ifying Weak
4-Compositions
In IR terms, the calculation (or determination) of the number of weak 4-compositions for
N that satisfy the conditionr1s0 > r0s1 (analogous to the conditionp > t) is equivalent to finding the number of collections, over all the possible collections of N documents, that satisfy the condition r1s0 > r0s1.
There are two basic ways that one can go about developing arithmetic expressions that compute the number of qualifying weak 4-compositions of N. One is direct, the other is indirect. The direct way concentrates on how to develop expressions for counting the number of weak 4-compositions of N that satisfy the restriction that is denoted by
r1s0 > r0s1. Satisfaction of Constraints 3.3.12 to 3.3.14 on page 94 is implicit in the definition of 4-compositions of N. The indirect way makes use of the fact that
C4(N) = Card4C(N, r1s0 < r0s1) + Card4C(N, r1s0 =r0s1) + Card4C(N, r1s0 > r0s1),
and that it is sometimes easier to calculate the number of weak 4-compositions that satisfy the restriction r1s0 = r0s1 than it is to compute the number that satisfy r1s0 > r0s1. The notation Card4C(N,restriction) denotes the number of 4-compositions of N after the set of 4-compositions of N has been subsetted by the condition that is denoted by
restriction. If, for any N, we can prove that the number of weak 4-compositions of N that satisfyr1s0 < r0s1 is equal to the number of weak 4-compositions of N that satisfies
r1s0 > r0s1, then the number of weak 4-compositions for N can be re-expressed as
C4(N) = 2·Card4C(N, r1s0 > r0s1) + Card4C(N, r1s0 =r0s1).
This means that if we can compute the number of weak 4-compositions of N that satisfy
r1s0 =r0s1, then we can determine the number of weak 4-compositions of N that satisfy
r1s0 > r0s1 rather easily. The next lemma establishes that the number of weak 4- compositions that satisfy the condition r1s0 > r0s1 in an N document collection for a query q has the same value as the number of weak 4-compositions that satisfy the condition r1s0 < r0s1. This helps us to prove that once we have a way to determine the number of weak 4-compositions that satisfyr1s0 =r0s1,we also have a way to determine how many satisfy the condition r1s0 > r0s1. This fact is useful later on in this chapter. Lemma 5.2.1. The number of weak 4-compositions of N that satisfies r1s0 > r0s1 is equal to the number of weak 4-compositions of N that satisfies r1s0 < r0s1.
Proof. Any weak 4-composition ofN that satisfiesr1s0 > r0s1 can be represented by a 4- tuple where the first component contains the value forr1, the second component contains the value for s0, the third component contains the value for r0, and the last component contains the value fors1. Let a, b, c, and drepresent an instance of respective values for these components such that these values satisfy r1s0 > r0s1. For any particular instance of a 4-tuple, say, (a, b, c, d), that is in the set of weak 4-compositions of N that satisfy
r1s0 > r0s1, the instances (a, b, d, c), (b, a, c, d), and (b, a, d, c) also satisfy this relation because the products of the values of the first 2 and last 2 components of each instance are a×b and c×d, respectively.
into an instance that satisfies r1s0 < r0s1 by the following two actions: interchange the value of its first component with that of its third component, then interchange the value of its second component with that of its fourth component. This yields the 4-tuples (a, b, c, d), (a, b, d, c), (b, a, c, d), and (b, a, d, c), from the 4-tuples (c, d, a, b), (d, c, a, b), (c, d, b, a), and (d, c, b, a), respectively. By definition of the set of weak 4-compositions forN, the former set of 4-tuple instances are also members of this set. Furthermore, the cardinality of the former set is exactly the same as that of the latter set.
5.3
The Number of Distinct 2-Partitions
Compositions, both weak and strong, are an integral part of the many formula derivations that occur later in this chapter. The two equations for determining the number of weak and strong compositions appear in Chapter 2, starting on page 26. In addition to needing these equations, we also need an equation that determines the number of 2-partitions for the derivations that occur later in Section 5.11.2 and Section 5.11.3 in this chapter.
Partitions and compositions are closely related. In fact, to a large degree, each of these mathematical structures can be defined in terms of the other one. A partition of a positive integer n is an unordered sum of positive integers that has the value n. By contrast, compositions are ordered sums of positive integers and weak compositions are ordered sums of natural numbers. By convention in mathematics, the value of partition parts are listed in non-increasing order. For example, when viewed as unordered sums, the seven partitions of the number 5 are:
5 4 + 1 3 + 2
3 + 1 + 1 2 + 2 + 1 2 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1.
There is one partition with 1 part, two partitions with 2 parts, two partitions with 3 parts, one partition with 4 parts, and one partition with 5 parts.
In this dissertation, our interest was in the number of distinct 2-partitions of n. A 2-partition ofn is a partition ofn that has exactly 2 parts. The distinct 2-partitions of n
can be obtained from the associated 2-partitions by simply discarding the, at most one, 2-partition where both parts have the same value. For example, the 2-partitions of the number 5 are:
4 + 1 3 + 2.
These are also the distinct 2-partitions of 5 because both parts of the sum have different values. As another example, consider the 2-partitions of the number 6:
5 + 1 4 + 2 3 + 3.
we discard this partition, we obtain the two distinct 4-partitions of 6, that is,
5 + 1 4 + 2.
In partition theory, the partition function Q(n, k) denotes the number of distinct k- partitions for the positive integer n. The general formula for this particular function is defined in terms of the partition function P(n, k), which is recursive in nature, and denotes the number ofk-partitions ofn.The work in this chapter does not need the use of the general version of eitherQ(n, k) orP(n, k),which is good, because relatively simple, and closed form, versions of these functions only exist for very small (e.g., k=1,2,3,4) values of k. Based on the work in Comtet (1974), the partition function Q(n,2), where
n≥1, can be defined as:
Q(n,2) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ n−2 2 , if n is even; n−1 2 , otherwise.
With the use of the greatest integer function, it can be defined even more succinctly as
Q(n,2) = n−1 2 .