[seqfan] Array of nondecreasing permutation downsteps

[Ron Hardin]

Mystery at end

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

Table starts
.1..2.......6..........24..............120.................720..................5040

.1..3......27.........410............10055..............353654..............17052210

.1..4.....112........6120...........738150...........148700748...........49096652080

.1..5.....453.......85035.........51149685.........57614883627.......130516916069715

.1..6....1818.....1130256.......3451956516......21241004664348....332469512259607296

.1..7....7279....14576404.....230141263315....7575106427737240.827196295133181978340

.1..8...29124...183919920...15258126049410.2638115823321645192......................

.1..9..116505..2282493365.1009051056050225..........................................

.1.10..466030.27960543720...........................................................

.1.11.1864131.......................................................................


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


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)


Mystery:
The coefficients of the line k recurrences are found in rows of table 
http://oeis.org/A169593

"Coefficients of characteristic polynomials of determinant equals trace matrices 
using Eulerian trace and factorial determinant."


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