A slightly less well known combinatorial interpretation of the unsigned Stirling number of the first kind is the number of permutations of $\{1,\dots,n\}$ with $k$ left-to-right maxima.

A left-to-right maximum of a permutation $\pi$ of $\{1,\dots,n\}$ is a position $i$ such that $\pi(j)<\pi(i)$ for all $j<i$. In particular, $n$ is always the value of the rightmost left-to-right maximum, and $1$ is always the position of the leftmost left-to-right maximum.

Identity (1) says that the number of permutations of $\{1,\dots,n\}$ with at least $m$ left-to-right maxima, $m$ of which are chosen as "special", is equal to the number of permutations of $\{1,\dots,n+1\}$ with exactly $m+1$ left-to-right maxima.

Here is a bijection between the two classes of permutations.

Let $\pi$ be a permutation of $\{1,\dots,n\}$, and suppose its $k\ge m$ left-to-right maxima are in positions $1=i_1<\dots<i_k\le n$. Note that $\pi(i_1)<\dots<\pi(i_k)=n$. Set $i_{k+1}=n+1$, and define the subblocks $$ \pi_{j}=\pi(i_j)\dots\pi(i_{j+1}-1), \quad i=1,\dots,k. $$ Then $\pi$ is the concatenation of subblocks $\pi_j$, $j=1,\dots,k$, i.e. $\pi=\pi_1\dots\pi_k$.

Choose $m$ left-to-right-maxima $i_{j_1}<\dots<i_{j_m}$ and declare them "special". Let $\pi'$ be the string $\pi$ with subblocks $\pi_{j_1},\dots,\pi_{j_m}$ deleted. Then our map $f$ is $$ \pi=\pi_1\dots\pi_k\mapsto f(\pi)=\pi_{j_1}\dots\pi_{j_m},n+1,\pi'. $$ In other words, we partition $\pi$ into maximal blocks starting on the left with each successive left-to-right minimum, choose $m$ of those blocks, then arrange them in their order from left to right, followed by $n+1$, followed by the rest of the blocks in their original order. Thus, all "special" blocks go on the left of $n+1$, and all "nonspecial" blocks go on the right of $n+1$.

The inverse of this map is as follows: mark the blocks before $n+1$ as "special", partition the string $\pi'$ to the right of $n+1$ into maximal blocks starting with the left-to-right minima of $\pi'$, then remove $n+1$ and arrange all the blocks (to the left and to the right of $n+1$) in the increasing order of their left-to-right minima.

A slightly less well known combinatorial interpretation of the unsigned Stirling number of the first kind is the number of permutations of $\{1,\dots,n\}$ with $k$ left-to-right maxima.

A left-to-right maximum of a permutation $\pi$ of $\{1,\dots,n\}$ is a position $i$ such that $\pi(j)<\pi(i)$ for all $j<i$. In particular, $n$ is always the value of the rightmost left-to-right maximum, and $1$ is always the position of the leftmost left-to-right maximum.

Identity (1) says that the number of permutations of $\{1,\dots,n\}$ with at least $m$ left-to-right maxima, $m$ of which are chosen as "special", is equal to the number of permutations of $\{1,\dots,n+1\}$ with exactly $m+1$ left-to-right maxima.

Here is a bijection between the two classes of permutations.

Let $\pi$ be a permutation of $\{1,\dots,n\}$, and suppose its $k\ge m$ left-to-right maxima are in positions $1=i_1<\dots<i_k\le n$. Note that $\pi(i_1)<\dots<\pi(i_k)=n$. Set $i_{k+1}=n+1$, and define the subblocks $$ \pi_{j}=\pi(i_j)\dots\pi(i_{j+1}-1), \quad i=1,\dots,k. $$ Then $\pi$ is the concatenation of subblocks $\pi_j$, $j=1,\dots,k$, i.e. $\pi=\pi_1\dots\pi_k$.

Choose $m$ left-to-right-maxima $i_{j_1}<\dots<i_{j_m}$ and declare them "special". Let $\pi'$ be the string $\pi$ with subblocks $\pi_{j_1},\dots,\pi_{j_m}$ deleted. Then our map $f$ is $$ \pi=\pi_1\dots\pi_k\mapsto f(\pi)=\pi_{j_1}\dots\pi_{j_m},n+1,\pi'. $$ In other words, we partition $\pi$ into maximal blocks starting on the left with each successive left-to-right minimum, choose $m$ of those blocks, then arrange them in their order from left to right, followed by $n+1$, followed by the rest of the blocks in their original order. Thus, all "special" blocks go on the left of $n+1$, and all "nonspecial" blocks go on the right of $n+1$.

The inverse of this map is as follows: mark the blocks before $n+1$ as "special", partition the string $\pi'$ to the right of $n+1$ into maximal blocks starting with the left-to-right minima of $\pi'$, then remove $n+1$ and arrange all the blocks (to the left and to the right of $n+1$) in the increasing order of their left-to-right minima.

For example, $\color{red}{21}\|43\|\color{red}{65}\|87\longleftrightarrow\color{red}{21}\|\color{red}{65}\|\mathbf{9}|43|87$. Here $n=8$, $m=2$, $k=4$.