# [seqfan] Crossed Partitions / Weighted Equal Row and Column Sums

Ron Hardin rhhardin at att.net
Mon Nov 18 01:33:29 CET 2013

```This just came up, equal rows sums and equal column sums for a binary matrix with elements in rows and columns weighted 1,2,3,...

It's unusually filled with gaps until both n and k are not small, which indicates some heavy constraint.

/tmp/dpk
T(n,k)=Number of (n+2)X(k+2) 0..1 arrays x(i,j) with every row sum{j*x(i,j), j=1..k+2} equal, and every column sum{i*x(i,j), i=1..n+2} equal, with upper left element zero

Table starts
..2..2...1...1......5.......8.....1......1.....36......63.....1.....1..362.658
..2..2...1...1......5.......8.....1......1.....36......63.....1.....1..362.658
..1..1...7...7......1......19....55......1....259.....613.....1..3571.8881...1
..1..1...7...7......1......19....55......1....259.....613..2761..7615.8881....
..5..5...1...1....186.....557.....1......1..31989..139905.19105.27769.........
..8..8..19..19....557....2004...571......1.209288.1044803.....................
..1..1..55..55......1.....571.27453..70897..66043.............................
..1..1...1...1......1.......1.70897.419803....................................
.36.36.259.259..31989..209288.66043...........................................
.63.63.613.613.139905.1044803.................................................

Some.solutions.for.n=5 k=5..
..0..1..1..1..1..0..0....0..1..0..0..1..0..1....0..0..1..0..1..1..0..
..0..1..1..1..1..0..0....0..1..0..0..1..0..1....0..1..0..0..1..0..1..
..1..0..0..0..0..1..1....1..0..1..1..0..1..0....1..1..0..1..0..0..1..
..1..1..0..1..0..0..1....0..0..1..0..1..1..0....0..1..0..0..1..0..1..
..0..0..1..0..1..1..0....1..1..0..1..0..0..1....1..1..0..1..0..0..1..
..0..0..1..0..1..1..0....1..1..0..1..0..0..1....1..0..1..1..0..1..0..
..1..1..0..1..0..0..1....0..0..1..0..1..1..0....0..0..1..0..1..1..0..

rhhardin at mindspring.com
rhhardin at att.net (either)
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