Enumeration of paths in Young--Fibonacci graph

TL;DR

This paper derives a polynomial formula for counting paths between vertices in the Young--Fibonacci graph, a structure related to Young lattices and modular lattices, enhancing understanding of its combinatorial properties.

Contribution

It provides the first explicit polynomial formula for path enumeration in the Young--Fibonacci graph, advancing combinatorial analysis of this lattice structure.

Findings

Derived a polynomial formula for path counts

Established connections to Young lattice structures

Enhanced understanding of Young--Fibonacci graph properties

Abstract

The Young--Fibonacci graph is the Hasse diagram of one of the two (along with the Young lattice) 1-differential graded modular lattices. This explains the interest to path enumeration problems in this graph. We obtain a formula for the number of paths between two vertices of the Young--Fibonacci graph which is polynomial with respect to the minimum of their ranks.