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. We also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, and a new Schröder subset of Baxter permutations.
Highlights
The sequence of Catalan numbers (A000108 in [14]) is arguably the most well-known combinatorial sequence
To give only a few examples, consider for instance lattice paths: the Dyck paths generalize into Schroder paths, but have to our knowledge no natural Baxter analogue; on the contrary, pairs of NILPs are counted by Catalan, whereas triples of NILPs are counted by Baxter, leaving Schroder aside
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 [14]) is arguably the most well-known combinatorial sequence. To give only a few examples, consider for instance lattice paths: the Dyck paths generalize into Schroder paths (by allowing an additional flat step of length 2), but have to our knowledge no natural Baxter analogue; on the contrary, pairs of NILPs are counted by Catalan, whereas triples of NILPs are counted by Baxter, leaving Schroder aside Consider another well-known Catalan family: that of binary trees. The Baxter and Schroder generalizations of Catalan objects are often independent and are not reconciled
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