  Data vectors generalise finite multisets: they are finitely supported
functions into a commutative monoid. We study the question if a given data
vector can be expressed as a finite sum of others, only assuming that 1) the
domain is countable and 2) the given set of base vectors is finite up to
permutations of the domain.
  Based on a succinct representation of the involved permutations as integer
linear constraints, we derive that positive instances can be witnessed in a
bounded subset of the domain.
  For data vectors over a group we moreover study when a data vector is
reversible, that is, if its inverse is expressible using only nonnegative
coefficients. We show that if all base vectors are reversible then the
expressibility problem reduces to checking membership in finitely generated
subgroups. Moreover, checking reversibility also reduces to such membership
tests.
  These questions naturally appear in the analysis of counter machines extended
with unordered data: namely, for data vectors over $(\mathbb{Z}^d,+)$
expressibility directly corresponds to checking state equations for Coloured
Petri nets where tokens can only be tested for equality. We derive that in this
case, expressibility is in NP, and in P for reversible instances. These upper
bounds are tight: they match the lower bounds for standard integer vectors
(over singleton domains).
