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For integers m, n $\ge$ 1, we describe a bijection sending dissections of the(mn + 2)-regular polygon into (m + 2)-sided polygons to a new basis of thequotient of the polynomial algebra in mn variables by an ideal generated bysome kind of higher quasi-symmetric functions. We show that divisibility of thebasis elements corresponds to a new partial order on dissections, which isstudied in some detail.

Poset structures on (m + 2)-angulations and polynomial bases of the quotient by G^m -quasisymmetric functions