Abstract
The ring of symmetric functions is introduced. The six standard bases for symmetric functions; namely, the monomial, elementary, homogeneous, power, forgotten, and Schur symmetric functions, are defined. Numerous relationships between these functions are proved.
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
Beck, D.A., Remmel, J.B., Whitehead, T.: The combinatorics of transition matrices between the bases of the symmetric functions and the B n analogues. In: Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics, Florence, 1993, vol. 153, pp. 3–27 (1996)
Eğecioğlu, Ö., Remmel, J.B.: Brick tabloids and the connection matrices between bases of symmetric functions. Discret. Appl. Math. 34(1–3), 107–120 (1991) [Combinatorics and Theoretical Computer Science (Washington, DC, 1989)]
Haglund, J.: The q,t-Catalan Numbers and the Space of Diagonal Harmonics. University Lecture Series, vol. 41. American Mathematical Society, Providence (2008) [With an appendix on the combinatorics of Macdonald polynomials]
Knuth, D.E.: Permutations, matrices, and generalized Young tableaux. Pac. J. Math. 34, 709–727 (1970)
LoBue Tiefenbruck, J., Remmel, J.B.: A Murnaghan-Nakayama rule for generalized Demazure atoms. In: 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013). Discrete Mathematics and Theoretical Computer Science Proceedings, AS, pp. 969–980. Association of Discrete Mathematics and Theoretical Computer Science, Nancy (2013)
Loehr, N.A.: Abacus proofs of Schur function identities. SIAM J. Discret. Math. 24(4), 1356–1370 (2010)
Loehr, N.A.: Bijective Combinatorics. Discrete Mathematics and Its Applications (Boca Raton). CRC Press, Boca Raton (2011)
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
Stanley, R.P.: Binomial posets, Möbius inversion, and permutation enumeration. J. Comb. Theory Ser. A 20(3), 336–356 (1976)
Wagner, J.: The permutation enumeration of wreath products C k §S n of cyclic and symmetric groups. Adv. Appl. Math. 30(1–2), 343–368 (2003) [Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001)]
White, D.: Orthogonality of the characters of S n . J. Comb. Theory Ser. A 40(2), 265–275 (1985)
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). Symmetric Functions. In: Counting with Symmetric Functions. Developments in Mathematics, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-23618-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-23618-6_2
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)