The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, states that the growth rate of every proper permutation class is singly exponential.
As Arratia (1999) observed, this is equivalent to the convergence of the limit The upper bound given by Marcus and Tardos for C is exponential in the length of β.
Kaiser and Klazar went on to establish every possible growth constant of a permutation class below 2; these are the largest real roots of the polynomials for an integer k ≥ 2.
Their results also imply that in the set of all growth rates of permutation classes, ξ is the least accumulation point from above.
That result was later improved by Bevan (2018), who proved that every real number at least 2.36 is the growth rate of a permutation class.