[seqfan] Tipped Array Contours

[Ron Hardin]

Take an nXk array, make the top left element 0, and the bottom right element n+k-2-Q.

How many nondecreasing ways are there to fill in the array?   As if the numbers were contours of a tipped array.

With Q=0, obviously there is only one way.

Wiith Q=1, you get (Empirical)  binomial(n+k,k)-2 ways.

Can the formula be guessed for Q=2, 3,  below?

The Q=1 case is in OEIS row- and column-wise, and with antidiagonals as rows.  The formula is stolen from one of the rows.


ets
T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right n+k-3 and value increasing by 0 or 1 with every step right or down
Table starts
..0...1...2....3....4....5....6....7....8....9...10..11..12.13
..1...4...8...13...19...26...34...43...53...64...76..89.103...
..2...8..18...33...54...82..118..163..218..284..362.453.......
..3..13..33...68..124..208..328..493..713..999.1363...........
..4..19..54..124..250..460..790.1285.2000.3001................
..5..26..82..208..460..922.1714.3001.5003.....................
..6..34.118..328..790.1714.3430.6433..........................
..7..43.163..493.1285.3001.6433...............................
..8..53.218..713.2000.5003....................................
..9..64.284..999.3001.........................................
.10..76.362.1363..............................................
.11..89.453...................................................
.12.103.......................................................
.13...........................................................

Empirical: T(n,k)=binomial(n+k,k)-2

Some.solutions.for.4X4..
..0..1..1..2....0..1..2..2....0..1..2..2....0..1..1..2....0..1..1..2..
..1..2..2..3....1..2..3..3....1..2..3..3....1..2..2..3....1..1..2..3..
..1..2..3..4....2..3..3..4....2..3..4..4....2..3..3..4....2..2..3..4..
..2..3..4..5....3..3..4..5....2..3..4..5....2..3..4..5....3..3..4..5..

ett
T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right n+k-4 and value increasing by 0 or 1 with every step right or down
Table starts
..0....0.....1......3......6......10......15......21......28.....36.....45
..0....1.....8.....26.....61.....120.....211.....343.....526....771...1090
..1....8....44....153....413.....949....1948....3676....6497..10894..17492
..3...26...153....615...1953....5281...12686...27805...56624.108549.197804
..6...61...413...1953...7313...23203...64920..164399..383735.836797.......
.10..120...949...5281..23203...85801..277585..806347.2142634..............
.15..211..1948..12686..64920..277585.1030330.3407823......................
.21..343..3676..27805.164399..806347.3407823..............................
.28..526..6497..56624.383735.2142634......................................
.36..771.10894.108549.836797..............................................
.45.1090.17492.197804.....................................................
.55.1496.27083............................................................
.66.2003..................................................................
.78.......................................................................

Some.solutions.for.4X4..
..0..0..1..1....0..0..1..2....0..0..1..1....0..1..2..3....0..1..2..3..
..0..1..2..2....1..1..2..2....1..1..1..2....1..2..2..3....1..2..3..4..
..1..2..3..3....2..2..3..3....1..1..2..3....1..2..3..3....1..2..3..4..
..1..2..3..4....2..2..3..4....1..2..3..4....2..3..3..4....1..2..3..4..

etu
T(n,k)=Number of nXk nonnegative integer arrays with upper left 0 and lower right n+k-5 and value increasing by 0 or 1 with every step right or down
Table starts
...0.....0......0.......1........4........10........20........35........56
...0.....0......1......13.......61.......192.......483......1050......2058
...0.....1.....18.....153......770......2859......8694.....22924.....54272
...1....13....153....1236.....6997.....30802....112877....359550...1024773
...4....61....770....6997....46812....248182...1100210...4230324..14477724
..10...192...2859...30802...248182...1592348...8528422..39423196.161160206
..20...483...8694..112877..1100210...8528422..54926890.303382053..........
..35..1050..22924..359550..4230324..39423196.303382053....................
..56..2058..54272.1024773.14477724.161160206..............................
..84..3732.118057.2667554.44951694........................................
.120..6369.239798.6438457.................................................
.165.10351.460207.........................................................
.220.16159................................................................
.286......................................................................

Some.solutions.for.4X4..
..0..1..1..2....0..1..2..2....0..0..1..2....0..1..2..2....0..0..0..0..
..0..1..2..2....0..1..2..2....1..1..1..2....1..1..2..2....0..0..0..1..
..0..1..2..2....0..1..2..3....1..1..2..2....1..2..2..2....0..1..1..2..
..1..2..3..3....1..1..2..3....1..2..3..3....1..2..2..3....1..1..2..3..




 
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