A221623 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.

1, 2, 1, 6, 3, 1, 24, 27, 4, 1, 120, 410, 112, 5, 1, 720, 10055, 6120, 453, 6, 1, 5040, 353654, 738150, 85035, 1818, 7, 1, 40320, 17052210, 148700748, 51149685, 1130256, 7279, 8, 1, 362880, 1075295220, 49096652080, 57614883627, 3451956516, 14576404

Offset: 1

Author

R. H. Hardin Jan 21 2013

Keywords

• nonn
• tabl

Comments

Table starts

.1..2.......6...........24.................120.........................720

.1..3......27..........410...............10055......................353654

.1..4.....112.........6120..............738150...................148700748

.1..5.....453........85035............51149685.................57614883627

.1..6....1818......1130256..........3451956516..............21241004664348

.1..7....7279.....14576404........230141263315............7575106427737240

.1..8...29124....183919920......15258126049410.........2638115823321645192

.1..9..116505...2282493365....1009051056050225.......902542985526634773509

.1.10..466030..27960543720...66655625407012320....304529313276100670030616

.1.11.1864131.338950264686.4400938611593606031.101620178879261858322711162

Examples

			Some solutions for n=3 k=4
..3..4..2..1....2..3..1..4....2..1..3..4....3..1..2..4....1..2..4..3
..2..1..4..3....2..3..1..4....4..1..2..3....3..2..4..1....3..4..1..2
..2..1..4..3....4..1..2..3....2..1..4..3....3..2..4..1....2..4..3..1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..138

Crossrefs

Column 3 is A014825(n+1)

Row 1 is A000142

Formula

Empirical for column k:

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

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

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

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

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

k=6: a(n) = 720*a(n-1) -164746*a(n-2) [phone]*a(n-3) -321204409*a(n-4) [phone]*a(n-5) -296321796*a(n-6)

k=7: a(n) = 5040*a(n-1) -8349390*a(n-2) [phone]*a(n-3) -936232732785*a(n-4) [phone]*a(n-5) -99624831647040*a(n-6) [phone]*a(n-7)