Counting parallelogram polyominoes with respect to wide rows 1
R. A. Sulanke

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.

In this figure a polyomino in is mapped to one in . The exchange of the initial and final sequences of squares serves to preserve the number of notches on each side when one of these sequences is empty. The figure indicates a bijection proving

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 [1], we have that is a Narayana number and has the symmetric Kreweras-Poupard distribution, namely,