| The generating function of weighted Lukasiewicz paths, which generalize Dyck paths by allowing the steps up to be arbitrarily large, is a generalized continued fraction. We use Lukasiewicz paths to encode labeled combinatorial objects---permutations, partitions, idempotent functions, increasing trees, and multipermutations---and express their ordinary generating functions as generalized continued fractions.;A restriction on the weights of the down steps allows for interpretation of the Lukasiewicz paths as differential operators. We use the encodings to count the objects using differential operators. We use the encoding of permutations to give a combinatorial proof to the identity expyd dx+fx =exp 0yft+x dtexpyd dx.;Finally, we consider the normal ordering problem for elements of a Weyl algebra, such as the algebra of differential operators generated by ddx and multiplication by x. We show that the normal order coefficients of a word are rook numbers on a Ferrers board. We use this result to give a new proof of the rook factorization theorem, give an explicit formula for the normal order coefficients of a word, and calculate the Weyl binomial coefficients. We extend these results to the q-analogue of the Weyl algebra. |
