A374985 - OEIS

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.

%I #23 Nov 20 2024 09:47:38

%S 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,

%T 197,4593,4593,197,1,1,1,1,903,126289,732963,126289,903,1,1,1,1,4279,

%U 3888343,155242003,155242003,3888343,4279,1,1,1,1,20793,130016393,40007492715,289599115433,40007492715,130016393,20793,1,1

%N 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.

%C 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

%H Andrew Howroyd, <a href="/A374985/b374985.txt">Table of n, a(n) for n = 0..230</a>

%H R. A. Sulanke, <a href="https://doi.org/10.37236/1807">Generalizing Narayana and Schroeder Numbers to Higher Dimensions</a>, Electron. J. Combin. 11 (2004), Research Paper 54.

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

%e Array begins:

%e =====================================================================

%e n/k | 0 1 2 3 4 5 6 ...

%e ----+----------------------------------------------------------------

%e 0 | 1 1 1 1 1 1 1 ...

%e 1 | 1 1 1 1 1 1 1 ...

%e 2 | 1 1 3 11 45 197 903 ...

%e 3 | 1 1 11 197 4593 126289 3888343 ...

%e 4 | 1 1 45 4593 732963 155242003 40007492715 ...

%e 5 | 1 1 197 126289 155242003 289599115433 723253222084867 ...

%e 6 | 1 1 903 3888343 40007492715 723253222084867 ...

%e ...

%e The T(2,3) = 11 matrices are:

%e [1 2 3] [1 2 3] [1 2 3] [1 2 3] [1 2 4] [1 2 4]

%e [2 3 4] [2 4 5] [3 4 5] [4 5 6] [2 3 5] [3 4 5]

%e .

%e [1 2 4] [1 2 5] [1 3 4] [1 3 4] [1 3 5]

%e [3 5 6] [3 4 6] [2 4 5] [2 5 6] [2 4 6]

%o (PARI) \\ See PARI link in A374514 for program code.

%o for(n=0, 7, print(vector(7, k, A374985(n, k-1))))

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

%Y Main diagonal is A374514.

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

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, Sep 16 2024