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A003170 Number of 4 X n Latin rectangles in which the first row is in order.
(Formerly M5172)
24, 1344, 393120, 155185920, 88390995840, 69761852246016, 74175958614030336, 103657593656495554560, 186355188348102566876160, 423073240119513285788344320, 1193404222275011001999025311744, 4123706289611916312851104783171584, 17237448791456599571078045378751528960 (list; graph; refs; listen; history; text; internal format)



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).


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


(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


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

Sequence in context: A187029 A186967 A068294 * A160310 A269271 A347857

Adjacent sequences:  A003167 A003168 A003169 * A003171 A003172 A003173




N. J. A. Sloane


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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Last modified May 28 18:24 EDT 2022. Contains 354122 sequences. (Running on oeis4.)