Enumeration of polyominoes determined by Catalan words avoiding the consecutive pattern 012

Abstract

We study Catalan words avoiding the consecutive pattern 0 1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , that is, Catalan words having no two consecutive strict rises, through their associated bottom-aligned polyominoes. This class is known to be enumerated by Motzkin numbers; however, the main contribution of the paper is the refined enumerative study of the associated polyomino statistics, which are not transported directly by the known Motzkin bijection and differ from those arising in other Motzkin classes, in particular the recently studied ( ≥ , ≥ ) case. We derive functional equations for multivariate generating functions that record length, semiperimeter, area, final symbol, and number of interior points. For the area and interior-point statistics, first-return decompositions lead to 𝑞 -continued fractions. We also obtain refined recurrences according to the height of the last column and closed formulas, in terms of central trinomial coefficients, for the total semiperimeter, total area, total sum of final symbols, and total number of interior points over all objects of a fixed length. Finally, we record two consequences tied to these statistics, namely a direct relation between semiperimeter, area, and interior points, and a Fibonacci specialization for polyominoes with no interior point.

The authors' abstract, as published at the source. Discrete Applied Mathematics, 2026 · DOI ↗