The generating polynomial of all n-permutations with respect to the number of alternating runs possesses a root at \(-1\) of multiplicity \(\lfloor (n-2)/2 \rfloor \) for \(n \ge 2\) . This fact can be deduced by combining the David–Barton formula for Eulerian polynomials with the Foata–Schützenberger \(\gamma \) -decomposition of these polynomials. Recently, Bóna provided a group—action proof of this result. In the present paper, we propose an alternative approach based on the Hetyei–Reiner action on binary trees, which yields a new combinatorial interpretation of Bóna’s quotient polynomial. Furthermore, we extend our study to analogous results for permutations of types B and D. As a consequence of our bijective framework, we also obtain combinatorial proofs of David–Barton type identities for permutations of types A and B.
