You can use https://www.findstat.org to obtain a candidate for a conjectural solution, see https://www.findstat.org/StatisticsDatabase/St000389oMp00093oMp00127oMp00066oMp00090 for details:
after adding 1 to every value
and applying
Mp00090: cycle-as-one-line notation: Permutations -> Permutations
Mp00066: inverse: Permutations -> Permutations
Mp00127: left-to-right-maxima to Dyck path: Permutations -> Dyck paths
Mp00093: to binary word: Dyck paths -> Binary words
to the objects (see `.compound_map()` for details)
your input matches
St000389: The number of runs of ones of odd length in a binary word.
I did check the first few generating polynomials. From the definition of the map, it is easy to see that the leading term is $2^{n-1}$, the number of compositions of $n$. It is also easy to see that the linear term vanishes for even $n$, because there is no Dyck path of semilength $n$ with a single odd ascent.
1
q
2*q^2
4*q^3 + 2*q
8*q^4 + 16*q^2
16*q^5 + 80*q^3 + 24*q
32*q^6 + 320*q^4 + 368*q^2
64*q^7 + 1120*q^5 + 3136*q^3 + 720*q
128*q^8 + 3584*q^6 + 19712*q^4 + 16896*q^2
256*q^9 + 10752*q^7 + 102144*q^5 + 209408*q^3 + 40320*q
512*q^10 + 30720*q^8 + 462336*q^6 + 1838080*q^4 + 1297152*q^2
1024*q^11 + 84480*q^9 + 1892352*q^7 + 12869120*q^5 + 21441024*q^3 + 3628800*q