Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Discrete Mathematics
The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees--History of Combinatorial Generation (Art of Computer Programming)
Optimal coding and sampling of triangulations
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Bijective counting of plane bipolar orientations and Schnyder woods
European Journal of Combinatorics
Bijections for Baxter families and related objects
Journal of Combinatorial Theory Series A
Intervals of balanced binary trees in the Tamari lattice
Theoretical Computer Science
Unified bijections for maps with prescribed degrees and girth
Journal of Combinatorial Theory Series A
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The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley order can be defined on the set of Dyck paths of size n as the relation of being above. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck paths. In a previous article, the second author defined a bijection @F between pairs of non-crossing Dyck paths and the realizers of triangulations (or Schnyder woods). We give a simpler description of the bijection @F. Then, we study the restriction of @F to Tamari and Kreweras intervals. We prove that @F induces a bijection between Tamari intervals and minimal realizers. This gives a bijection between Tamari intervals and triangulations. We also prove that @F induces a bijection between Kreweras intervals and the (unique) realizers of stack triangulations. Thus, @F induces a bijection between Kreweras intervals and stack triangulations which are known to be in bijection with ternary trees.