  Inspired by Yakoubov's 2015 investigation of pattern avoiding linear
extensions of the posets called combs, we study pattern avoiding linear
extensions of rectangular posets. These linear extensions are closely related
to standard tableaux. For positive integers $s$ and $t$ we consider two natural
rectangular partial orders on $\{1,2,\ldots,s t\}$, which we call the NE
rectangular order and the EN rectangular order. First we enumerate linear
extensions of both rectangular orders avoiding most sets of patterns of length
three. Then we use both a generating tree and a bijection to show that the
linear extensions of the EN rectangular order which avoid 1243 are counted by
the Fuss-Catalan numbers. Next we use the transfer matrix method to enumerate
linear extensions of the EN rectangular order which avoid 2143. Finally, we
open an investigation of the distribution of the inversion number on pattern
avoiding linear extensions.
