Ewens Sampling and Invariable Generation

Abstract

We study the number of random permutations needed to invariably generate the symmetric group, Sn, when the distribution of cycle counts has the strong α-logarithmic property. The canonical example is the Ewens sampling formula, for which the number of k-cycles relates to a conditioned Poisson random variable with mean α/k. The special case α = 1 corresponds to uniformly random permutations, for which it was recently shown that exactly four are needed.For strong α-logarithmic measures, and almost every α, we show that precisely (1 − α log 2) −1 permutations are needed to invariably generate Sn. A corollary is that for many other probability measures on Sn no bounded number of permutations will invariably generate Sn with positive probability. Along the way we generalize classic theorems of Erdős, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula. arXiv:1610.04212v2 [math.PR]

The invariably generating graph of the alternating and symmetric groups

“…We pause for a moment and we show that d(Ξ(S 12 )) = 5. Let z be a vertex of Λ(S 12 ); we prove that it is either isolated or connected to one among (1,11), (3,9), (5,7), (1,4,7), (2,3,7), (12). I claim that these vertices have pairwise distance at most 3, from which d(Ξ(S 12 )) 5.…”

“…(1 3 , 5 3 ) is contained in S 6 ≀S 3 : it is not connected to (9 2 ). Instead, by Theorem 2.3 it is connected to (4,14). On the other hand, (1,2,4,11) is connected to (8,10) and to (9 2 ); the diameter is 4 because (4,14) is connected to none of these vertices (S 2 ≀ S 9 ).…”

“…Instead, by Theorem 2.3 it is connected to (4,14). On the other hand, (1,2,4,11) is connected to (8,10) and to (9 2 ); the diameter is 4 because (4,14) is connected to none of these vertices (S 2 ≀ S 9 ). The reader will have noted that the proof of Lemma 5.8 works perfectly also for symmetric groups of even degree (values contemplated there).…”

“…Therefore, since z is odd, z is indeed connected to (3, n − 3). For n = 16, define z = (1 2 , 4 2 , 6); for n = 22, z = (1 2 , 5, 6, 9) (the latter is not connected to (4,18) because of S 11 ≀ S 2 ).…”

“…These problems have been considered first in [5]. Then, several results have been obtained, for example in [17], [20], [7], [8], [4]. In [20] it is shown that 4 random permutations invariably generate S n with probability bounded away from zero, whilst in [8] it is shown that, if we replace 4 with 3, the same probability tends to zero as n → ∞.…”