# Residues modulo powers of two in the Young-Fibonacci lattice

## TL;DR

This paper investigates the structure of a subgraph in the Young-Fibonacci lattice induced by elements with odd f-statistics, revealing it forms a binary tree with residues equidistributed modulo powers of two, linking combinatorics and number theory.

## Contribution

It demonstrates that the subgraph of the Young-Fibonacci lattice with odd f-statistics is a binary tree and establishes residue equidistribution modulo powers of two, connecting combinatorics with number theory.

## Key findings

- The subgraph induced by elements with odd f-statistics is a binary tree.
- Residues of f-statistics in this tree are equidistributed modulo powers of two.
- Number theoretic results on residues of products of odd numbers are derived.

## 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.