Symmetric colorings of Hales-Jewett cubes coincide with one-weight colorings, reducing the symmetric lower-bound problem to 1D Gallai homothety coloring and yielding HJ(3,3)≥22 and HJ(4,2)≥14.
A dual of Dilworth’s decomposition theorem
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Disappear-Sort on random sequences has expected passes equal to the expected first-column length of a Plancherel Young diagram, hence asymptotically 2√n with Tracy-Widom fluctuations.
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One-Weight Colorings, the Symmetric Class, and Lower Bounds for Hales--Jewett Numbers
Symmetric colorings of Hales-Jewett cubes coincide with one-weight colorings, reducing the symmetric lower-bound problem to 1D Gallai homothety coloring and yielding HJ(3,3)≥22 and HJ(4,2)≥14.
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Recursive Record Filtering and Longest Decreasing Subsequences
Disappear-Sort on random sequences has expected passes equal to the expected first-column length of a Plancherel Young diagram, hence asymptotically 2√n with Tracy-Widom fluctuations.