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On Young diagrams of maximum dimension

Vasilii Duzhin, Egor Smirnov-Maltsev

Abstract

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.

On Young diagrams of maximum dimension

Abstract

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 in each row and column. It is proven that when moving boxes to symmetric positions , 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.
Paper Structure (7 sections, 15 equations, 7 figures)

This paper contains 7 sections, 15 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Dimensions of asymmetric Young diagrams
  4. Theorem.
  5. Algorithm for searching Young diagrams with large and maximum dimensions
  6. Computer experiments
  7. Conclusion

Figures (7)

  • Figure 1: Transformation of an asymmetric Young diagram to a Young diagram with a larger dimension
  • Figure 2: Different types of boxes of Young diagrams for the case $k<s$
  • Figure 3: Different types of boxes of Young diagrams for the case $k>s$
  • Figure 4: Transformation of an asymmetric Young diagram to a Young diagram with a larger dimension
  • Figure 5: Different types of boxes of Young diagrams
  • ...and 2 more figures

Theorems & Definitions (1)

  • proof