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Topic: Fibonacci, compositions, history
Replies: 6   Last Post: Feb 21, 2012 10:22 AM

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Brian Hopkins

Posts: 9
Registered: 12/4/04
Fibonacci, compositions, history
Posted: Mar 27, 2011 1:48 PM
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There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets):

a) compositions with parts from {1,2}
(e.g., 2+2 = 2+1+1 = 1+2+1 = 1+1+2 = 1+1+1+1)

b) compositions that do not have 1 as a part
(e.g., 6 = 4+2 = 3+3 = 2+4 = 2+2+2)

c) compositions that only have odd parts
(e.g., 5 = 3+1+1 = 1+3+1 = 1+1+3 = 1+1+1+1+1)

The connection between (a) & the Fibonacci numbers traces back to the analysis of Vedic poetry in the first millennium C.E., at least (Singh, Hist. Math. 12, 1985). I believe Cayley is responsible for the connection to (b). Who first established the connection with (c), odd-part compositions? It was known by the early years of the Fibonacci Quarterly (1960s), but I suspect it was done before that. Thanks for any assistance, especially with citations. --Brian Hopkins

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