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

1,2

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

Links

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

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) +12686988*a(n-3) -321204409*a(n-4) +605003244*a(n-5) -296321796*a(n-6)

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)

Example

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

Crossrefs

Column 3 is A014825(n+1)

Row 1 is A000142

Sequence in context: A249619 A345462 A222159 * A096334 A107867 A158442

Adjacent sequences: A221620 A221621 A221622 * A221624 A221625 A221626

Keyword

nonn,tabl

Author

R. H. Hardin Jan 21 2013

Status

approved