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%I M5172 #26 Jan 31 2022 01:20:05
%S 24,1344,393120,155185920,88390995840,69761852246016,
%T 74175958614030336,103657593656495554560,186355188348102566876160,
%U 423073240119513285788344320,1193404222275011001999025311744,4123706289611916312851104783171584,17237448791456599571078045378751528960
%N Number of 4 X n Latin rectangles in which the first row is in order.
%D S. M. Kerawala, The enumeration of the Latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc., 33 (1941), 119-127.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Douglas Stones, <a href="/A003170/b003170.txt">Table of n, K(4,n) for n=4..80</a>
%H F. W. Light, Jr., <a href="http://www.fq.math.ca/Scanned/11-3/light.pdf">A procedure for the enumeration of 4 X n Latin rectangles</a>, Fib. Quart., 11 (1973), 241-246.
%H B. D. McKay and E. Rogoyski, <a href="http://www.combinatorics.org/Volume_2/volume2.html#N3">Latin squares of order ten</a>, Electron. J. Combinatorics, 2 (1995) #N3.
%H Douglas Stones, <a href="http://code.google.com/p/latinrectangles/downloads/list">Doyle's formula for the number of reduced 6xn Latin rectangles</a>
%H Douglas Stones, <a href="http://combinatoricswiki.org/wiki/Enumeration_of_Latin_Squares_and_Rectangles">Enumeration Of Latin Squares And Rectangles</a>
%H D. S. Stones, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1a1/0">The many formulas for the number of Latin rectangles</a>, Electron. J. Combin 17 (2010), A1.
%H D. S. Stones and I. M. Wanless, <a href="http://dx.doi.org/10.1016/j.jcta.2009.03.019">Divisors of the number of Latin rectangles</a>, J. Combin. Theory Ser. A 117 (2010), 204-215.
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>
%o (GAP) ChooseList:=function(a,B) local x,p,i; x:=a; p:=1; for i in B do p:=p*Binomial(x,i); x:=x-i; od; return p; end;;
%o DoylePartitions:=function(n) return Union(List(Partitions(n+8,8)-1,P->PermutationsList(P))); end;;
%o DoyleF1:=function(A) return A[1]+A[3]+A[2]+A[4]; end;;
%o DoyleF2:=function(A) return A[1]+A[2]+A[5]+A[6]; end;;
%o DoyleF3:=function(A) return A[1]+A[3]+A[5]+A[7]; end;;
%o DoyleF12:=function(A) return A[1]+A[2]; end;;
%o DoyleF23:=function(A) return A[1]+A[5]; end;;
%o DoyleF13:=function(A) return A[1]+A[3]; end;;
%o DoyleF123:=function(A) return A[1]; end;;
%o DoyleG:=function(A) return DoyleF1(A)*DoyleF2(A)*DoyleF3(A) -DoyleF12(A)*DoyleF3(A) -DoyleF23(A)*DoyleF1(A) -DoyleF13(A)*DoyleF2(A) +2*DoyleF123(A); end;;
%o DoyleGProduct:=function(A) local i,p,B; p:=1; for i in [1..8] do B:=List(A,j->j); B[i]:=B[i]-1; B[8]:=B[8]+1; p:=p*DoyleG(B)^A[i]; od; return p; end;;
%o NrFourLineNormalisedLatinRectanglesDoyle:=function(n) local count,A; count:=0; for A in DoylePartitions(n) do count:=count+(-1)^(A[2]+A[3]+A[5]+2*(A[4]+A[6]+A[7])+3*A[8])*ChooseList(n,A)*DoyleGProduct(A); od; return count; end;; # Douglas Stones (douglas.stones(AT)sci.monash.edu.au), Apr 01 2009, Sep 05 2009
%Y Equals A000573*(n-1)!/(n-4)!.
%K nonn,nice
%O 4,1
%A _N. J. A. Sloane_
%E Doron Zeilberger pointed out that was an error in a(10), which has now been corrected.
%E More terms from Douglas Stones (douglas.stones(AT)sci.monash.edu.au), Apr 01 2009
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