Linear Extension Numbers of n-Element Posets
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Abstract
We address the following natural but hitherto unstudied question: what are the possible linear extension numbers of an n-element poset? Let LE(n) denote the set of all positive integers that arise as the number of linear extensions of some n-element poset. We show that LE(n) skews towards the “small” end of the interval [1, n!]. More specifically, LE(n) contains all of the positive integers up to
exp
c
n
log
n
$\exp \left (c\frac {n}{\log n}\right )$
for some absolute constant c, and |LE(n) ∩ ((n − 1)!, n!]| < (n − 3)!. The proof of the former statement involves some intermediate number-theoretic results about the Stern-Brocot tree that are of independent interest.
Date issued
2020-05-11Department
Massachusetts Institute of Technology. Department of MathematicsPublisher
Springer Netherlands