Similar to, but different from, superfactorial Pascal triangle
A009963.
A009963(n,m) = (Product_{p=0..m-1} (n-p)!)/superfac(m) with n >= m >= 0, otherwise 0.
Denoting this sequence as the superbinomial sb(n,k), the hook length formula for a j X k rectangular Young tableau states the number of configurations of j*k distinct numbers such that each row and column is strictly increasing is (j*k)!/sb(j+k,j), ie. 1/sb(j+k,j) is the probability that a random permutation is a Young tableau.
Meanwhile, if the numbers are placed into the array with repetition, but the columns are still strictly increasing, there are c(n,j,k) = sb(n+1,j+k)/(sb(n+1-j,k)*sb(n+1-k,j)) configurations.
If the strict criterion is relaxed to monotonic, this becomes C(n,j,k) = sb(n-1+j+k,j+k)/(sb(n-1+j,j)*sb(n-1+k,k)).
By proposition 13.2(i) of Stanley's PhD thesis, for fixed j,k, c(n,j,k) and C(n,j,k) are polynomials in n of degree j*k, and c(n,j,k) = (-1)^(j*k)*C(-n,j,k).
For example, c(n,1,k)=(n choose k) and C(n,1,k)=(n+k-1 choose k), while c(n,2,k) = N(n,k+1) and C(n,2,k) = N(n+k,k+1), so the binomial coefficients and Narayana numbers N=
A001263 obey the dualities (under continuation as polynomials) (n choose k) = (-1)^k*(k-1-n choose k) and N(n,k) = N(k-1-n,k).
(End)
