Andrews–Gordon Type Series for Kanade–Russell Conjectures

Kağan Kurşungöz

doi:10.1007/s00026-019-00470-7

Annals of Combinatorics

November 2019, Volume 23, Issue 3–4, pp 835–888 | Cite as

Andrews–Gordon Type Series for Kanade–Russell Conjectures

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Kağan Kurşungöz

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First Online: 20 November 2019

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Abstract

We construct Andrews–Gordon type positive series as generating functions of partitions satisfying certain difference conditions in six conjectures by Kanade and Russell. Thus, we obtain q-series conjectures as companions to Kanade and Russell’s combinatorial conjectures. We construct generating functions for missing partition enumerants as well, without claiming new partition identities.

Keywords

Partition generating function Andrews–Gordon identities Kanade–Russell conjectures

Mathematics Subject Classification

05A17 05A15 11P84

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Notes

Acknowledgements

We thank George E. Andrews, Alexander Berkovich, Karl Mahlburg and Dennis Stanton for useful discussions, suggesting references or terminology during the preparation of the manuscript. The term prestidigitation and the story in the Appendix is due to the historian and my friend Emre Erol of Sabancı University. We also thank the anonymous referees for their time and their suggestions for improvements.

Appendix

Now, let us try to visualize this process with a metaphor.

Imagine a person walking into a fancy cupcake store to taste the delicacies that he heard so much about from his colleagues at work. The cupcakes are neatly arranged in a large display case with one shelf over another. Each shelf has different kinds of cupcakes put into boxes of different sizes. There is certain logic to the way the boxes are displayed. The shelves have boxes with three cupcakes at the first two rows followed by a box with a single cupcake or two cupcakes at the back of the shelves.

The hypothetical cupcake enthusiast starts gazing colorful cupcakes of various types until his eyes are fixated towards a single box with a single cupcake in it. The box is located behind two bigger boxes with three cupcakes in each at a middle shelve as per the logic of display and there is hardly any space for one to grab the box with the single cupcake from the back of the shelf. The cupcake enthusiast is certain of his choice and makes a move towards the box in the back to grab it. The shop owner at the register sees the customer’s move and immediately interrupts him: ‘I am afraid you can’t move the box at the back of the self without my help sir! It’s impossible for you to squeeze your hand through the narrow space between the shelves without ruining the cupcakes.

The cupcake enthusiast stops for a brief moment, listens to the shop owner’s warning and then he confidently keeps moving towards the box with the single cupcake behind the two larger boxes with three cupcakes in each. He thrusts his hand towards the narrow middle shelf and magic happens in the blink of an eye. The customer is able bring both the single-size box and the single cupcake of his choice to the front of the shelf albeit separately. The customer turned out to be a prestidigitator and performed some masterly sleight-of-hand. He retrieved the single cupcake of his choice by relocating it through the two other boxes with three cupcakes. The cupcake was swiftly put in an out of these larger boxes and united at the very front of the shelf with its original box in the end. The impossible became possible under this rare circumstance that allowed different cake to be put in and out of the boxes of three.

The shop owner was awed. He asked if the same trick could be done with another middle shelf that had a box with two cupcakes at the back as well. The cupcake enthusiast tried his trick there too and it worked again! The box of two and the cupcakes are separately delivered to the front while the to cupcakes got in an out of the boxes of three. Not only that, he was able to put back all the boxes that he retrieved from the back of the middle shelf to their original places reversing his trick. The shopkeeper, now amused, decided to offer his cupcakes free of charge to the customer.

Needless to say, each cupcake represents individual numbers and each box represents a free cluster of a particular size in this metaphor. I can only hope that the ‘prestidigitator cupcake enthusiast’s proof of his ‘sleight-of-hand’ would also prove to be as ‘amusing’ for his fellow mathematicians in real life as it does in the metaphor.

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Kağan Kurşungöz
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1.Faculty of Engineering and Natural SciencesSabancı UniversityİstanbulTurkey

Andrews–Gordon Type Series for Kanade–Russell Conjectures

Abstract

Keywords

Mathematics Subject Classification

Notes

Acknowledgements

References

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