Large Hypercube Intervals in Bruhat Order
Prior state unknown→proved
Asymptotically optimal for powers of 2; the exact extremal answer for general n stays open.
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combinatorics / Coxeter combinatorics
Large Hypercube Intervals in Bruhat Order
How large can a Bruhat interval in $S_n$ that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension $O(n \log n)$ for $n$ a power of 2, matching the largest possible dimension up to a constant - in the problem circle around the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.
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How large can a Bruhat interval in $S_n$ that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension $O(n \log n)$ for $n$ a power of 2, matching the largest possible dimension up to a constant - in the problem circle around the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.
Asymptotically optimal for powers of 2; the exact extremal answer for general n stays open.
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