Recall a sequence $a_0,\ldots,a_n$ of positive integers is *unimodal* if $a_0 \leq \cdots \leq a_m \geq \cdots \geq a_n$ for some $0 \leq m \leq n$. Unimodal integer sequences are abundant in combinatorics. For instance, see the classic survey <a href="https://math.mit.edu/~rstan/pubs/pubfiles/72.pdf">"Log-concave and unimodal sequences in algebra, combinatorics, and geometry"</a> by Stanley.

For lack of a better term, let me call a sequence $a_0,\ldots,a_n$ of positive integers *upside-down unimodal* if $a_0 \geq \cdots \geq a_m \leq \cdots \leq a_n$ for some $0 \leq m \leq n$.

**Question**: What are natural examples of upside-down unimodal sequences in combinatorics?

Here is one that I know. Let $b_{n,k}$ be the number of words $w=w_1, w_2,\ldots, w_{2n}$ with $w_i = \pm 1$ such that the greatest $m$ with $w_1 + \cdots + w_m = 0$ is $m=2k$. Then $b_{n,k} = \binom{2k}{k}\binom{2(n-k)}{n-k}$, and the sequence $b_{n,0},b_{n,1},\ldots,b_{n,n}$ is (symmetric and) upside-down unimodal (<a href="https://oeis.org/A067804">OEIS: A067804</a>). Evidently, $b_{n,k}$ is intimately related to the famous identity $4^n =\sum_{k=0}^{n} \binom{2k}{k}\binom{2(n-k)}{n-k}$ and to the last return time for a simple random walk on $\mathbb{Z}$; see <a href="https://mathoverflow.net/a/266757/25028">this MO answer</a>.