Combinatorics for Metrical Feet

Halle & Vergnaud (1987) propose a convention on the parsing of elements into metrical feet — the Exhaustivity Condition — that requires all elements to belong to some foot, except for certain principled cases of extrametricality. However, the general consensus now prevailing is that even internal metrical elements can remain unparsed, failing to belong to any foot, generalizing the notion of extrametricality. Hayes (1995), Halle & Idsardi (1995), and Kager (1999), among many others, explicitly reject the Exhaustivity Condition. Hayes’s comments are given in (1), Halle & Idsardi’s are given in (2).


Introduction
propose a convention on the parsing of elements into metrical feet -the Exhaustivity Condition -that requires all elements to belong to some foot, except for certain principled cases of extrametricality.However, the general consensus now prevailing is that even internal metrical elements can remain unparsed, failing to belong to any foot, generalizing the notion of extrametricality.Hayes (1995), Halle &Idsardi (1995), andKager (1999), among many others, explicitly reject the Exhaustivity Condition.Hayes's comments are given in (1), Halle & Idsardi's are given in (2).
(1) "The upshot seems to be that in our present state of knowledge, it would be aprioristic to adhere firmly to a rigid principle of exhaustive prosodic parsing […]." (Hayes 1995: 110) (2) "We also deviate from previous metrical theories by not requiring exhaustive parsing of the sequence of elements, that is we do not require that every element belong to some constituent […]." (Halle & Idsardi 1995: 440) In this squib I will prove that the number of possible metrical parsings into feet under these assumptions for a string of n elements is Fib(2n) where Fib(n) is the n th Fibonacci number.

Initial Observations
Disregarding prominence relations within the feet (that is, headedness), the possible footings for strings up to a length of three elements are shown in (3).Feet are indicated here by matching parentheses; elements not contained within parentheses are unfooted (that is, 'unparsed' in Optimality Theory terminology).
3 elements, 13 possible parsings: The number of possible footings is equal to every other member of the Fibonacci sequence, illustrated and defined as a recurrence relation in (4); see, for example, Cameron (1994).

Proof
Let f(n) be the number of parsings of a string of n elements into metrical feet, not subject to the Exhaustivity Condition.We can derive a recurrence relation for the number of metrical feet in a string of length n+1 by dividing the string after the places where an initial foot could occur, as shown in ( 5).

A Corollary
Given the above proof, substituting into the footing recurrence relation gives ( 14). (

Conclusion
The number of non-exhaustive parsings of n elements into metrical feet (i.e. the number of non-exhaustive partitions of n elements) has been proven to be equal to Fib(2n), the 2n th Fibonacci number.