A **parallelogram polyomino**
is an array of unit squares bounded by two lattice paths that intersect
only initially and terminally. (Paths have
positively
directed vertical and horizontal steps.)
A row of a polyomino is **wide** when it
contains more than one square.

Let
denote the set of parallelogram polyominoes having
perimeter 2*n*+2 and *r* rows.
Let
denote the subset of those in
having *i* left notches and *j* right notches.
Let
denote the set of parallelogram
polyominoes
having
perimeter 2*n*+2 and *w* wide rows.
Let
denote the subset of those
in
having *i* left notches and *j* right notches.

In this

These results were motivated by a recent one of
Emeric Deutsch that
the number of Catalan paths (Dyck words) of length 2*n*
with *k* high peaks (i.e., peaks which exceed the constraint by
more than one step)
equals the
number of Catalan paths of length 2*n* with *k*+1 peaks. It is well known that the latter is counted by the
Narayana number,

Since the right-hand sides of (1) and (2) have been counted previously [1], we have that is a Narayana number and has the symmetric Kreweras-Poupard distribution, namely,