Abstract
We introduce two definitions of Schröder numberq-analogs and show some combinatorial interpretations of theseq-numbers. We use the following combinatorial objects for these interpretations: Schröder paths, 1-colored parallelogram polyominoes and permutations with forbidden subsequences (4231, 4132). We enumerate these objects according to various parameters by means of a recentq-counting technique. We prove that the firstq-Schröder number enumerates of Schröder paths with respect to area and the number of permutation inversions, while the second one counts the 1-colored parallelogram polyominoes according to their width and area. Finally, we illustrate some relations among the parameters characterizing the combinatorial objects.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, A methodology for plane tree enumeration, Discrete Math.180 (1998) 45–64.
R.J. Baxter, Exactly, Solved Models in Statistical Mechanics, Academic Press, New York, 1982.
J. Bonin, L. Shapiro, and R. Simion, Someq-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths, J. Stat. Planning and Inference34 (1993) 35–55.
M. Bousquet-Mélou, A method for the enumeration of various classes of column-convex polygons, Discrete Math.154 (1996) 1–25.
M. Bousquet-Mélou, Percolation models and animals, Europ. J. Combin.17 (1996) 343–369.
R. Brak, J.W. Essam, and A. Owczarek, New results for directed vesicles and chains near an attractive wall, J. Stat. Phys.93 (1998) 155–192.
R. Brak, A. Owczarek, and T. Prellberg, Exact scaling behaviour of partially convex vesicles, J. Stat. Phys.76 (1994) 1101–1128.
A.R. Conway, R. Brak, and A.J. Guttmann, Directed animals on two-dimensional lattices, J. Phys. A26 (1993) 3085–3091.
M. Delest and J.M. Fédou, Enumeration of skew Ferrers diagrams, Discrete Math.112 (1993) 65–79.
M.E. Fisher, Walks, wall, wetting and melting, J. Stat. Phys.34 (1984) 667–729.
I.P. Goulden and D.M. Jackson, Combinatorial Enumeration, John Wiley & Sons, New York, 1983.
D. Gouyou-Beauchamps and B. Vauquelin, Deux propriétés combinatoires des nombres de Schröder, Theoretical Informatics and Applications22 (1988) 361–388.
O. Guibert, Combinatoire des permutations à motifs exclus en liaison avec mots, cartes planaires et tableaux de Young, Thèse de l'Université de Bordeaux I, 1996.
T. Prellberg and R. Brak, Critical exponents from non-linear functional equations for directed cluster models, J. Stat. Phys. (1994).
D.G. Rogers and L.W. Shapiro, Some correspondences involving the Schröder numbers and relations, In: Combinatorial Mathematics, D.A. Holton and J. Seberry, Eds., Lecture Notes in Mathematics, Vol. 686, Springer-Verlag, 1978, pp. 267–274.
J. West, Permutations with forbidden subsequences and stack-sortable permutations, Ph. D. Thesis, MIT, Cambridge, 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Barcucci, E., Del Lungo, A., Pergola, E. et al. Some combinatorial interpretations ofq-analogs of Schröder numbers. Annals of Combinatorics 3, 171–190 (1999). https://doi.org/10.1007/BF01608782
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01608782