This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A374985 #23 Nov 20 2024 09:47:38 %S A374985 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 A374985 197,4593,4593,197,1,1,1,1,903,126289,732963,126289,903,1,1,1,1,4279, %U A374985 3888343,155242003,155242003,3888343,4279,1,1,1,1,20793,130016393,40007492715,289599115433,40007492715,130016393,20793,1,1 %N 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. %C A374985 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 A374985 Andrew Howroyd, <a href="/A374985/b374985.txt">Table of n, a(n) for n = 0..230</a> %H A374985 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 A374985 T(n,k) = T(k,n). %e A374985 Array begins: %e A374985 ===================================================================== %e A374985 n/k | 0 1 2 3 4 5 6 ... %e A374985 ----+---------------------------------------------------------------- %e A374985 0 | 1 1 1 1 1 1 1 ... %e A374985 1 | 1 1 1 1 1 1 1 ... %e A374985 2 | 1 1 3 11 45 197 903 ... %e A374985 3 | 1 1 11 197 4593 126289 3888343 ... %e A374985 4 | 1 1 45 4593 732963 155242003 40007492715 ... %e A374985 5 | 1 1 197 126289 155242003 289599115433 723253222084867 ... %e A374985 6 | 1 1 903 3888343 40007492715 723253222084867 ... %e A374985 ... %e A374985 The T(2,3) = 11 matrices are: %e A374985 [1 2 3] [1 2 3] [1 2 3] [1 2 3] [1 2 4] [1 2 4] %e A374985 [2 3 4] [2 4 5] [3 4 5] [4 5 6] [2 3 5] [3 4 5] %e A374985 . %e A374985 [1 2 4] [1 2 5] [1 3 4] [1 3 4] [1 3 5] %e A374985 [3 5 6] [3 4 6] [2 4 5] [2 5 6] [2 4 6] %o A374985 (PARI) \\ See PARI link in A374514 for program code. %o A374985 for(n=0, 7, print(vector(7, k, A374985(n, k-1)))) %Y A374985 Columns k=1..4 are A000012, A001003, A105124, A374985. %Y A374985 Main diagonal is A374514. %Y A374985 Cf. A060854 (case all values also distinct), A299968. %K A374985 nonn,tabl %O A374985 0,13 %A A374985 _Andrew Howroyd_, Sep 16 2024