A000573 -id:A000573 - cp's OEIS frontend

A001009 Triangle giving number L(n,k) of normalized k X n Latin rectangles.

1, 1, 1, 1, 1, 1, 1, 3, 4, 4, 1, 11, 46, 56, 56, 1, 53, 1064, 6552, 9408, 9408, 1, 309, 35792, 1293216, 11270400, 16942080, 16942080, 1, 2119, 1673792, 420909504, 27206658048, 335390189568, 535281401856, 535281401856, 1, 16687, 103443808

Offset: 1

Brendan McKay

CRC Handbook of Combinatorial Designs, 1996, p. 104.

Herman Jamke, Table of n, a(n) for n = 1..66
Thomas Bloom, Problem 725, Erdős Problems.
Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. (2023) Vol 28, 105-119.
Brendan D. McKay and Eric Rogoyski , Latin squares of order 10, Electron. J. Combinatorics, 2 (1995) #N3.
Douglas S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
Eric Weisstein's World of Mathematics, Latin Rectangle.
Index entries for sequences related to Latin squares and rectangles

Rows include A001623, A000573. Diagonals include A000576.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 12 2010

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

Original entry on oeis.org

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

Offset: 4

N. J. A. Sloane

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

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

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

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

cp's OEIS Frontend

A001009 Triangle giving number L(n,k) of normalized k X n Latin rectangles.

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