Combinatorics of diagonally convex directed polyominoes

A new bijection between the diagonally convex directed (dai-) polyominoes and ternary trees makes it possible to enumerate the dcd-polyominoes according to several parameters (sources, diagonals, horizontal and vertical edges, target cells). For a part of these results we also give another proof, which is based on Raney’s generalized lemma. Thanks to the fact that the diagonals of a dcd-polyomiuo can grow at most by one, the problem of 4~n~eration of this object can be solved by an application of Gessel’s q-analog of the Lagrange inversion formula. Une nouvelle bijection entre les polyominos dirigbs diagonalement convexes (polyominos d.d.c.) et les arbres ternaires permet l%num&ration des polyominos d.d.c. suivant plusieur paramhtres (sources, diagonales, a&es horizontales et verticales, cellules cibles). Pour une partie de ces r&Mats nous douuons une preuve supp~~rnen~~, qui est bash sur le lemme g&nkralis&e & Raney. G&e au fait que les diagonales d’un polyomino d.d.c. croissent au plus d’une unit& leur qdnum~ration peut &re r&solue en utilisant le q-analogue de la formule d’inversion de Lagrange dO B Gessel. 1. Definitions, conventions and notations 1.1. Binomiai coefficients Generally, we adopt the convention: if a binomial coefficient has a negative numerator or denominator, then the value of the coefficient is zero. Exceptionally, for those binomial coefficients which are indicated by an arrow Y we stipulate: (I:), = 1. *Corresponding author. ~12-36SX/96/~15.~ @I 1996 F!lsevier Science B.V. All rights reserved SSDJ 0012-365X(95)00260-X 148 S. FeretiC. D. ~~rta~~~iscreie ~~tke~tics 157 (1994) 147-168 The Gaussian polynomials are defined by k [I = (1 #)(l qk-‘)...(l qk-‘+I) r (1 -q)(l -$)“‘(l --Cjr) * If k -C 0 or r < 0, we agree that [i] = 0. Again, the only exception is [I i] I = 1.