Record-dependent measures on the symmetric groups

A probability measure Pn on the symmetric group Sn is said to be record-dependent if Pn(σ) depends only on the set of records of a permutation σ ∈ Sn. A sequence P = (Pn)n∈N of consistent recorddependent measures determines a random order on N. In this paper we describe the extreme elements of the convex set of such P . This problem turns out to be related to the study of asymptotic behavior of permutation-valued growth processes, to random extensions of partial orders, and to the measures on the Young-Fibonacci lattice.