Abstract
We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.
Highlights
The sequence of Catalan numbers (a000108 in [20]) is arguably the most well-known combinatorial sequence
The set of triples of non-intersecting lattice paths (NILPs) contains all pairs of NILPs (which are in essence parallelogram polyominoes, see Figure 1(a) and the blue and red paths of Figure 1(c)); and Baxter permutations, defined by the avoidance of the vincular1 patterns 2 41 3 and 3 14 2, include τ -avoiding permutations, for any τ ∈ {132, 213, 231, 312}
As we demonstrate in this work, for the known generating trees associated with the Schroder and Baxter numbers, when they can be seen as generalizations of the generating tree of Catalan numbers, these two generalizations go in two opposite directions
Summary
The sequence of Catalan numbers (a000108 in [20]) is arguably the most well-known combinatorial sequence. The Baxter and Schroder generalizations of Catalan objects are often independent and are not reconciled This includes triples of NILPs, permutations and mosaic floorplans. This results in two families of subclasses of Baxter slicings: the m-skinny slicings and the m-row-restricted slicings. In view of our method, we offer the conjecture that the generating functions for m-skinny slicings and m-row-restricted slicings are algebraic, for all m
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