Why Is Pi Less Than Twice Phi?

Abstract

We give a proof of the inequality in the title in terms of Fibonacci numbers and Euler numbers via a combinatorial argument and asymptotics for these numbers. The result is motivated by Sidorenko’s theorem on the number of linear extensions of a partially ordered set and its complement. We conclude with some open problems.

Key words:

• Primary 05A20
• Secondary 05A05
• 05A16

Acknowledgments

We are grateful to Alexey Glazyrin, Richard Stanley, and Damir Yeliussizov for helpful conversations, and to Jordan Ellenberg for suggesting Exercise 8. We thank the anonymous referees for helpful suggestions and comments on previous versions of the article, including suggesting Exercise 6. We also thank Alexia Guuinic for the photo used in Figure 1. The first author was partially supported by an AMS–Simons travel grant. The second author was partially supported by NSF grant DMS:1363193. The third author was partially supported by NSF grant DMS:1500834.

Additional information

Notes on contributors

Alejandro H. Morales

Alejandro H. Morales is an Assistant Professor at the University of Massachusetts, Amherst. He was a Hedrick Assistant Adjunct Professor at the University of California, Los Angeles, and a CRM–ISM postdoctoral fellow at the Université du Québec à Montréal. He got his Ph.D. from MIT and did undergraduate studies at the University of Waterloo. He is originally from Colombia.

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 02001

[email protected]

Igor Pak

Igor Pak is a Professor in the Mathematics Department at the University of California, Los Angeles. Before coming to UCLA, he moved around proving theorems all over the world. He held long-term positions at University of Minnesota, MIT, and Yale University. He got his Ph.D. from Harvard University and did undergraduate studies at Moscow State University. He is originally from Russia.

Department of Mathematics, University of California, Los Angeles, CA 90095

[email protected]

Greta Panova

Greta Panova is an Associate Professor at the University of Pennsylvania and was a von Neumann Fellow at the Institute for Advanced Study during the 2017–2018 academic year. She was a Simons Postdoctoral Fellow at UCLA. She obtained her Ph.D. in mathematics in 2011 from Harvard and did undergraduate studies at MIT. She is originally from Bulgaria.

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104

Institute for Advanced Study, Princeton, NJ 08540

[email protected]