We prove that there is a constant C ≤ 6.614 such that every Boolean function of degree at most d (as a polynomial over R) is a C ·2 d -junta, i.e. it depends on at most C ·2 d variables. This improves the d · 2 d−1 upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)].The bound of C · 2 d is tight up to the constant C as a lower bound of 2 d − 1 is achieved by a read-once decision tree of depth d. We slightly improve the lower bound by constructing, for each positive integer d, a function of degree d with 3 · 2 d−1 − 2 relevant variables. A similar construction was independently observed by Shinkar and Tal.
This paper presents a collection of experimental results regarding permutation pattern avoidance, focusing on cases where there are "many" patterns to be avoided.
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