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Multiset Structures Derived from Vector Spaces
Tung-Shan Fu1 and Zhe-Xian Wan2, 3
1National Pingtung Institute of Commerce, 51 Min-Shen E. Road, Pingtung 900, Taiwan
tsfu@npic.edu.tw
2Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080,P.R. China
3Department of Information Technology, Lund University, Box 118, Lund 22100, Sweden
wan@it.lth.se
Annals of Combinatorics 5 (3) p.305-318 September, 2001
AMS Subject Classification: 05A05, 05A19, 05E20, 06A07
Abstract:
Let be an n-dimensional vector space over a division ring D and let be positive integers such that . Denoted by the subgroup of , consisting of upper triangular block matrices of the form

where , , and is an submatrix over D, . How the subspaces of are partitioned into orbits under the action of is determined. An isomorphism between the lattice of these orbits and the lattice of submultisets of the multiset is established. The length of each orbit is also enumerated in case . Moreover, a combinatorial interpretation of a multiset Mahonian statistics is given and two identities of the q-multinomial coefficients are obtained.
Keywords: multisets, vector spaces, lattice isomorphism, Mahonian statistics

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