Packing densities of patterns.
Packing patterns into words.
Partition identities. I: Sandwich theorems and logical 0-1 laws.
Partitions with parts in a finite set and with parts outside a finite set.
Pattern avoidance classes and subpermutations.
Permutations with inversions.
Probability that an element of a finite group has a square root
Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - ⌊√|G|⌋/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.
Profiles of permutations.