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A geometric form for the extended patience sorting algorithm

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Bibliographic Details
Title: A geometric form for the extended patience sorting algorithm
Authors: Burstein, Alexander1 burstein@math.iastate.edu, Lankham, Isaiah2 issy@math.ucdavis.edu
Source: Advances in Applied Mathematics. Feb2006, Vol. 36 Issue 2, p106-117. 12p.
Subjects: Algorithms, Combinatorial probabilities, Configurations (Geometry), Permutations, Lattice paths
Abstract: Abstract: Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors extended Patience Sorting to a full bijection between the symmetric group and certain pairs of combinatorial objects (called pile configurations) that are most naturally defined in terms of generalized permutation patterns and barred pattern avoidance. This Extended Patience Sorting Algorithm is very similar to the Robinson–Schensted–Knuth (or RSK) Correspondence, which is itself built from repeated application of the Schensted Insertion Algorithm. In this work we introduce a geometric form for the Extended Patience Sorting Algorithm that is in some sense a natural dual algorithm to G. Viennot''s celebrated Geometric RSK Algorithm. Unlike Geometric RSK, though, the lattice paths coming from Patience Sorting are allowed to intersect. We thus also give a characterization for the intersections of these lattice paths in terms of the pile configurations associated with a given permutation under the Extended Patience Sorting Algorithm. [Copyright &y& Elsevier]
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Abstract:Abstract: Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors extended Patience Sorting to a full bijection between the symmetric group and certain pairs of combinatorial objects (called pile configurations) that are most naturally defined in terms of generalized permutation patterns and barred pattern avoidance. This Extended Patience Sorting Algorithm is very similar to the Robinson–Schensted–Knuth (or RSK) Correspondence, which is itself built from repeated application of the Schensted Insertion Algorithm. In this work we introduce a geometric form for the Extended Patience Sorting Algorithm that is in some sense a natural dual algorithm to G. Viennot''s celebrated Geometric RSK Algorithm. Unlike Geometric RSK, though, the lattice paths coming from Patience Sorting are allowed to intersect. We thus also give a characterization for the intersections of these lattice paths in terms of the pile configurations associated with a given permutation under the Extended Patience Sorting Algorithm. [Copyright &y& Elsevier]
ISSN:01968858
DOI:10.1016/j.aam.2005.08.001