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
%Y A000903 Cf. A000142, A037223, A037224, A000085, A005635.
%Y A000903 Sequence in context: A073344 A038119 A006986 this_sequence A049021 A002494 
               A032264
%Y A000903 Adjacent sequences: A000900 A000901 A000902 this_sequence A000904 A000905 
               A000906
%K A000903 nonn,nice
%O A000903 1,3
%A A000903 N. J. A. Sloane (njas(AT)research.att.com).
%E A000903 More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jul 13 
               2003

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