Bijections between Pattern-avoiding Binary Fillings of Young Diagrams

Pattern-avoiding binary fillings of Young diagrams were first defined and studied by A. Postnikov. Important examples are Γ -diagrams, that are related to decorated permutations and positive Grassman cells. Other examples are acyclic orientations of a graph defined from the Young diagram. Using reccurence relations, he could prove that the numbers of such fillings are equal, for these two examples in any Young diagram. A. Spiridonov extended this recurrence relation and proved that many pattern pairs are equivalent, in the sense that for any Young diagram the numbers of the corresponding pattern-avoiding fillings are the same. We give here new bijective proofs of this fact for some pattern pairs, including the one first proved by Postnikov. Our bijections preserve the parameters ”number of zero columns” and ”number of unrestricted rows”.