Abstract
Recently we interpreted five q-series identities of Rogers combinatorially by using partitions with “n +t cop-ies of n” of Agarwal and Andrews (J. Combin. Theory Ser.A, 45(1987), No.1, 40-49). In this paper we use lattice paths of Agarwal and Bressoud (Pacific J. Math. 136(2) (1989), 209-228) to provide new combinatorial interpretations of the same identities. This results in five new 3-way combinatorial identities.
Highlights
Definitions and the Main ResultsIn the literature we find that several q -identities such as given in Slater’s compendium [3] have been interpreted combinatorially using ordinary partitions by several authors
In this paper we prove the following combinatorial interpretations of the identities (1.1)-(1.5) in terms of lattice paths: Theorem 6
We establish a 1 1 correspondence between the lattice paths enumerated by E1 and the n -color partitions enumerated by A1
Summary
In the literature we find that several q -identities such as given in Slater’s compendium [3] have been interpreted combinatorially using ordinary partitions by several authors (for example, see Connor [4], Subbarao [5], Subbarao and Agarwal [6] and Agarwal and Andrews [7]). PLAIN: A section of path consisting of only horizontal steps which starts either on the y-axis or at a vertex preceded by a southeast step and ends at a vertex followed by a northeast step. Example: The following path has five peaks, three valleys, three mountains and one plain. In this paper we prove the following combinatorial interpretations of the identities (1.1)-(1.5) in terms of lattice paths: Theorem 6. Let E1 denote the number of lattice paths of weight which start from 0, 0 , have no valley above height 0, the lengths of the plains, if any, are 0 mod 4 and the height of each peak is greater than 2.
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