On q -Series and Split Lattice Paths
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ORIGINAL PAPER
On q-Series and Split Lattice Paths Megha Goyal1 Received: 24 April 2016 / Revised: 10 May 2020 / Published online: 13 August 2020 Ó Springer Japan KK, part of Springer Nature 2020
Abstract In this paper a natural question which arise to study the graphical aspect of split ðn þ tÞ-color partitions, is answered by introducing a new class of lattice paths, called split lattice paths. A direct bijection between split ðn þ tÞ-color partitions and split lattice paths is proved. This new combinatorial object is applied to give new combinatorial interpretations of two basic functions of Gordon-McIntosh. Some generalized q-series are also discussed. We further explore these paths by providing combinatorial interpretations of some Rogers-Ramanujan type identities which reveal their rich structure and great potential for further research. Keywords q-series Split ðn þ tÞ-color partitions Split lattice paths Combinatorial identities
Mathematics Subject Classification 05A15 05A17 05A19 11P81
1 Introduction The graphical prospect of partitions and compositions has always drawn the attention of mathematicians. Graphical representation is very useful when applications of partitions and compositions are considered [9, 19, 20]. In 1987, Agarwal and Andrews [1] defined ðn þ tÞ-color partitions. To study the graphical aspect of this new set of partitions, Agarwal and Bressoud [2] introduced weighted lattice paths. Several basic series identities had been interpreted combinatorially using ordinary partitions, colored partitions, Frobenius partitions, lattice paths, associated lattice paths etc. in [1–5, 8, 11, 12, 14, 15, 18]. In 2014, Agarwal and Sood [6] defined a new class of partitions, called split ðn þ tÞ-color partitions. Using & Megha Goyal [email protected] 1
Department of Mathematical Sciences, IK Gujral Punjab Technical University Jalandhar, Main Campus, Kapurthala 144603, India
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Graphs and Combinatorics (2020) 36:1273–1295
this new combinatorial object, Agarwal and Sood gave combinatorial meaning to two basic functions of Gordon–McIntosh [10]. This new set of partitions has the potential to provide combinatorial interpretations to q-series which cannot be interpreted combinatorially using ordinary partitions or ðn þ tÞ-color partitions, for instance see [13, 17]. But the graphical aspect of split ðn þ tÞ-color partitions is still remained an unexplored concept. The present article is a contribution to the quest for settling the graphical representation of split ðn þ tÞ-color partitions. We have originally started this problem by defining a new combinatorial object which we call split lattice paths. These objects are natural combinatorial structures associated with basic series. It turned out, however, that the split lattice paths provide not only the graphical representation of split ðn þ tÞ-color partitions but split lattice paths will also provide graphical representations of all ordered split ðn þ tÞ-color partitions which may be studied in future. The next section is dedi
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