  We exhibit explicit and easily realisable bijections between Hecke--Kiselman
monoids of type $A_n$/$\widetilde{A}_n$; certain braid diagrams on the
plane/cylinder; and couples of integer sequences of particular types. This
yields a fast solution of the word problem and an efficient normal form for
these HK monoids. Yang--Baxter type actions play an important role in our
constructions.
