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In this paper, we present a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements. Our method allows us not only to generalize previous results on refined enumeration of general interval orders, but also to enumerate self-dual interval orders with respect to analogous statistics. Using the newly derived generating function formulas, we are able to prove a bijective relationshipdoi:10.1016/j.jcta.2011.11.010 fatcat:mctpy3r3erhcrfrkjtrmjolfvy