This module contains a function to generate (equivalence classes of) triangular tableaux of size k, strictly increasing to the right and to the bottom. For example

 1  
 2  4  
 3  5  8  
 6  7  9  10 

is such a tableau of size 4. The numbers filling a tableau always consist of an interval [1..c]; c is called the content of the tableaux. There is a unique tableau of minimal content 2k-1:

 1  
 2  3  
 3  4  5 
 4  5  6  7 

Let us call the tableaux with maximal content (that is, m = binomial (k+1) 2) standard. The number of such standard tableaux are

1, 1, 2, 12, 286, 33592, 23178480, ...

OEIS:A003121, "Strict sense ballot numbers", https://oeis.org/A003121.

See R. M. Thrall, A combinatorial problem, Michigan Math. J. 1, (1952), 81-88.

The number of tableaux with content c=m-d are

 d=  |     0      1      2      3    ...
-----+----------------------------------------------
 k=2 |     1
 k=3 |     2      1
 k=4 |    12     18      8      1
 k=5 |   286    858   1001    572    165     22     1
 k=6 | 33592 167960 361114 436696 326196 155584 47320 8892 962 52 1 

We call these "GT simplex tableaux" (in the lack of a better name), since they are in bijection with the simplicial cones in a canonical simplicial decompositions of the Gelfand-Tsetlin cones (the content corresponds to the dimension), which encode the combinatorics of Kostka numbers.