Bijections between pattern-avoiding fillings of Young diagrams

TL;DR

This paper establishes bijections between different classes of pattern-avoiding fillings of Young diagrams, connecting various combinatorial objects and extending previous recurrence-based results to more general polyominoes.

Contribution

It provides explicit bijections demonstrating the equivalence in count between different pattern-avoiding fillings, extending known results to broader classes of diagrams and patterns.

Findings

Bijections between pattern-avoiding fillings of Young diagrams

Equivalence in enumeration of different pattern-avoiding classes

Extension of results to general polyominoes

Abstract

The pattern-avoiding fillings of Young diagrams we study arose from Postnikov's work on positive Grassman cells. They are called Le-diagrams, and are in bijection with decorated permutations. Other closely-related diagrams are interpreted as acyclic orientations of some bipartite graphs. The definition of the diagrams is the same but the avoided patterns are different. We give here bijections proving that the number of pattern-avoiding filling of a Young diagram is the same, for these two different sets of patterns. The result was obtained by Postnikov via a reccurence relation. This relation was extended by Spiridonov to obtain more general results about other patterns and other polyominoes than Young diagrams, and we show that our bijections also extend to more general polyominoes.