On the Relationship between Pipe Dreams and Permutation Words

Pipe dreams represent permutations pictorially as a series of crossing pipes. Recent applications of pipe dreams include the calculation of Schubert polynomials, fillings of moon polyominoes, and in the combinatorics of antidiagonal simplicial complexes. These applications associate pipe dreams to words of elementary symmetric transpositions via a canonical mapping. However, this canonical mapping is by no means the only way of mapping pipe dreams to permutation words. We define sensical mappings from pipe dreams to words and prove sensical mappings are in bijection with standard shifted tableaux of triangular shape. We characterize the set of pipe dreams associated to a given word (under any sensical map) using step ladder moves. These moves induce a partial order on the set of pipe dreams mapping to a given word, yielding a distributive lattice.