Residues modulo powers of two in the Young-Fibonacci lattice

Abstract

We study the subgraph of the Young-Fibonacci graph induced by elements with odd f-statistic (the f-statistic of an element w of a differential graded poset is the number of saturated chains from the minimal element of the poset to w). We show that this subgraph is a binary tree. Moreover, the odd residues of the f-statistics in a row of this tree equidistibute modulo any power two. This is equivalent to a purely number theoretic result about the equidistribution of residues modulo powers of two among the products of distinct odd numbers less than a fixed number.