Counting the Restricted Gaussian Partitions of a Finite Vector Space

A subspace partitionof a finite vector space V = V (n,q) of dimension n over GF(q) is a collection of subspaces of V such that their union is V , and the intersection of any two subspaces inis the zero vector. The multiset Tof dimensions of subspaces inis called the type of �, or, a Gaussian partition of V. Previously, we showed that subspace partitions of V and their types are natural, combinatorial q-analogues of the set partitions of {1,...,n} and integer partitions of n respectively. In this paper, we connect all four types of partitions through the concept of "basic" set, subspace, and Gaussian partitions, corresponding to the integer partitions of n. In particular, we combine Beutelspacher's classic construction of subspace partitions with some additional conditions to derive a special subset G of Gaussian partitions of V. We then show that the cardinality of G is a rational polynomial R(q) in q, with R(1) = p(n), where p is the integer partition function.

[1]  Albrecht Beutelspacher,et al.  Partial spreads in finite projective spaces and partial designs , 1975 .

[2]  R. C. Bose,et al.  The construction of translation planes from projective spaces , 1964 .

[3]  Johannes André,et al.  Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe , 1954 .

[4]  Frank R. Kschischang,et al.  Coding for Errors and Erasures in Random Network Coding , 2007, IEEE Transactions on Information Theory.

[5]  J. Schönheim,et al.  Group Partition, Factorization and the Vector Covering Problem , 1972, Canadian Mathematical Bulletin.

[6]  Tor Bu.  Partitions of a vector space , 1980, Discret. Math..

[7]  Olof Heden,et al.  The supertail of a subspace partition , 2013, Des. Codes Cryptogr..

[8]  Olof Heden,et al.  A survey of perfect codes , 2008, Adv. Math. Commun..

[9]  Se June Hong,et al.  A General Class of Maximal Codes ror Computer Applications , 1972, IEEE Transactions on Computers.

[10]  David A. Drake,et al.  Partial t-spreads and group constructible (s,r,μ)-nets , 1979 .

[11]  Papa A. Sissokho,et al.  Partitions of finite vector spaces into subspaces , 2008 .

[12]  Simon Thomas,et al.  Designs over finite fields , 1987 .

[13]  Olof Heden,et al.  A Survey of the Different Types of Vector Space Partitions , 2011, Discret. Math. Algorithms Appl..

[14]  Guido Zappa,et al.  Partitions and other coverings of finite groups , 2003 .

[15]  R. Baer.  Partitionen endlicher Gruppen , 1961 .

[16]  Heather Jordon,et al.  The maximum size of a partial 3-spread in a finite vector space over GF(2) , 2010, Des. Codes Cryptogr..

[17]  Fusun Akman,et al.  The lattice of finite subspace partitions , 2012, Discret. Math..

[18]  Albrecht Beutelspacher.  Partitions of finite vector spaces: An application of the frobenius number in geometry , 1978 .

[19]  Bernt Lindström.  Group Partitions and Mixed Perfect Codes , 1975, Canadian Mathematical Bulletin.

[20]  Olof Heden,et al.  Some necessary conditions for vector space partitions , 2011, Discret. Math..

[21]  Papa A. Sissokho,et al.  Partitions of V(n, q) into 2- and s-Dimensional Subspaces , 2012 .

[22]  Vaclav Smil,et al.  Book of numbers , 1996, Nature.

[23]  Leo Storme,et al.  On the Spectrum of the Sizes of Maximal Partial Line Spreads in PG(2n,q), n ≥ 3 , 2005, Des. Codes Cryptogr..

[24]  Olof Heden.  On the length of the tail of a vector space partition , 2009, Discret. Math..

[25]  G. A. Miller.  Groups in which all the operators are contained in a series of subgroups such that any two have only identity in common , 1906 .