The monoids of the patience sorting algorithm

@article{Cain2019TheMO,
  title={The monoids of the patience sorting algorithm},
  author={Alan J. Cain and Ant{\'o}nio Malheiro and F{\'a}bio M. Silva},
  journal={Int. J. Algebra Comput.},
  year={2019},
  volume={29},
  pages={85-125}
}
The left patience sorting ([Formula: see text][Formula: see text]PS) monoid, also known in the literature as the Bell monoid, and the right patient sorting ([Formula: see text]PS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences. Presentations for these monoids are given. Each finite-rank [Formula: see text]PS monoid is shown to have polynomial growth and to satisfy a… 

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References

SHOWING 1-10 OF 71 REFERENCES
Combinatorics of patience sorting piles
TLDR
This work exploits the similarities between Patience Sorting and the Schensted Insertion Algorithm in order to do things that include defining an analog of the Knuth relations and extending Patiences Sorting to a bijection between permutations and certain pairs of set partitions.
A note on identities in plactic monoids and monoids of upper-triangular tropical matrices
This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy
Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids
TLDR
This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoid, hypoplactic monoids, and sylvester monoids; the monoid algebras corresponding to monoids of these classes are automaton algeBRas in the sense of Ufnarovskij.
String Rewriting and Homology of Monoids
  • D. Cohen
  • Mathematics
    Math. Struct. Comput. Sci.
  • 1997
TLDR
A simplified version of Brown's proof, which is conceptual, in contrast with the other proofs, which are computational, is given.
Length Considerations in Context-Free Languages
  • D. Raz
  • Computer Science
    Theor. Comput. Sci.
  • 1997
The algebra of binary search trees
...
...