# Algebraic Combinatorics and Applications: Proceedings of the by George E. Andrews, Peter Paule, Axel Riese (auth.), Anton

By George E. Andrews, Peter Paule, Axel Riese (auth.), Anton Betten, Axel Kohnert, Reinhard Laue, Alfred Wassermann (eds.)

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**Extra resources for Algebraic Combinatorics and Applications: Proceedings of the Euroconference, Algebraic Combinatorics and Applications (ALCOMA), held in Gößweinstein, Germany, September 12–19, 1999**

**Example text**

S,. •. Ar8 ' := Ao,... ,o . s,. \'s are killed by the n = operator which, alternatively, can be viewed as a constant term Operator. As already pointed out by MacMahon [12, Vol. 2, Sect. VIII, p. 104], this Operator is related to f! \-variables from n =-expressions. However, 32 George E. 4 is much more convenient - especially with respect to efficiency of computer algebra implementation. Despite having developed his theory long time before the age of computers, this was exactly the program carried out by MacMahon in his book.

P. Stanley, Linear homogeneaus diophantine equations and magic labelings of graphs, Duke Math. J. 40 (1973), 607-632. 16. P. Stanley, Combinatorics and Commutative Algebra, Birkhäuser, Boston, 1983. 17. P. Stanley, Enumerative Combinatorics - Volume 1, Wadsworth , Monterey, California, 1986. 18. J . Yee, On Combinatorics of Lecture Hall Partitions, (to appear). Note on the Proper Linear Spaces on 18 Points A. Betten 1 and D. de Mathematisches Seminar der Universität Kiel Ludewig-Meyn-Str. de Abstract In [4] we constructed and enumerated all proper linear spaces on 17 points using the so-called TDO-method.

2 for r = (r 1 , r 2 , . . , rn) E Nn the monomial 24 George E. Andrews, Peter Paule, Axel Riese, and Volker Strehl then we have from the semilinear representation of lecture hall partitions fn(YI. Y2, . . 'Yn) = L)Yb : b E Ln = R~ E9 &~} L{YR: RE R~} fl1::;j::;n(l- Y'i) . Multiplying the numerator and the denominator of this last fraction by 1- Yn = 1- yw(no") Ieads (thanks to a finite geometrie series) to where &n = w(Vn), and where Vn is the free semimodule generated by <5 1 , . , §n- 1 and n §n = (0, ..