A374985 Array read by antidiagonals: T(n,k) is the number of n X k matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 45, 197, 45, 1, 1, 1, 1, 197, 4593, 4593, 197, 1, 1, 1, 1, 903, 126289, 732963, 126289, 903, 1, 1, 1, 1, 4279, 3888343, 155242003, 155242003, 3888343, 4279, 1, 1, 1, 1, 20793, 130016393, 40007492715, 289599115433, 40007492715, 130016393, 20793, 1, 1

Offset: 0

Author

Andrew Howroyd, Sep 16 2024

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of normal generalized Young tableaux with all rows and columns strictly increasing whose shape is a rectangle of size n X k (cf. A299968). - Ludovic Schwob, Nov 18 2024

Examples

			Array begins:
=====================================================================
n/k | 0 1   2       3           4               5               6 ...
----+----------------------------------------------------------------
  0 | 1 1   1       1           1               1               1 ...
  1 | 1 1   1       1           1               1               1 ...
  2 | 1 1   3      11          45             197             903 ...
  3 | 1 1  11     197        4593          126289         3888343 ...
  4 | 1 1  45    4593      732963       155242003     40007492715 ...
  5 | 1 1 197  126289   155242003    289599115433 723253222084867 ...
  6 | 1 1 903 3888343 40007492715 723253222084867 ...
...
The T(2,3) = 11 matrices are:
  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 4]  [1 2 4]
  [2 3 4]  [2 4 5]  [3 4 5]  [4 5 6]  [2 3 5]  [3 4 5]
.
  [1 2 4]  [1 2 5]  [1 3 4]  [1 3 4]  [1 3 5]
  [3 5 6]  [3 4 6]  [2 4 5]  [2 5 6]  [2 4 6]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..230
• R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54.

Crossrefs

Columns k=1..4 are A000012, A001003, A105124, A374985.

Main diagonal is A374514.

Cf. A060854 (case all values also distinct), A299968.

Programs

PARI

\\ See PARI link in A374514 for program code.
for(n=0, 7, print(vector(7, k, A374985(n, k-1))))

Formula

T(n,k) = T(k,n).