Quasi symmetric functions are a generalisation of symmetric functions and are closely related to noncommutative symmetric functions.

Let $X$ be a totally ordered set of indeterminants. Let $R$ be a ring. A polynomial in $R[X]$ or a power series in $R[ [X] ]$ is said to be **quasi-symmetric** if whenever $X_1 \lt X_2 \lt \dots \lt X_n$ and $Y_1 \lt Y_2 \lt \dots \lt Y_n$ are finite sets of indeterminants then the coefficients of $X_1^{i_1} X_2^{i_2} \cdots X_n^{i_n}$ and $Y_1^{i_1} Y_2^{i_2} \cdots Y_n^{i_n}$ are the same.

The ring $\QSymm^{\hat{}}$ is defined as the ring of quasi-symmetric power series over $\mathbb{Z}$ in countably many variables. Its subring $\QSymm$ is defined as the ring of quasi-symmetric polynomials (meaning, power series of bounded degree).

(Copied from noncommutative symmetric function as the two concepts are often studied together.)

- G. Duchamp, F. Hivert, J.-Y. Thibon,
*Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras*, Internat. J. Alg. Comput. 12 (2002), 671–717. - I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, J.-Y. Thibon,
*Noncommutative symmetric functions*, Adv. in Math.**112**(1995), 218–348, hep-th/9407124 - Jean-Christophe Novelli, Jean-Yves Thibon,
*Noncommutative symmetric functions and Lagrange inversion*, math.CO/0512570;*Noncommutative symmetric functions and an amazing matrix*arxiv/1109.1184 - Lenny Tevlin,
*Noncommutative Monomial Symmetric Functions*, Formal Power Series and Algebraic Combinatorics Nankai University, Tianjin, China, 2007, proceedings pdf - D. Krob, J.-Y. Thibon,
*Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at $q = 0$*, pdf - Christos A. Athanasiadis,
*Power sum expansion of chromatic quasisymmetric functions*, arxiv/1409.2595

- Michael Hazewinkel,
*Symmetric functions, noncommutative symmetric functions and quasisymmetric functions*, pdf - V. Retakh and R. Wilson, Advanced Course on Quasideterminants and Universal Localization: pdf (see the part
*Factorization of Noncommutative Polynomials*and Noncommutative Symmetric Functions_)

- Mike Zabrocki,
*Non-commutative symmetric functions II: Combinatorics and coinvariants*, slides from a talk pdf, III: A representation theoretical approach pdf - Lenny Tevlin,
*Introduction to quasisymmetric and noncommutative symmetric functions*, slides, Fields Institute 2010 pdf

category: combinatorics

Last revised on August 23, 2015 at 02:46:50. See the history of this page for a list of all contributions to it.