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Counting with the Elementary and Homogeneous Symmetric Functions

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Counting with Symmetric Functions

Part of the book series: Developments in Mathematics ((DEVM,volume 43))

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Abstract

The relationship between the elementary and homogeneous symmetric functions, specifically the expansion involving brick tabloids, is used to find an assortment of generating functions. We are able to count and refine permutations according to restricted appearances of descents and prove a variety of results about words.

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References

  1. André, D.: Developpements de sec x et de tang x. C. R. Acad. Sci. Paris 88, 965–967 (1879)

    MATH  Google Scholar 

  2. André, D.: Mémoire sur les permutations alternées. J. Math. Pure Appl. 7, 167–184 (1881)

    MATH  Google Scholar 

  3. Bach, Q., Remmel, J.: Generating functions for descents over permutations which avoid sets of consecutive patterns. Aust. J. Comb. (to appear)

    Google Scholar 

  4. Beck, D.: Permutation enumeration of the symmetric and hyperoctahedral groups and the combinatorics of symmetric functions. Ph.D. thesis, University of California, San Diego (1993)

    Google Scholar 

  5. Beck, D.: The combinatorics of symmetric functions and permutation enumeration of the hyperoctahedral group. Discret. Math. 163(1–3), 13–45 (1997)

    Article  MATH  Google Scholar 

  6. Beck, D., Remmel, J.: Permutation enumeration of the symmetric group and the combinatorics of symmetric functions. J. Comb. Theory Ser. A 72(1), 1–49 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brenti, F.: Unimodal polynomials arising from symmetric functions. Proc. Am. Math. Soc. 108(4), 1133–1141 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brenti, F.: Permutation enumeration symmetric functions, and unimodality. Pac. J. Math. 157(1), 1–28 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carlitz, L.: Enumeration of up-down permutations by number of rises. Pac. J. Math. 45, 49–58 (1973)

    Article  Google Scholar 

  10. Carlitz, L.: Permutations and sequences. Adv. Math. 14, 92–120 (1974)

    Article  MATH  Google Scholar 

  11. Carlitz, L.: Permutations, sequences and special functions. SIAM Rev. 17, 298–322 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Carlitz, L., Scoville, R.: Generalized Eulerian numbers: combinatorial applications. J. Reine Angew. Math. 265, 110–137 (1974)

    MathSciNet  MATH  Google Scholar 

  13. Carlitz, L., Scoville, R.: Enumeration of up-down permutations by upper records. Monatsh. Math. 79, 3–12 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carlitz, L., Scoville, R., Vaughan, T.: Enumeration of pairs of permutations. Discret. Math. 14(3), 215–239 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Enlarged edn. D. Reidel Publishing Co., Dordrecht (1974)

    Book  MATH  Google Scholar 

  16. Entringer, R.: A combinatorial interpretation of the Euler and Bernoulli numbers. Nieuw Arch. Wisk. (3) 14, 241–246 (1966)

    Google Scholar 

  17. Euler, L.: Foundations of Differential Calculus. Springer, New York (2000) [Translated from the Latin by John D. Blanton]

    Book  MATH  Google Scholar 

  18. Fédou, J.M., Rawlings, D.: More statistics on permutation pairs. Electron. J. Comb. 1, Research Paper 11, approx. 17 pp. (electronic) (1994)

    Google Scholar 

  19. Fédou, J.M., Rawlings, D.: Statistics on pairs of permutations. Discret. Math. 143(1–3), 31–45 (1995)

    Article  MATH  Google Scholar 

  20. Flajolet, P.: Combinatorial aspects of continued fractions. Ann. Discret. Math. 9, 217–222 (1980) [Combinatorics 79 (Proc. Colloq., Université de Montréal, Montreal, 1979), Part II]

    Google Scholar 

  21. Foata, D., Schützenberger, M.P.: Théorie Géométrique des Polynômes Eulériens. Lecture Notes in Mathematics, vol. 138. Springer, Berlin (1970)

    Google Scholar 

  22. Fuller, E., Remmel, J.: Symmetric functions and generating functions for descents and major indices in compositions. Ann. Comb. 14(1), 103–121 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Garsia, A., Gessel, I.: Permutation statistics and partitions. Adv. Math. 31(3), 288–305 (1979)

    Article  MathSciNet  Google Scholar 

  24. Gessel, I.: Generating functions and enumeration of sequences. Ph.D. thesis, Massachusetts Institute of Technology (1977)

    Google Scholar 

  25. Harmse, J., Remmel, J.: Patterns in column strict fillings of rectangular arrays. Pure Math. Appl. 22(2), 141–171 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Jones, M.E., Remmel, J.: A reciprocity approach to computing generating functions for permutations with no pattern matches. In: 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011). Discrete Mathematics and Theoretical Computer Science Proceedings, AO, pp. 551–562. Association of Discrete Mathematics and Theoretical Computer Science, Nancy (2011)

    Google Scholar 

  27. Kulikauskas, A., Remmel, J.: Lyndon words and transition matrices between elementary, homogeneous and monomial symmetric functions. Electron. J. Comb. 13(1), Research Paper 18, 30 (2006)

    Google Scholar 

  28. Langley, T.: The plethysm of two schur functions at hook, near-hook, and two row shapes. Ph.D. thesis, University of California, San Diego (2001)

    Google Scholar 

  29. Langley, T.: Alternate transition matrices for Brenti’s q-symmetric functions and a class of (q, t)-symmetric functions on the hyperoctahedral group. In: Proceedings of Formal Power Series and Algebraic Combinatorics (FPSAC), Melbourne (2002)

    Google Scholar 

  30. Langley, T.M., Remmel, J.B.: Enumeration of m-tuples of permutations and a new class of power bases for the space of symmetric functions. Adv. Appl. Math. 36(1), 30–66 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liese, J., Remmel, J.: Generating functions for permutations avoiding a consecutive pattern. Ann. Comb. 14(1), 123–141 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mendes, A., Remmel, J.: Permutations and words counted by consecutive patterns. Adv. Appl. Math. 37(4), 443–480 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nakayama, T.: On some modular properties of irreducible representations of symmetric groups, II. Jpn. J. Math. 17, 411–423 (1941)

    Google Scholar 

  34. Pólya, G.: Kombinatorische anzahlbestimmungen für gruppen, graphen und chemische verbindungen. Acta Math. 68(1), 145–254 (1937)

    Article  MathSciNet  Google Scholar 

  35. Remmel, J.: Permutation statistics and (k, ℓ)-hook schur functions. Discret. Math. 67, 271–298 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  36. Remmel, J., Riehl, M.: Generating functions for permutations which contain a given descent set. Electron. J. Comb. 17(1), Research Paper 27, 33 (2010)

    Google Scholar 

  37. Riddell, R., Uhlenbeck, G.: On the theory of the virial development of the equation of state of monoatomic gases. J. Chem. Phys. 21, 2056–2064 (1953)

    Article  MathSciNet  Google Scholar 

  38. Stanley, R.P.: Theory and application of plane partitions, I, II. Stud. Appl. Math. 50, 167–188 (1971); Stud. Appl. Math. 50, 259–279 (1971)

    Google Scholar 

  39. Stanley, R.P.: Enumerative Combinatorics, vol. 1, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  40. Viennot, G.: Une forme géométrique de la correspondence de Robinson-Schensted. In: Combinatoire et Représentation du Groupe Symétrique (Actes Table Ronde CNRS, Université Louis-Pasteur Strasbourg, Strasbourg, 1976). Lecture Notes in Mathematics, vol. 579, pp. 29–58. Springer, Berlin (1977)

    Google Scholar 

  41. Wagner, J.: The combinatorics of the permutation enumeration of wreath products between cyclic and symmetric groups. Ph.D. thesis, University of California, San Diego (2000)

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, California Polytechnic State University, San Luis Obispo, CA, USA

    Anthony Mendes

  2. Department of Mathematics, University of California at San Diego, La Jolla, CA, USA

    Jeffrey Remmel

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  1. Anthony Mendes
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  2. Jeffrey Remmel
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Mendes, A., Remmel, J. (2015). Counting with the Elementary and Homogeneous Symmetric Functions. In: Counting with Symmetric Functions. Developments in Mathematics, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-23618-6_3

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