What is the name of this combinatorics problem?

Question

What is the name of this problem? The number of ways of ordering a string of $m$ zeros and $n$ ones such that there are never more zeros than ones in the first $k$ digits (with $k \leq m+n$). I have seen the Wikipedia page before, so I know that it is a named problem, but I can't remember the name. It has some relation to taxicab geometry.

Answer 1

As mentioned in the comments, this is the Ballot problem and such strings are called ballot sequences. More generally, words $w_1 w_2 \ldots w_l$ on the alphabet $\mathbb{N} = \{1,2,\ldots\}$ that satisfy the property that, for each $k$ and $i$, $$\left|\{j \leq k: w_j = i\}\right| \geq \left|\{j \leq k: w_j = i+1\}\right|$$ are called Yamanouchi words.