Baseball Experiment: Good Or Bad

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The bijection between Binary trees and Young diagrams using Dyck paths are well known. We studied bijections between various well-known sets having a Catalan number of objects, such as binary trees, Young diagrams, torsion classes and 213-avoiding permutations. In this chapter a new function between Young diagram and binary trees is defined, which we call ’the bookshelf construction’. To get the final Young diagram described by a binary tree, push the boxes row-by-row from right to left such that the left-most box at each row is aligned, thus yielding a Young diagram. Notice that prior to tilting the tree, any shelf is a branch of the binary tree with negative slope, excluding the right most one which is the ceiling. This is because we have defined the corresponding binary tree to have size such that putting the top-left corner of the diagram fits the appropriate fixed tree exactly. In this section, it will be proven by induction that the following diagram commutes, i.e the Young Diagram given using the bookshelf construction is equal to the one given by the well known mappings between binary trees and Dyck paths and Young diagrams. This process yields the final corresponding Young diagram with gaps

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