Given natural numbers $k$ and $n$, I am interested in what it is known about how many k-ary monotone maps there are from $\{1,2,...,n,n+1\}$ to $\{1,2,...,n,n+1\}$. I suppose the answer must be known (and perhaps there are even explicit recurrences to compute the number), but I have not succeeded to find any information (one of the problems is how to search for this).
Looking at the online integer encyclopedia I have located the sequence 1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, etc. (see https://oeis.org/A001700 ) which corresponds to the particular case that k=1.
So let me rephrase the above question as:
how many binary monotone maps there are from $\{1,2,...,n,n+1\}$ to $\{1,2,...,n,n+1\}$?
how many 3-ary monotone maps there are from $\{1,2,...,n,n+1\}$ to $\{1,2,...,n,n+1\}$?
etc.
Some clarification: Such functions go from $\{1,2,...,n,n+1\}^k$ to $\{1,2,...,n,n+1\}$. The monotonicity is based in using the following partial orders:
$\{1,2,...,n,n+1\}$ with the natural (i.e., standard) one,
$\{1,2,...,n,n+1\}^k$ with the direct product of the natural one.
In particular, when $k \geq 2$ this last partial order is not linear.