Geodesic
Slicings of parallelogram polyominoes: Catalan, Schröder, Baxter, and other sequences
Nicholas R. Beaton1 ; Mathilde Bouvel2 ; Veronica Guerrini3 ; Simone Rinaldi4
1University of Melbourne
2University of Zurich
3University of Siena
4University of Sienna
The electronic journal of combinatorics, Tome 26 (2019) no. 3
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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 $m$.
DOI : 10.37236/7375
Classification : 05A05, 05A15
Mots-clés : generating tree, Schröder numbers, Baxter classes

Nicholas R. Beaton  1   ; Mathilde Bouvel  2   ; Veronica Guerrini  3   ; Simone Rinaldi  4

1 University of Melbourne
2 University of Zurich
3 University of Siena
4 University of Sienna 
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     author = {Nicholas R. Beaton and Mathilde Bouvel and Veronica Guerrini and Simone Rinaldi},
     title = {Slicings of parallelogram polyominoes: {Catalan,} {Schr\"oder,} {Baxter,} and other sequences},
     journal = {The electronic journal of combinatorics},
     year = {2019},
     volume = {26},
     number = {3},
     doi = {10.37236/7375},
     zbl = {1423.05004},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/7375/}
}
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Nicholas R. Beaton; Mathilde Bouvel; Veronica Guerrini; Simone Rinaldi. Slicings of parallelogram polyominoes: Catalan, Schröder, Baxter, and other sequences. The electronic journal of combinatorics, Tome 26 (2019) no. 3. doi: 10.37236/7375

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