Abstract
The RSK algorithm is introduced and used to find generating functions for permutation statistics. Connections are made to increasing subsequences in permutations and words and the Schur symmetric functions. A q-analogue of the hook length formula is proved, and the Hillman-Grassl algorithm is introduced.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Désarménien, J., Foata, D.: Fonctions symétriques et séries hypergéométriques basiques multivariées. Bull. Soc. Math. Fr. 113(1), 3–22 (1985)
Désarménien, J., Foata, D.: Fonctions symétriques et séries hypergéométriques basiques multivariées, II. In: Combinatoire énumérative, Montreal, 1985/Quebec, 1985. Lecture Notes in Mathematics, vol. 1234, pp. 68–90. Springer, Berlin (1986)
Désarménien, J., Foata, D.: Statistiques d’ordre sur les permutations colorées. Discret. Math. 87(2), 133–148 (1991)
Frame, J.S., Robinson, G.d.B., Thrall, R.M.: The hook graphs of the symmetric groups. Can. J. Math. 6, 316–324 (1954)
Garsia, A., Gessel, I.: Permutation statistics and partitions. Adv. Math. 31(3), 288–305 (1979)
Gessel, I.: Counting permutations by descents, greater index, and cycle structure. Preprint, Department of Mathematics M.I.T., Cambridge (1977)
Greene, C.: An extension of Schensted’s theorem. Adv. Math. 14, 254–265 (1974)
Hillman, A.P., Grassl, R.M.: Reverse plane partitions and tableau hook numbers. J. Comb. Theory Ser. A 21(2), 216–221 (1976)
Kitaev, S.: Patterns in Permutations and Words. Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2011) [With a foreword by Jeffrey B. Remmel]
Redfield, J.H.: The theory of group-reduced distributions. Am. J. Math. 49(3), 433–455 (1927)
Riordan, J.: An Introduction to Combinatorial Analysis. Dover, Mineola (2002) (Reprint of the 1958 original [Wiley, New York; MR0096594 (20 #3077)])
Robinson, G.d.B.: On the representations of the symmetric group. Am. J. Math. 60(3), 745–760 (1938). doi:10.2307/2371609. http://www.dx.doi.org/10.2307/2371609
Sagan, B.E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edn. Graduate Texts in Mathematics, vol. 203. Springer, New York (2001)
Schensted, C.: Longest increasing and decreasing subsequences. Can. J. Math. 13, 179–191 (1961)
Touchard, J.: Sur les cycles des substitutions. Acat. Math. 70, 243–297 (1939)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Mendes, A., Remmel, J. (2015). Counting with RSK. In: Counting with Symmetric Functions. Developments in Mathematics, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-23618-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-23618-6_5
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23617-9
Online ISBN: 978-3-319-23618-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)