  A poset is said to be (2+2)-free if it does not contain an induced subposet
that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two
elements in a poset are indistinguishable if they have the same strict up-set
and the same strict down-set. Being indistinguishable defines an equivalence
relation on the elements of the poset. We introduce the statistic maxindist,
the maximum size of a set of indistinguishable elements. We show that, under a
bijection of Bousquet-Melou et al., indistinguishable elements correspond to
letters that belong to the same run in the so-called ascent sequence
corresponding to the poset. We derive the generating function for the number of
(2+2)-free posets with respect to both maxindist and the number of different
strict down-sets of elements in the poset. Moreover, we show that (2+2)-free
posets P with maxindist(P) at most k are in bijection with upper triangular
matrices of nonnegative integers not exceeding k, where each row and each
column contains a nonzero entry. (Here we consider isomorphic posets to be
equal.) In particular, (2+2)-free posets P on n elements with maxindist(P)=1
correspond to upper triangular binary matrices where each row and column
contains a nonzero entry, and whose entries sum to n. We derive a generating
function counting such matrices, which confirms a conjecture of Jovovic, and we
refine the generating function to count upper triangular matrices consisting of
nonnegative integers not exceeding k and having a nonzero entry in each row and
column. That refined generating function also enumerates (2+2)-free posets
according to maxindist. Finally, we link our enumerative results to certain
restricted permutations and matrices.
