Increasing and decreasing subsequences and their variants

• Autores: Richard P. Stanley
• Localización: Proceedings oh the International Congress of Mathematicians: Madrid, August 22-30,2006 : invited lectures / coord. por Marta Sanz Solé, Javier Soria de Diego, Juan Luis Varona Malumbres, Joan Verdera, Vol. 1, 2006, ISBN 978-3-03719-022-7, págs. 545-579
• Idioma: inglés
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• Resumen

  • We survey the theory of increasing and decreasing subsequences of permutations.

    Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2, . . . , n was obtained by Vershik�Kerov and (almost) by Logan�Shepp.

    The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson.

    These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions.