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Following a result of Eriksen and Sj\"{o}strand (2014) we detail a techniqueto construct structures following the Fishburn distribution from appropriateMahonian structures.  This technique is introduced on a bivincular pattern of Bousquet-M\'elou etal. (2010) and then used to introduce a previously unconsidered class ofmatchings; explicitly, zero alignment matchings according to the number of arcswhich are both right-crossed and left nesting.  We then define a statistic on the factorial posets of Claesson and Linusson(2011) counting the number of features which we refer to as mislabelings anddemonstrate that according to the number of mislabelings that factorial posetsfollow the Fishburn distribution.  As a consequence of our approach we find an identity for the Fishburn numbersin terms of the Mahonian numbers.

Sieved Enumeration of Interval Orders and Other Fishburn Structures