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 "Log-concave and unimodal sequences in algebra, combinatorics, and geometry" 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 (OEIS: A067804). 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 this MO answer.

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 "Log-concave and unimodal sequences in algebra, combinatorics, and geometry" 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 (OEIS: A067804). 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 this MO answer.

EDIT: In light of the examples from Chris McDaniel, I should amend to say that I am mostly interested in sequences where $a_0 > \cdots > a_m < \cdots < a_n$ for some $0 < m < n$.

Recall a sequence $a_1,\ldots,a_n$ of positive integers is unimodal if $a_1 \leq \cdots \leq a_m \geq \cdots \geq a_n$ for some $1 \leq m \leq n$. Unimodal integer sequences are abundant in combinatorics. For instance, see the classic survey "Log-concave and unimodal sequences in algebra, combinatorics, and geometry" by Stanley.

For lack of a better term, let me call a sequence $a_1,\ldots,a_n$ of positive integers upside-down unimodal if $a_1 \geq \cdots \geq a_m \leq \cdots \leq a_n$ for some $1 \leq m \leq n$.

Question: What are some natural examples of upside-down 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 upside-down unimodal. This is https://oeis.org/A067804. (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 this MO answer.)

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 "Log-concave and unimodal sequences in algebra, combinatorics, and geometry" 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 (OEIS: A067804). 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 this MO answer.