Abstract
The plactic monoid is the quotient of the free monoid by the congruence generated by Knuth’s well-celebrated rules. It is well-known that the set of Young tableaux is a cross-section of this congruence which happens to be regular. The main result of this work shows that the set of alphabetically minimal elements in the congruence classes is also regular. We give a full combinatorial characterization of these minimal elements and show that constructing them is as fast as constructing a tableau.
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References
Anisimov, A.V., Knuth, D.E.: Inhomogeneous sorting. International Journal of Computer and Information Sciences 8(4), 255–260 (1979)
Arnold, A., Kanta, M., Krob, D.: Recognizable subsets of the two letter plactic monoid. Information Processing Letters 64, 53–59 (1997)
Cartier, P., Foata, D.: Problèmes combinatoires de commutation et réarrangements. Lecture Notes in Mathematics, vol. 85 (1969)
Choffrut, C., Mercaş, R.: Contextual partial commutations. Discrete Mathematics and Theoretical Computer Science 12(4), 59–72 (2010)
Diekert, V., Rozenberg, G.: The Book of Traces. World Scientific Publishing Co., Singapore (1995)
Duboc, C.: On some equations in free partially commutative monoids. Theoretical Computer Science 46(2-3), 159–174 (1986)
Eilenberg, S.: Automata, Languages and Machines, vol. A. Academic Press (1974)
Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word processing in groups. Jones and Bartlett (1992)
Greene, C.: An extension of Schensted’s Theorem. Advances in Mathematics 14(2), 254–265 (1974)
Knuth, D.E.: The art of computer programming, vol. 3. Addison Wesley (1973)
Lallement, G.: Semigroups and Combinatorial Applications. John Wiley & Sons (1979)
Lascoux, A., Leclerc, B., Thibon, J.-Y.: The plactic monoid. In: Lothaire, M. (ed.) Algebraic Combinatorics on Words, pp. 144–173. Cambridge University Press (2002)
Lascoux, A., Schützenberger, M.-P.: Le monoïde plaxique. In: de Luca, A. (ed.) Non-commutative Structures in Algebra and Geometric Combinatorics, vol. 109, pp. 129–156. C.N.R. (1981)
Lentin, A., Schützenberger, M.-P.: A combinatorial problem in the theory of free monoids. In: Bose, R.C., Bowlings, T.E. (eds.) Combinatorial Mathematics, pp. 112–144. North Carolina Press, Chapel Hill (1967)
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Choffrut, C., Mercaş, R. (2013). The Lexicographic Cross-Section of the Plactic Monoid Is Regular. In: Karhumäki, J., Lepistö, A., Zamboni, L. (eds) Combinatorics on Words. Lecture Notes in Computer Science, vol 8079. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40579-2_11
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DOI: https://doi.org/10.1007/978-3-642-40579-2_11
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