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Cyclic Tableaux and Symmetric Functions
Author(s) -
Chen William Y. C.,
Lih KoWei,
Yeh YeongNan
Publication year - 1995
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1995943327
Subject(s) - mathematics , symmetric function , combinatorics , combinatorial proof , lattice (music) , duality (order theory) , pfaffian , theta function , pure mathematics , multinomial distribution , physics , acoustics , statistics
We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function p n in terms of the elementary symmetric function e n and the homogeneous symmetric function h n . The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.