Vasilii Duzhin ; Egor Smirnov-Maltsev - On Young diagrams of maximum dimension

cm:12641 - Communications in Mathematics, December 22, 2023, Volume 31 (2023), Issue 3 (Special issue: in memory of Sergei Duzhin) - https://doi.org/10.46298/cm.12641
On Young diagrams of maximum dimensionArticle

Authors: V. S. Duzhin ; E. D. Smirnov-Maltsev

    We study the problem of finding Young diagrams of maximum dimension, i. e. those with the largest number of Young tableaux of their shapes. Consider a class of Young diagrams that differ from a symmetric diagram by no more than one box $(i,j)$ in each row and column. It is proven that when moving boxes $(i,j), i>j$ to symmetric positions $(j,i)$, the original diagram is transformed into another diagram of the same size, but with a greater or equal dimension. A conjecture is formulated that generalizes the above fact to the case of arbitrary Young diagrams. Based on this conjecture, we developed an algorithm applied to obtain new Young diagrams of sizes up to 42 thousand boxes with large and maximum dimensions.


    Volume: Volume 31 (2023), Issue 3 (Special issue: in memory of Sergei Duzhin)
    Published on: December 22, 2023
    Accepted on: December 5, 2023
    Submitted on: December 4, 2023
    Keywords: Mathematics - Combinatorics

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