On Wild Frieze Patterns

ABSTRACT

In this article, among other things, we show: (1) There are periodic wild -frieze patterns whose entries are positive integers. (2) There are non-periodic -frieze patterns whose entries are positive integers. (3) There is an -frieze pattern whose entries are positive integers and with infinitely many different entries.

KEYWORDS:

• frieze pattern
• -frieze
• wild friezes

2010 AMS SUBJECT CLASSIFICATION:

• 05E
• 13F60

Acknowledgments

I would like to thank C. Bessenrodt, T. Holm, P. Jørgensen, and S. Morier-Genoud for calling my attention to some of the questions addressed here and for many further valuable comments. I am also very grateful to the referee for suggesting Example 1.2 and for many other useful remarks.

Notes

1 There are 5, 51, 868 tame integral positive -friezes of heights 1,2,3 resp. The proof that there are exactly 868 -friezes of height 3 is in an unpublished manuscript by Plamondon and the author.