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We show that very simple continued fractions can be obtained for the ordinarygenerating functions enumerating permutations or D-permutations with a largenumber of independent statistics, when each cycle is given a weight $-1$. Theproof is based on a simple lemma relating the number of cycles modulo 2 to thenumbers of fixed points, cycle peaks (or cycle valleys), and crossings.

A remark on continued fractions for permutations and D-permutations with a weight $-1$ per cycle