Partially restricted stacks as functions

We further generalize Berlow's stack sorting map s T to s (T,k) , where instead of avoiding all permutations in T , the stack contains at most k permutations from a restricted set T .We introduce the (T, k)-machine as well, defined as the composition of West's stack sorting map s and s (T,k) , as a parallel to Cerbai, Claesson, and Ferrari's -machine.We show that for every s (T,k) , there exists an equivalent s T and give an explicit construction of T .We then characterize the permutations in the preimage of id n in s ({12,21},1) and prove that the size of the preimage is the (n -1) th Catalan number.We also demonstrate that the s ({12,21},1) map and the ({12, 21}, 1)-machine sort all permutations after a finite number of repeated applications.