Proofs from THE BOOK

Abstract. In this paper we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices Lu = 1 0 u 1 and Rv = 1 v 0 1 , where u and v are nonnegative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numeratordenominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree.

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“…. , 2 n − 1, we have c(n, i + 1) = 1 2[c(n, i)] + 1 − c(n, i) (1) where [x] denotes the integer part of x.…”

Section: Introductionmentioning

confidence: 99%

The (u,v)-Calkin–Wilf forest

Han

Masuda

Singh

et al. 2016

Int. J. Number Theory

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“…This proves that S(A) is an antichain. Now, by a theorem of Sperner [1], [13], the largest size of an antichain in a set of m elements is …”

Section: Claim There Exists An Edge Uv Such That Eithermentioning

confidence: 99%

Strict Sign-Central Matrices

Brualdi¹,

Dahl²

2015

SIAM J. Matrix Anal. & Appl.

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“…A proof of Turán's eorem 8.11 can be found in Proofs from THE BOOK, 1998 [4]. A somewhat weaker version of eorem 8.13 appears in Hagit Attiya, Faith Ellen Fich, and Yaniv Kaplan, "Lower Bounds for Adaptive Collect and Related Objects," PODC, 2004 [13].…”

Section: Proofmentioning

confidence: 99%