[seqfan] Re: Latin-sum Squares

Ron Hardin rhhardin at att.net
Thu Nov 26 23:42:45 CET 2015


Oops, I had a longer version of the first table
/tmp/fio
T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with
Table starts
.1.....1.......1.........1........1.......1.....1.....1.1.1
.0.....2.......0.........6........0......20.....0....70.0..
.0.....6......12........90......360....2040.10080.54810....
.0....24.......0......4464........0.2850360.....0..........
.0...120....1080....493920.75502080........................
.0...720.......0.128269440.................................
.0..6300.3135720...........................................
.0.45360...................................................
.0.........................................................
Some.solutions.for.n=4.k=4..
..3..0..0..3....1..0..2..3....0..3..2..1....2..0..1..3....3..1..0..2..
..0..3..2..1....0..3..2..1....3..0..3..0....2..3..1..0....1..0..2..3..
..1..2..3..0....3..0..2..1....1..2..0..3....2..0..3..1....2..2..1..1..
..2..1..1..2....2..3..0..1....2..1..1..2....0..3..1..2....0..3..3..0..
 rhhardin at mindspring.com rhhardin at att.net (either)
 
      From: Ron Hardin <rhhardin at att.net>
 To: "seqfan at list.seqfan.eu" <seqfan at list.seqfan.eu> 
 Sent: Thursday, November 26, 2015 5:35 PM
 Subject: [seqfan] Latin-sum Squares
   
Take the same elements as a Latin square but require only that the row, column, and possibly one or both diagonals have equal sums.
That's the diagonal of (preliminary but probably difficult to extend much) :


/tmp/fio
T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with row sums equal and column sums equal
Table starts
.1....1....1......1...1.......1.....1..1.1
.0....2....0......6...0......20.....0.70..
.0....6...12.....90.360....2040.10080.....
.0...24....0...4464...0.2850360...........
.0..120.1080.493920.......................
.0..720....0..............................
.0.6300...................................
.0........................................
Some.solutions.for.n=4.k=4..
..3..0..0..3....1..0..2..3....0..3..2..1....2..0..1..3....3..1..0..2..
..0..3..2..1....0..3..2..1....3..0..3..0....2..3..1..0....1..0..2..3..
..1..2..3..0....3..0..2..1....1..2..0..3....2..0..3..1....2..2..1..1..
..2..1..1..2....2..3..0..1....2..1..1..2....0..3..1..2....0..3..3..0..

/tmp/fip
T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with row sums equal, column sums equal and full-length antidiagonal sums equal to the short dimension sums
Table starts
.1.1...1......1.......1.....1..1..1.1.1
.0.0...0......0.......0.....0..0..0.0..
.0.0...8......6......10....26.42.70....
.0.0...0....624.......0.10110..0.......
.0.0..16...6592.8140944................
.0.0...0.126764........................
.0.0.168...............................
.0.0...................................
.0.....................................
Some.solutions.for.n=4.k=4..
..0..3..1..2....0..3..1..2....2..1..3..0....3..3..0..0....1..2..2..1..
..2..1..3..0....3..0..2..1....2..1..3..0....0..1..3..2....0..3..3..0..
..3..0..0..3....3..2..1..0....1..2..0..3....2..2..1..1....3..0..0..3..
..1..2..2..1....0..1..2..3....1..2..0..3....1..0..2..3....2..1..1..2..

/tmp/fiq
T(n,k)=Number of nXk arrays of permutations of k copies of 0..n-1 with row sums equal, column sums equal and full-length diagonal and antidiagonal sums equal to the short dimension sums
Table starts
.1.1.1....1......1...1.1.1.1.1
.0.0.0....0......0...0.0.0.0..
.0.0.4....2......0...2.0.2....
.0.0.0..256......0.412.0......
.0.0.4..448.836864............
.0.0.0.2336...................
.0.0.4........................
.0.0..........................
.0............................
Some.solutions.for.n=4.k=4..
..3..1..2..0....3..0..3..0....1..0..3..2....0..2..1..3....1..0..2..3..
..0..2..1..3....0..1..2..3....2..2..1..1....3..3..0..0....3..2..0..1..
..0..2..1..3....2..3..0..1....3..3..0..0....1..1..2..2....0..1..3..2..
..3..1..2..0....1..2..1..2....0..1..2..3....2..0..3..1....2..3..1..0..
   



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