Let $S_n$ be the set of permutations on $\{1,2,\dotsc,n\}$. A peak-value of $\pi$ is some $\pi_i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$, where $1<i<n$. Let $PV(\pi)$ denote the set of peak-values.
A run of $\pi \in S_n$ is a maximal subsequence of elements in increasing order. Given $\pi$, we can list the runs increasingly based the first entries. Let us call this operation runsort. A run of $\pi \in S_n$ is a maximal subsequence of elements in increasing order. For example, $$ \mathrm{runsort}(458 \, 23 \, 6 \, 17) = 17\,23\,458\,6. $$
O. Nabawanda and I have with some effort (arxiv preprint) been able to show that $$ \sum_{ \pi \in S_n} t^{|PV(\pi)|} = \sum_{ \pi \in S_n} t^{|PV(\mathrm{runsort}(\pi))|}. $$ We actually show that a multivariate identity (which not only keeps track of the number of peaks, but the actual set of peak-values) holds. We construct a bijection through a recursive process, and the bijection is not canonical.
I wonder if perhaps there is some easy bijective argument which proves the univariate identity above, where the bijection is natural or canonical in some sense.
