# [seqfan] Matrix of 1..n*k permutation ordered by rows, columns and diagonals

Ron Hardin rhhardin at att.net
Sat Oct 2 02:57:56 CEST 2010

```What I'm looking at today.  The empirical column recurrences have a predictable
first term; and a power of 2 for the last term, but I guess nothing simple is
evolving.

1,1,1,1,1,1,1,2,1,1,1,5,4,1,1,1,14,29,8,1,1,1,42,290,169,16,1,1,1,132,
3532,6392,985,32,1,1,1,429,49100,352184,141696,5741,64,1,1,1,1430,
750325,25097600,36372976,3142704,33461,128,1,1,1,4862,12310294
T(n,k)=Number of nXk matrices containing a permutation of 1..n*k in increasing
order rowwise, columnwise, diagonally and (downwards) antidiagonally
Table starts
.1.1...1.......1...........1..............1................1.................1
.1.1...2.......5..........14.............42..............132...............429
.1.1...4......29.........290...........3532............49100............750325
.1.1...8.....169........6392.........352184.........25097600........2152061145
.1.1..16.....985......141696.......36372976......14083834704.....7372392431849
.1.1..32....5741.....3142704.....3777546912....8092149471168.26791156423752069
.1.1..64...33461....69705920...392658046912.4673805856338368..................
.1.1.128..195025..1546100352.40820345224064...................................
.1.1.256.1136689.34293030016..................................................
.1.1.512.6625109..............................................................
All solutions for 3X4
..0..1..2..3....0..1..2..3....0..1..2..3....0..1..2..3....0..1..2..3
..4..5..6..7....4..5..6..8....4..5..6..9....4..5..7..8....4..5..7..9
..8..9.10.11....7..9.10.11....7..8.10.11....6..9.10.11....6..8.10.11

..0..1..2..5....0..1..2..5....0..1..2..5....0..1..2..5....0..1..2..5
..3..4..6..7....3..4..6..8....3..4..6..9....3..4..7..8....3..4..7..9
..8..9.10.11....7..9.10.11....7..8.10.11....6..9.10.11....6..8.10.11

..0..1..3..5....0..1..3..5....0..1..3..5....0..1..3..5....0..1..3..5
..2..4..6..7....2..4..6..8....2..4..6..9....2..4..7..8....2..4..7..9
..8..9.10.11....7..9.10.11....7..8.10.11....6..9.10.11....6..8.10.11

..0..1..2..4....0..1..2..4....0..1..2..4....0..1..2..4....0..1..2..4
..3..5..6..7....3..5..6..8....3..5..6..9....3..5..7..8....3..5..7..9
..8..9.10.11....7..9.10.11....7..8.10.11....6..9.10.11....6..8.10.11

..0..1..3..4....0..1..3..4....0..1..3..4....0..1..3..4....0..1..3..4
..2..5..6..7....2..5..6..8....2..5..6..9....2..5..7..8....2..5..7..9
..8..9.10.11....7..9.10.11....7..8.10.11....6..9.10.11....6..8.10.11

..0..1..2..6....0..1..2..6....0..1..3..6....0..1..3..6
..3..4..7..8....3..4..7..9....2..4..7..8....2..4..7..9
..5..9.10.11....5..8.10.11....5..9.10.11....5..8.10.11
Empirical column 1: a(n)=a(n-1)
Empirical column 2: a(n)=a(n-1)
Empirical column 3: a(n)=2*a(n-1)
Empirical column 4: a(n)=6*a(n-1)-a(n-2)
Empirical column 5: a(n)=24*a(n-1)-40*a(n-2)-8*a(n-3)
Empirical column 6:
a(n)=120*a(n-1)-1672*a(n-2)+544*a(n-3)-6672*a(n-4)+256*a(n-5)
Empirical column 7:
a(n)=720*a(n-1)-84448*a(n-2)+1503360*a(n-3)-17912224*a(n-4)-318223104*a(n-5)+564996096*a(n-6)+270471168*a(n-7)-11373824*a(n-8)+65536*a(n-9)

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

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