Gandhi \(q\)-polynomials and Denert statistic
The authors use Hahn's \(q\)-difference operator to derive \(q\)-analogues of the Gandhi polynomials as \(y\) specializations of certain polynomials denoted \(D_n(x,y,q)\). They give a continued fraction expansion for the generating function of the polynomials \(D_n(x,y,q)\), and, inspired by the work of Dumont in this area, they also construct two different combinatorial interpretations of \(D_n(x,y,q)\) as a trivariate generating function on \(G_n\), the set of Genocchi permutations. The first interpretation is based on an intermediate interpretation using the set of Dyck paths with \(2n+ 1\) steps. The second is based on a statistic-preserving bijection between \(G_n\) and the set \(P_n\) of ``pistolets'', maps \(f\) from \([2n]\) onto \(2[n]\) with the property that \(i\leq f(i)\) for all \(i\). In both cases a key ingredient is the Denert statistic (as described by D. Foata and D. Zeilberger). The construction of an involution on \(G_n\) which directly relates the two trivariate statistics remains open.
scientific article; zbMATH DE number 1334682