Search for Young diagrams of maximum dimension

In this study, we investigate the problem of nding Young diagrams with maximum dimensions, i. e. containing the greatest number of Young tableaux. We propose an algorithm that transforms the initial diagram λ into another diagram λt of the same size. We prove that if diagram λ di ers from the symmetrical one by no more than one box in each row and each column, then the dimension of λt is greater than or equal to the dimension of λ. We propose a hypothesis that extends this fact to the case of arbitrary Young diagrams. We then reformulate the problem under study as nding the shortest path in the Young graph, focusing exclusively on the subgraph containing diagrams that satisfy the aforementioned hypothesis. We have developed a program to nd Young diagrams of large dimensions, based on the A∗ algorithm. This program has enabled us to identify new, previously unknown Young diagrams with large and maximum dimensions.