4,1

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

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.

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

Contribution from Douglas Stones (douglas.stones(AT)sci.monash.edu.au), Apr 01 2009, Sep 05 2009: (Start)

(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)-D\ oyleF13(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; ;

(End)

Equals A000573*(n-1)!/(n-4)!.

Sequence in context: A035025 A074652 A068294 this_sequence A160310 A010797 A099060

Adjacent sequences: A003167 A003168 A003169 this_sequence A003171 A003172 A003173

nonn,nice

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

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