A lift of West's stack-sorting map to partition diagrams (Q6165151)

In this paper, the author introduces a lifting of West's stack-sorting map \(s\) to partition diagrams, which are combinatorial objects indexing bases of partition algebras. The lifting \(\mathcal{S}\) of \(s\) is such that \(\mathcal{S}\) behaves in the same way as \(s\) when restricted to diagram basis elements in the order-\(n\) symmetric group algebra as a diagram subalgebra of the partition algebra \(\mathcal{P}^{\xi}_{n}\). A lifting of the notion of 1-stack-sortability, using the lifting of \(s\) is introduced. By direct analogy with \textit{D. E. Knuth}'s famous result that a permutation is 1-stack-sortable if and only if it avoids the pattern 231 [The art of computer programming. Vol. 1: Fundamental algorithms. London: Addison-Wesley Publishing Company (1968; Zbl 0191.17903)], a related pattern-avoidance property for partition diagrams, as opposed to permutations, is proved according to what the author refers to as stretch-stack-sortability. An open problem concludes the paper.