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A003170
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Number of 4 X n Latin rectangles in which the first row is in order.
(Formerly M5172)
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2
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24, 1344, 393120, 155185920, 88390995840, 69761852246016, 74175958614030336, 103657593656495554560, 186355188348102566876160, 423073240119513285788344320, 1193404222275011001999025311744, 4123706289611916312851104783171584, 17237448791456599571078045378751528960
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listen;
history;
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OFFSET
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4,1
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REFERENCES
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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.
F. W. Light, Jr., A procedure for the enumeration of 4 X n Latin rectangles, Fib. Quart., 11 (1973), 241-246.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. S. Stones, The many formulae for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.
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LINKS
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Douglas Stones, Table of n, K(4,n) for n=4..80
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
Douglas Stones, Doyle's formula for the number of reduced 6xn Latin rectangles
Douglas Stones, Enumeration Of Latin Squares And Rectangles
Index entries for sequences related to Latin squares and rectangles
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PROG
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(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;;
DoylePartitions:=function(n) return Union(List(Partitions(n+8, 8)-1, P->PermutationsList(P))); end;;
DoyleF1:=function(A) return A[1]+A[3]+A[2]+A[4]; end;;
DoyleF2:=function(A) return A[1]+A[2]+A[5]+A[6]; end;;
DoyleF3:=function(A) return A[1]+A[3]+A[5]+A[7]; end;;
DoyleF12:=function(A) return A[1]+A[2]; end;;
DoyleF23:=function(A) return A[1]+A[5]; end;;
DoyleF13:=function(A) return A[1]+A[3]; end;;
DoyleF123:=function(A) return A[1]; end;;
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;;
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;;
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
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CROSSREFS
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Equals A000573*(n-1)!/(n-4)!.
Sequence in context: A187029 A186967 A068294 * A160310 A187852 A010797
Adjacent sequences: A003167 A003168 A003169 * A003171 A003172 A003173
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Doron Zeilberger pointed out that was an error in a(10), which has now been corrected.
More terms from Douglas Stones (douglas.stones(AT)sci.monash.edu.au), Apr 01 2009
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STATUS
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approved
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