  Standard set-valued Young tableaux are a generalization of standard Young
tableaux where cells can contain unordered sets of integers, with the added
condition that every integer at position $(i,j)$ must be smaller that every
integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore properties
of standard set-valued Young tableaux with three rows and a fixed number of
integers in every cell of each row (referred to as set-valued tableaux with
row-constant density). Our primary focus is on standard set-valued Young
tableaux with $1$ integer in each first-row cell, $k-1$ integers in each
second-row cell, and $1$ integer in each third-row cell. For rectangular shapes
$\lambda=n^3$, such tableaux are placed in bijection with closed $k$-ary
product-coproduct prographs: directed plane graphs that correspond to finite
compositions involving a $k$-ary product operator and a $k$-ary coproduct
operator. That bijection is extended to three-row set-valued Young tableaux of
non-rectangular and skew shape, and it is shown that a set-valued analogue of
the Sch\"utzenberger involution on tableaux corresponds to $180$-degree
rotation of the associated prographs. As a set-valued analogue of the
hook-length formula is currently lacking, we also present direct enumerations
of three-row standard set-valued Young tableaux for a variety of row-constant
densities and a small number of columns. We then argue why the numbers of
tableaux with the row-constant density $(1,k-1,1)$ should be interpreted as a
one-parameter generalization of the three-dimensional Catalan numbers that
mirrors the generalization of the (two-dimensional) Catalan numbers provided by
the $k$-Catalan numbers.
