  Following a result of Eriksen and Sj\"{o}strand (2014) we detail a technique
to construct structures following the Fishburn distribution from appropriate
Mahonian structures.
  This technique is introduced on a bivincular pattern of Bousquet-M\'elou et
al. (2010) and then used to introduce a previously unconsidered class of
matchings; explicitly, zero alignment matchings according to the number of arcs
which 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 and
demonstrate that according to the number of mislabelings that factorial posets
follow the Fishburn distribution.
  As a consequence of our approach we find an identity for the Fishburn numbers
in terms of the Mahonian numbers.
