Young–Fibonacci lattice

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Young–Fibonacci graph and Young–Fibonacci lattice, named after <a href="/facts/Alfred_Young_(mathematician)/L3R6cM6D">Alfred Young</a> and <a href="/facts/Leonardo_Fibonacci/d5rA6p8c">Leonardo Fibonacci</a>, are two closely related structures involving sequences of the digits 1 and 2. Any digit sequence of this type can be assigned a rank, the sum of its digits: for instance, the rank of 11212 is 1 + 1 + 2 + 1 + 2 = 7. As was already known in ancient India, the number of sequences with a given rank is a <a href="/facts/Fibonacci_number/mqxpAUQD">Fibonacci number</a>. The Young–Fibonacci lattice is an infinite <a href="/facts/Modular_lattice/CLe2aDq7">modular lattice</a> having these digit sequences as its elements, compatible with this rank structure. The Young–Fibonacci graph is the <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> of this lattice, and has a vertex for each digit sequence. As the graph of a modular lattice, it is a <a href="/facts/Modular_graph/dqqxdlct">modular graph</a>.
The Young–Fibonacci graph and the Young–Fibonacci lattice were both initially studied in two papers by Fomin (1988) and Stanley (1988). They are named after the closely related <a href="/facts/Young%2527s_lattice/X90jpt0r">Young's lattice</a> and after the Fibonacci number of their elements at any given rank.

Young–Fibonacci lattice open-in-new

Young–Fibonacci lattice