Constrained exchangeable partitions

Under a partition of the set N we shall mean a sequence (b1, b2, . . .) of subsets of N such that (i) the sets bj are disjoint, (ii) ∪jbj = N, (iii) if bk = ∅ then also bk+1 = ∅ and (iv) if bk+1 6= ∅ then min bk < min bk+1. Condition (iv) says that the sequence of minimal elements of the blocks is increasing. One can think of partition as a mapping which sends a generic element j ∈ N to one of the infinitely many blocks, in such a way that conditions (iii) and (iv) are fulfilled. A random partition Π = (Bk) of N (so, with random blocks Bk) is a random variable with values in the set of partitions of N. This concept can be made precise by means of a projective limit construction and the measure extension theorem. To this end, one identifies Π with consistent partitions Πn := Π|[n] (n = 1, 2, . . .) of finite sets [n] := {1, . . . , n}. Note that the restriction Πn, which is obtained by removing all elements not in [n], still has the blocks in the order of increase of their least elements. There is a well developed theory of exchangeable partitions [1; 13; 17]. Recall that Π = (Bj) is exchangeable if the law of Π is invariant under all bijections σ : N → N. Partitions with weaker symmetry properties have also been studied. Pitman [16] introduced partially exchangeable random partitions of N with the property that the law of Π is invariant under all bijections σ : N → N that preserve the order of blocks, meaning that the sequence of the least elements of the sets σ(B1), σ(B2), . . . is also increasing. Pitman [16] derived a de Finetti-type representation for partially exchangeable partitions and established a criterion for their exchangeability. Kerov [14] studied a closely related structure of virtual permutations of N, which may be seen as partially exchangeable partitions with some total ordering of elements within each of the blocks. Kallenberg [13] characterised spreadable partitions whose law is invariant under increasing injections σ : N → N. In this note we consider constrained random partitions of N which satisfy the condition that, for a fixed integer sequence ρ = (ρ1, ρ2, . . .) with ρk ≥ 1, each block Bk contains ρk least elements of ∪j≥kBj , for every k with Bk 6= ∅. It is easy to check that this condition holds if and anly if the sequence comprised of ρ1 least elements of B1, followed by ρ2 least elements of B2 and so on, is itself an increasing sequence. We shall focus on the constrained partitions with the following symmetry property. Definition 1 For a given sequence ρ, we call Π constrained exchangeable if Π is a constrained partition with respect to ρ and the law of Π is invariant under all bijections σ : N → N that preserve this property.