A neat solution

We show that the generating function approach leads to a neat formula for the number of ordered bounded shapes.

Define

Thus #_n = c_n= gn⁽ⁿ⁾(0)

A size-partition can be interpreted as a bounded-shape partition with uniform bounds Li = 1, Ui = n. Then

Corollary 2.2.2. The number of ordered size-partitions is ¡_n−1

p−1

Chapter 3

The Bounded-shape

Sum-partition Problem in R¹ with Schur Convex Objective Function

In this chapter, we consider the bounded-shape sum-partition problem in R¹ with Schur convex objective function. We will show that the θ_i’s are one-sided, one can restrict attention to (reverse) size-consecutive partitions with a nonmajorized shape. As a (reverse) size-consecutive partition with a given shape is easy to determine(see (1.2.5) and (1.2.6)), the problem of finding an optimal partition is reduced to the task of identifying a set of shapes that contains all nonmajorized ones. Since Schur convex functions are symmetric, they do not differentiate between partitions that are obtained by part-permutations as long as the corresponding coordinate-permutations of the shapes are feasible. Thus, we may restrict attention to representatives of shape-types which are the equivalence classes of shapes with respect to coordinate-permutations. We will study nonmajorized shapes, bound their numbers and develop algorithms to enumerate them, too. Our study extends the analysis of a previous paper [15] which discussed the above problem assuming the existence of a majorizing shape.

3.1 Nonmajorized shapes

We explore the relation between shape-majorization and the optimization problem with Schur-convex objective function over partitions introduced in the Introduction. In particular, we explore the role of nonmajorized shapes, with respect to Γ(L, U ).

Corollary 3.1.1. Suppose f and Γ are as in Theorem 1.2.1, but no majoriz-ing shape exists.

(a) If θ_i ≥ 0 for i = 1, . . . , n, then there is a nonmajorized shape in Γ such that any corresponding size-consecutive partition is optimal.

(b) If θ_i ≤ 0 for i = 1, . . . , n, then there is a nonmajorized shape in Γ such that any corresponding reverse-size-consecutive partition is optimal.

Corollary 3.1.1 implies that when f is Schur convex and the θ_i’s are one-sided, it suffices to restrict attention to (reverse) size-consecutive par-titions whose shapes are nonmajorized. Of course, the symmetry of Schur convex functions implies that all size-consecutive partitions with the same shape have the same objective value F (as determined by (1.2.4)). We con-clude that the underlying optimization problem can be solved by obtaining a list that contains all nonmajorized shapes, determining corresponding size-consecutive partitions, and evaluating the right-hand side of (1.2.4) for each one of them to select the best. Further, it suffices to consider only repre-sentatives of all nonmajorized shape-types. The remainder of our paper will focus on studying and identifying nonmajorized shapes and shape-types with respect to sets of the form Γ(L, U ).

In the bounded-shape case which the majoring shape doesn’t exist [4], consider a vector a ∈ IR^p and J ⊆ {1, . . . , p}, let a_J denote the subvector of a consisting of the coordinates indexed by J.

Lemma 3.1.2. Consider vectors a and b in IR^p with P^p

i=1

a_i =P^p

i=1

b_i and a set J ⊆ {1, . . . , p} for which a_i = b_i for each i ∈ {1, . . . , p} \ J. Then

[a_J majorizes b_J] ⇔ [a majorizes b]; (3.1.1)

further (3.1.1) holds with “majorizes” replaced by “strictly majorizes”.

The freedom in selecting H and k allows us to conclude that aJ majorizes bJ.

The strict version of (3.1.1) follows directly from the weak version and the observation that a vector u strictly majorizes another vector v if and only if u majorizes v and v does not majorize u.

Lemma 3.1.2 will be used particularly with sets J consisting of two ele-ments.

Throughout the remainder of this section, let L and U be nonnegative integer p-vectors that satisfy (1.1.1)–(1.1.2). In particular, we refer to a nonmajorized shape under Γ(L, U ) as a nonmajorized shape. We next explore the properties of such shapes.

Lemma 3.1.3. Consider the following properties of a shape s = (n₁, . . . , n_p):

(a) s is nonmajorized;

(b) there exist no distinct i and j such that

L_j < n_i < U_i and L_j < n_j < U_i, (3.1.2) (c) if for distinct i and j, L_j < n_j and n_i < U_i, then n_i < n_j; and

(d) there exists at most one index i with L_i < n_i < U_i. Then (a) ⇒ (b) ⇒ (c) ⇒ (d).

Proof. (a) ⇒ (b). Suppose n_i and n_j satisfy (3.1.2) where i 6= j. Without loss of generality, assume that n_i ≥ n_j. Then s is majorized by the shape obtained from s by increasing n_i to max{n_i, n_j} + 1, and decreasing n_j to min{n_i, n_j} − 1 (see Lemma 3.1.2).

(b) ⇒ (c). Suppose condition (b) holds, and i and j are indices satisfying L_j < n_j, n_i < U_i and i 6= j. By condition (b), either L_j ≥ n_i or n_j ≥ U_i. In the former case, n_i ≤ L_j < n_j and in the latter case n_j ≥ U_i > n_i.

(c) ⇒ (d). Suppose condition (c) holds, and i and j are indices satisfying L_i < n_i < U_i, L_j < n_j < U_j and i 6= j. We will establish a contradiction.

Indeed, if n_i ≥ n_j we get a direct violation of (c) and if n_i < n_j we get a violation of (c) with the roles of i and j reversed.

The following examples shows that condition (b) of Lemma 3.1.3 does not imply condition (a).

Example 3. Let U = (5, 5, 5, 2), L = (1, 4, 3, 1), s = (5, 4, 3, 1) and s⁰ = (2, 5, 5, 1). It is easy to verify that s is majorized by s⁰. To see that there exist no i and j satisfying (3.1.2), observe that the only coordinate of s that is strictly larger than the lower bound is the first one, so if (3.1.2) is satisfied necessarily j = 1. But, n₁ is not strictly below any upper bound.

For a given shape s, call part i an upper part, a middle part or a lower part if, respectively, n_i = U_i, L_i < n_i < U_i, n_i = L_i. If part i has L_i = U_i, each shape (n₁, . . . , n_p) ∈ Γ(L, U ) has n_i = L_i = U_i. Thus, in search of nondominated shapes under (L, U ), one can ignore such parts. Of course, when L ¿ U (i.e., L_i < U_i for each i), the parts are classified uniquely.

Lemma 3.1.3 shows that a nonmajorized shape can have at most one middle part.

Suppose L ¿ U. Given a shape s = (n₁, . . . , n_p), let B(s) stand for the p-vector whose elements are the symbols L, M and U constructed in the following way: For a permutation i₁, . . . , i_p of the coordinates for which n_i1 ≥ n_i2 ≥ · · · ≥ n_ip, let B(s)_t for t = 1, . . . , p be L, M, U according to i_t being an upper, middle or lower part. The next result shows that no ambiguity can arise in the definition of B(s), i.e., it is uniquely defined, and that B(s) has a simple structure.

Lemma 3.1.4. Suppose L ¿ U and s = (n₁, . . . , n_p) is a nonmajorized shape. Let (i₁, . . . , i_p) be a permutation of (1, . . . , p) such that n_i1 ≥ n_i2 ≥

· · · ≥ n_ip. Then:

(a) n_ir = n_it for r, t ∈ {1, . . . , p} implies i_r and i_t are either both upper parts or both lower parts.

(b) B(s) has the form (U, . . . , U, M, L, . . . , L) or (U, . . . , U, L, . . . , L).

Proof. (a) If n_ir = n_it, i_r is a lower-part and i_t is not, then L_it < n_it = n_ir = L_ir < U_ir, in contradiction to implication (a) ⇒ (b) in Lemma 3.1.3.

A similar argument applies to prove that if i_r is an upper-part, so is i_t. (b) The implication (a) ⇒ (c) in Lemma 3.1.3 assures that if n_j = U_j > L_j and n_i < U_i, then n_i < n_j, and that if n_i = L_i < U_i and n_j > L_j, then

n_i < n_j. It follows that for every permutation i₁, . . . , i_p of 1, . . . , p with n_i1 ≥ · · · ≥ n_ip and r, t ∈ {1, . . . , p}

[n_ir = U_ir and n_it < U_it] ⇒ [r < t]

and

[n_it = L_it and n_ir > L_ir] ⇒ [r < t].

These implications establish the asserted structure of B(s).

We conclude this section with an observation about a necessary difference between two nonmajorized shapes.

Lemma 3.1.5. Two distinct nonmajorized shapes s = (n₁, . . . , n_p) and s⁰ = (n⁰₁, . . . , n⁰_p) must differ in at least two coordinates; further, if such s and s⁰ differ in exactly two coordinates, say coordinates i and coordinate j, where n_i > n⁰_i, then s⁰ is obtained from s by permuting these coordinates,

n_i = U_i or n_j = L_j (3.1.3) and

n⁰_i = Li or n⁰_j = Uj. (3.1.4) Proof. Suppose shapes s and s⁰ differ in only one part, then P

in_i 6=P

in⁰_i, contradicting the fact that both are shapes and their coordinate sum is n.

Next, assume that s = (n₁, . . . , n_p) and s⁰ = (n⁰₁, . . . , n⁰_p) are nonma-jorized shapes that differ only in coordinates i and j. As neither strictly dominates the other (they are nonmajorized), we have that s⁰ is obtained from s by permuting two coordinates, say coordinates i and j. Now, suppose n_i < n⁰_i = n_j. As L_j ≤ n_j = n⁰_i < n_i ≤ U_i, the implication (a) ⇒ (b) in Lemma 3.1.3 assures that either n_i = U_i or n_j = L_j, and (applying the result on s⁰ with the roles of i and j reversed), either n⁰_j = U_j or n⁰_i = L_i.

We say that two shapes are equivalent if one is obtained from the other by coordinate-permutation. Of course, not all coordinate-permutations of a shape in Γ(L, U ) are necessarily in Γ(L, U ).

Corollary 3.1.6. If s and s⁰ are nonmajorized shapes which are not equiv-alent, then they differ in at least 3 coordinates.