cp's OEIS Frontend

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

%I A000903 M1761 N0698 #70 Feb 16 2025 08:32:21
%S A000903 1,1,2,7,23,115,694,5282,46066,456454,4999004,59916028,778525516,
%T A000903 10897964660,163461964024,2615361578344,44460982752488,
%U A000903 800296985768776,15205638776753680,304112757426239984,6386367801916347184
%N A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the 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 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="/A000903/b000903.txt">Table of n, a(n) for n = 1..100</a>
%H A000903 P. J. Cameron, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000903 L. C. Larson, <a href="/A000900/a000900_1.pdf">The number of essentially different nonattacking rook arrangements</a>, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
%H A000903 E. Lucas, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k29021h">Théorie des Nombres</a>, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
%H A000903 E. Lucas, <a href="/A000899/a000899.pdf">Théorie des nombres</a> (annotated scans of a few selected pages)
%H A000903 R. C. Read, <a href="/A000684/a000684_1.pdf">Letter to N. J. A. Sloane, Oct. 29, 1976</a>
%H A000903 R. W. Robinson, <a href="http://dx.doi.org/10.1007/BFb0097382">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
%H A000903 R. W. Robinson, <a href="/A000899/a000899_1.pdf">Counting arrangements of bishops</a>, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
%H A000903 Zvezdelina Stankova-Frenkel and Julian West, <a href="http://arxiv.org/abs/math/0103152">A new class of Wilf-equivalent permutations</a>, arXiv:math/0103152 [math.CO], 2001. See Fig. 9.
%H A000903 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookNumber.html">Rook Number.</a>
%H A000903 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RooksProblem.html">Rooks Problem.</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_, 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
%t A000903 c[n_] := Floor[n/2]! 2^Floor[n/2];
%t A000903 r[n_] := If[Mod[n, 4] > 1, 0, m = Floor[n/4]; If[m == 0, 1, (2 m)!/m!]];
%t A000903 d[0] = d[1] = 1; d[n_] := d[n] = (n - 1)d[n - 2] + d[n - 1];
%t A000903 a[1] = 1; a[n_] := (n! + c[n] + 2 r[n] + 2 d[n])/8;
%t A000903 Array[a, 21] (* _Jean-François Alcover_, Apr 06 2011, after _Matthias Engelhardt_, further improved by _Robert G. Wilson v_ *)
%Y A000903 Cf. A000142, A037223, A037224, A000085, A005635, A099952, A263685.
%K A000903 nonn,nice
%O A000903 1,3
%A A000903 _N. J. A. Sloane_
%E A000903 More terms from _David W. Wilson_, Jul 13 2003