A215297 T(n,k) = number of permutations of 0..floor((n*k-2)/2) on odd squares of an n X k array such that each row and column of odd squares is increasing.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 6, 6, 1, 1, 10, 30, 30, 10, 1, 1, 20, 70, 280, 70, 20, 1, 1, 35, 420, 2100, 2100, 420, 35, 1, 1, 70, 1050, 23100, 23100, 23100, 1050, 70, 1, 1, 126, 6930, 210210, 1051050, 1051050, 210210, 6930, 126, 1, 1, 252, 18018, 2522520, 14294280

Offset

1,5

Comments

Table starts

.1...1.....1........1...........1..............1.................1

.1...2.....3........6..........10.............20................35

.1...3.....6.......30..........70............420..............1050

.1...6....30......280........2100..........23100............210210

.1..10....70.....2100.......23100........1051050..........14294280

.1..20...420....23100.....1051050.......85765680........5703417720

.1..35..1050...210210....14294280.....5703417720......577185873264

.1..70..6930..2522520...814773960...577185873264...337653735859440

.1.126.18018.25729704.12547518984.48236247979920.43364386933948080

Even columns match A215292.

The first column is number of symmetric standard Young tableaux of shape (n), the second column is number of symmetric standard Young tableaux of shape (n,n) and the third column is number of symmetric standard Young tableaux of shape (n,n,n). - Ran Pan, May 21 2015

Links

R. H. Hardin, Table of n, a(n) for n = 1..1000

Hodge, Jonathan K.; Krines, Mark; Lahr, Jennifer, Preseparable extensions of multidimensional preferences, Order 26, No. 2, 125-147 (2009), Table 1.

Ran Pan, Exercise P, Problem 4, Project P.

Formula

f1=floor(k/2), f2=floor((k+1)/2), f3=floor((n+1)/2), f4=floor(n/2);

T(n,k) = A060854(f1,f3)*A060854(f2,f4)*binomial(f1*f3+f2*f4,f1*f3).

Example

Some solutions for n=5, k=4:
..x..0..x..4....x..0..x..1....x..1..x..3....x..0..x..6....x..0..x..1
..1..x..2..x....4..x..7..x....0..x..8..x....3..x..5..x....3..x..7..x
..x..3..x..8....x..2..x..3....x..2..x..5....x..1..x..7....x..2..x..5
..6..x..7..x....5..x..9..x....4..x..9..x....4..x..9..x....6..x..8..x
..x..5..x..9....x..6..x..8....x..6..x..7....x..2..x..8....x..4..x..9

Crossrefs

Column 2 is A001405. Column 4 is A215288. Column 6 is A215290.

Sequence in context: A216956 A285522 A181039 * A225910 A215292 A124975

Adjacent sequences: A215294 A215295 A215296 * A215298 A215299 A215300

Keyword

nonn,tabl

Author

R. H. Hardin, Aug 07 2012

Status

approved