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

The word problem for Hecke--Kiselman monoids of type $A_n$ and $\widetilde{A}_n$