Abstract: By  using combinatorial bijective method, the author  studied the compositions of the mutual conjugate partition of positive integers without  2’s, and obtained a  relation between the number of these compositions and the Fibonacci number. Consequently, the author gave several identities about the number of these compositions, the number of compositions into parts equal to 1 or 2, the number of compositions into odd parts and the number of compositions into parts greater than 1.

Key words: compositions; , Fibonacci number, identity, combinatorial bijective