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 2n+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 2n+2 and w wide rows. Let denote the subset of those in having i left notches and j right notches.
These results were motivated by a recent one of Emeric Deutsch that the number of Catalan paths (Dyck words) of length 2n with k high peaks (i.e., peaks which exceed the constraint by more than one step) equals the number of Catalan paths of length 2n 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 , we have that is a Narayana number and has the symmetric Kreweras-Poupard distribution, namely,