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 $n$. Since the first map sorts the cycles by their minimal elements, its image starts with $1$, which implies that the first ascent of the Dyck path has length one. Thus, the constant terms of the generating polynomials vanish. The linear terms vanish 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