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T(n,k)=Number of nXk arrays with each row a permutation of 1..k having at least as many downsteps as the preceding row
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%I #4 Jan 21 2013 07:27:00

%S 1,2,1,6,3,1,24,27,4,1,120,410,112,5,1,720,10055,6120,453,6,1,5040,

%T 353654,738150,85035,1818,7,1,40320,17052210,148700748,51149685,

%U 1130256,7279,8,1,362880,1075295220,49096652080,57614883627,3451956516,14576404

%N T(n,k)=Number of nXk arrays with each row a permutation of 1..k having at least as many downsteps as the preceding row

%C Table starts

%C .1..2.......6...........24.................120.........................720

%C .1..3......27..........410...............10055......................353654

%C .1..4.....112.........6120..............738150...................148700748

%C .1..5.....453........85035............51149685.................57614883627

%C .1..6....1818......1130256..........3451956516..............21241004664348

%C .1..7....7279.....14576404........230141263315............7575106427737240

%C .1..8...29124....183919920......15258126049410.........2638115823321645192

%C .1..9..116505...2282493365....1009051056050225.......902542985526634773509

%C .1.10..466030..27960543720...66655625407012320....304529313276100670030616

%C .1.11.1864131.338950264686.4400938611593606031.101620178879261858322711162

%H R. H. Hardin, <a href="/A221623/b221623.txt">Table of n, a(n) for n = 1..138</a>

%F Empirical for column k:

%F k=1: a(n) = a(n-1)

%F k=2: a(n) = 2*a(n-1) -a(n-2)

%F k=3: a(n) = 6*a(n-1) -9*a(n-2) +4*a(n-3)

%F k=4: a(n) = 24*a(n-1) -166*a(n-2) +264*a(n-3) -121*a(n-4)

%F k=5: a(n) = 120*a(n-1) -4345*a(n-2) +52950*a(n-3) -93340*a(n-4) +44616*a(n-5)

%F k=6: a(n) = 720*a(n-1) -164746*a(n-2) +12686988*a(n-3) -321204409*a(n-4) +605003244*a(n-5) -296321796*a(n-6)

%F k=7: a(n) = 5040*a(n-1) -8349390*a(n-2) +5234439280*a(n-3) -936232732785*a(n-4) +51206316902496*a(n-5) -99624831647040*a(n-6) +49349521382400*a(n-7)

%e Some solutions for n=3 k=4

%e ..3..4..2..1....2..3..1..4....2..1..3..4....3..1..2..4....1..2..4..3

%e ..2..1..4..3....2..3..1..4....4..1..2..3....3..2..4..1....3..4..1..2

%e ..2..1..4..3....4..1..2..3....2..1..4..3....3..2..4..1....2..4..3..1

%Y Column 3 is A014825(n+1)

%Y Row 1 is A000142

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_ Jan 21 2013