Sieved Enumeration of Interval Orders and Other Fishburn Structures

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.