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

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**Example text**

Math. 95 (1981), 251-256. 3. E. E. ), Prog. Math. 161, Birkhäuser, Boston, 1998, pp. 1-22. 4. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III: The Omega package, (to appear). 5. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis VI: The fast algorithm, (in preparation). 6. C. Bessenrodt, A bijection for Lebesgue's identity in the spirit of Sylvester, Discrete Math. 132 (1994), 1-10. 7. M. Bousquet-Melou and K. Eriksson, Lecture hall partitions, Ramanujan J. 1 (1997), 101-111.

The TDA-scheme is a bit finer than t he TDO-scheme. In the natural action on four points, the generat ors can be identified as A = (12)(34) , B = (13)(24), C = (124) and D = (12) , for example. 00 0 0 0 0 0 0 00 0 00 0 0 0 00 00 0 0 0 0 0 0 0 0 00 0 0 0 00 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 00 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 TDO-scheme: 1 12 18 4 6 1 2 3 0 12 0 3 3 1 TDA-scheme: 1 6 12 12 12 6 4 2 1 0 1 2 0 3 2 1 1 A 2 = id, B 2 = id,A 8 = A , C3 = id,A 0 = B ,B 0 D2 = id, A 0 = A ,B 0 = A B,C 0 = C 2 = AB , 54 A.

14] (see also [17, Ch. 15]): find allnonnegative integer solutions (a1,a2,a3,a4) E N4 of (37) Encoding the corresponding generating function as an 0 =-expression results in 1 = ~ (1- ,\xi)(1 - ,\xz)(1- T)(1- y) The ,\-elimination rule that is needed for this situation is also to find in MacMahon's book; see [12, Vol. 2, Sect. 351 , p. 105]. OEqR OEqR[expr, z] applies the OmegaEq operator to expr eliminating the variable z. In[3] := f = Out [3] = 1 I ((1-x1 ,\)(1-x2 ,\)(1-x3/,\)(1-x4/,\)) 1 (1- ,\x1) (1- ,\x2) (1 - xn (1 - x,\4 ) 34 George E.