A003170 Number of 4 X n Latin rectangles in which the first row is in order.

24, 1344, 393120, 155185920, 88390995840, 69761852246016, 74175958614030336, 103657593656495554560, 186355188348102566876160, 423073240119513285788344320, 1193404222275011001999025311744, 4123706289611916312851104783171584, 17237448791456599571078045378751528960

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

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

Links

• Douglas Stones, Table of n, K(4,n) for n=4..80
• F. W. Light, Jr., A procedure for the enumeration of 4 X n Latin rectangles, Fib. Quart., 11 (1973), 241-246.
• 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
• D. S. Stones, The many formulas 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.
• Index entries for sequences related to Latin squares and rectangles

Crossrefs

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

Programs

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

Extensions

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