Restricted stacks as functions

The stack sort algorithm has been the subject of extensive study over years. In this paper we explore a generalized version where instead avoiding single decrease, avoids set T permutations. We let sT denote map. classify for which sets map is bijective. A corollary to answers question Baril, Cerbai, Khalil, and Vajnovszki about composed with s{σ,τ}, known as (σ,τ)-machine. This fully classifies σ τ preimage identity under (σ,τ)-machine counted by Catalan numbers. also prove that number preimages permutation bounded numbers, shift indices. For size 1, exactly when bound sharp. periodic points maximum various containing two length 3