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\iteman{io-port 00014564}
\itemau{Fouch\'e, W.L.}
\itemti{Cyclic decompositions of multisets.}
\itemso{Q. J. Math., Oxf. II. Ser. 42, No.167, 293-308 (1991).}
\itemab
Cyclic decompositions are what Richard Stanley has called multiset permutations. We begin with a finite word in the alphabet of positive integers (with repetitions allowed). Two words are equivalent if one is a cyclic permutation of the other. A cycle is an equivalence class of words. A cyclic decomposition is an element of the free abelian monoid over the set of cycles. The author establishes a bijective correspondence between cyclic decompositions and certain Young tableaux. This correspondence implies the identity $$\sum\sb{\gamma\in C} w(\gamma) = \prod\sb{k=1}\sp{\infty}[1-(x\sb 1\sp k + x\sb 2\sp k + \cdots) ]\sp{-1}, $$ where $C$ is the set of cyclic decompositions and $w(a\sb 1a\sb 2\ldots a\sb n) = x\sb{a\sb 1}x\sb{a\sb 2}\ldots x\sb{a\sb n}$. It is also demonstrated that this bijection provides interpretations of identities such as the Rogers-Ramanujan identity in terms of cyclic decompositions.
\itemrv{D.M.Bressoud (University Park)}
\itemcc{}
\itemut{word; multiset permutations; cyclic decomposition; Young tableaux; Rogers-Ramanujan identity}
\itemli{doi:10.1093/qmath/42.1.293}
\end