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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2 Partition Analysis and Cayley Compositions The following is the only strictly Partition Analysis identity that is required: n ~ 1+B (1 - A)(1 - Aß2) 0 >. ) (30) While (30) is not in MacMahon's fundamental list [12, po102], it is easily proved: n = n """" >. 2r+1-s r , s~O =L r~O 2r+1 L ArEs s=O 1 (1 - A)(1 - B) B2 (1 - Aß2)(1 - B) (1- AB 2 ) - B 2 (1- A) (1- A)(1- B)(1- Aß2) 1 +B (1 - A)(l - Aß2) 0 We remark that applying the Omega package would give (30) in one strokeo Let us now consider a j + 1 variable generating function for Cayley compositions: n1 ,n2 , ..

L {c 1+1(2m), CJ+t(2m- 1), if n =2m, if n =2m- 1 First, suppose that n = 2m and let be the set of Cayley compositions with j + 1 parts ending in 2m. Its cardinality is c1+1(2m). Each part of a Cayley composition is less or equal twice the preceding part. Hence, if we omit the last entry 2m from all these tuples, the resulting set is running through all Cayley compositions with j parts ending in elements n 1 2:: m. The cardinality ofthisset is exactly C1 (1) - 2:;:~ 1 c1 (l) . The case n = 2m - 1 is analogous.

Stanley's interest in the problern of solving linear homogeneaus equations for nonnegative integers is also reftected by his book [16] that contains many further references to this problern area. An additional reference is the chapter on rational generating function in Stanley's textbook [17]. 2. Weillustrate the use ofrule (35) by choosing an example from [16, Ex. x2)(1- Xf) _ - 1 + X1X2X3 2 2 (1- x 1x3)(1- x 2x3) MacMahon's Partition Analysis V 33 By geometric series expansion we obtain the desired parametrized representation of the solution set of (36), namely This means, {(2, 0,1) , (0,2, 1), (1, 1, 1)} is the set of fundamental solutions, whereas {(2,0, 1) , (0,2, 1)} is called the set of completely fundamental solutions; note that 2(1, 1, 1) (2, 0, 1) + (0, 2, 1).

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