PARTE II: REQUERIMIENTOS FORMALES DEL PROCESO DE CONTRATACIÓN
Capítulo 7. REQUERIMIENTOS FORMALES DE LA CONTRATACIÓN
(a) either x > y at all interior lattice points or y > x at all interior lattice points ; and (b) y ≤ x at every lattice point on the path, and
(c) the path never crosses the line y = x.
Solution. (a) The number of paths of this type will be twice the number of good paths from
(1, 0) to (n, n – 1), or 2Cn.
(b) Let A be the point (n, n).
Suppose the origin O(0, 0) is transferred to O′(– 1, 0). The new coordinates are O′(0, 0), O(1, 0), and A(n + 1, n). The number of good paths from O to A, namely, Cn + 1 is equal to the number of paths from O to A in which y ≤ x at every lattice point.
(c) By reflectional symmetry, the required number is twice the number found in (b), or 2Cn + 1.
1.8 GROUP
A non empty set G with a binary operation • defined on it constitutes a group (G, o) if the following four properties hold.
(i) For all x and y in G, x o y is in G. (In multiplicative notation one writes xy instead of x • y) (ii) There exists an identity element e in G such that x o e = e o x = x for all x in G.
(iii) Corresponding to each element x in G, there exists an inverse element x– 1 in G such that
x o x– 1 = x– 1ox = e.
(iv) For every x, y and z in G the elements x • (y • z) and (x • y) • z are identical.
The associativity property (iv) allows us to write x • y • z for the triple product. We usually write a • b as ab and (G, o) as G if there is no risk of ambiguity.
1.8.1 Subgroup
A subset H of G is called a subgroup of (G, o), if (H, o) is a group.
1.8.2 Finite group
If G is a finite set with | G | = n then (G, o) is a finite group of order n.
For example, the symmetric difference of sets A and B is defined by A * B = (A ∪ B) – (A ∩ B)
that is, A * B is the set of elements that belong to A or to B but not to both.
1.8.3 Permutation
Suppose that G is a fixed subgroup of the symmetric group of a finite set X and x is a given element of X.
Let Gx ≡ {g(x) : g ∈ G} Gx≡ {g ∈ G : g(x) = x} F(g) ≡ {z ∈ X : g(z) = z}.
In words, Gx (the orbit of x with respect to G) is the set of all images of the given element x
under the permutations in G ; Gx (the stabilizer of x in G) is the set of all permutations in G that have x as a fixed point ; F(g) (the permutation character of g in X) is the set of all fixed points of a given permutation g ∈ G.
1.8.4 Permutation Groups and Their Cycle Indices
A permutation of a finite set X is a bijective (one-to-one and onto) mapping from X to X. Suppose f is a permutation of X and x is any element of X. Define recursively, f ′(x) ≡ f (x), f 2(x) ≡ f (f ′(x)), ....
f i(x) ≡ f (f i – 1(x)), ... since X is finite, there exists a positive integer r such that f r(x) = x.
The sequence 〈 x, f 1(x), f 2(x), ..., f r – 1(x) 〉 is called a cycle of order (or length) r of the permutation f.
Obviously, every permutation of X can be represented as a composition of k disjoint cycles,
where k is atleast 1 and atmost the cardinality of X.
The concept of the cycle representation of a permutation f of X = {1, 2, ... n}. The following algorithm produces this representation :
(i) Choose an element i of X (usually i = 1). Find the image of i under the mapping f, then the image of the image, then ..., until the image j appears such that f (j) = i. Thus the cycle (i ... j) has been generated.
(ii) Choose an element of X not found in any one of the cycles already generated, and use this element as element i in step (i), thereby generating a new cycle.
(iii) Repeat step (ii) until X has been exhausted.
The cycle representation of a permutation is unique upto the order of the cycles in the composi- tion and upto the choice, within each cycle, of the leading element.
For example, Given X = {1, 2, ..., 8} and 1 2 3 4 5 6 7 8 ⎯⎯→f 3 2 5 1 4 8 6 7 Starting with 1 : f (1) = 3, f (3) = 5, f (5) = 4, f (4) = 1.
Thus we have a cycle of length 4 which may be denoted by (1 3 5 4) [or (3 5 4 1) or (5 4 1 3) or (4 1 3 5)] Starting with 2 : f (2) = 2. We have the cycle (2) of length 1.
Starting with 6 : f (6) = 8, f (8) = 7, f (7) = 6. Now we have a cycle of length 3 which may be denoted by (6 8 7) [or (8 7 6) or (7 6 8)].
The sum of the lengths has reached 4 + 1 + 3 = 8 = | X | which means we are finished : the cycle representation of f is (1 3 5 4) (2) (6 8 7) (or ...).
1.8.5 Weight
Let the cycle representation of f, a permutation of an n set, consist of a1 cycles of length 1, a2 cycles of length 2, ... ai cycles of length i, ... . Then the type of f is the vector [a1 a2 ... an], and the weight of the type is the positive integer W = 1a1 2a2 ... nan.
For example, the permutation of above example, has 1 cycle of length 1, 1 cycle of length 3, and
1 cycle of length 4. The type of this permutation is [1 0 1 1 0 0 0 0]. The weight of this type is 1′ 3′ 4′ = 12.
1.8.6 Cycle index
Let G denote a group, of order m, of permutations of an n-set and length g ∈ G be of type [a1, a2,
Z (g ; x1, x2, ..., xn) ≡ 1 2
1a 2a ... axn
x x x and the cycle index of G is the multinomial.
Z(G ; x1, x2, ..., xn) ≡ G 1 Z ∈
∑
g m (g ; x1, x2, ..., xn)For example, suppose the 4 vertices of a square are labeled 1, 2, 3 and 4, clockwise. A clockwise
rotation through an angle of 0°, 90°, 180° or 270° takes the square into itself. Thus there are 4 circular or cyclic symmetries. In addition, there are 4 dihedral symmetries that are obtained by reflection of the square in the 2 diagonals and in the 2 lines bisecting opposite sides.
Conversely, the symmetries of the square compose a subgroup G of order 8 of S4; the elements of G are as follows :
(i) The permutation induced by rotating the square clockwise through 0° is g1 = e = (1) (2) (3) (4), with cycle index x14.
(ii) The permutation induced by rotating the square clockwise through 90° is g2 = (1 2 3 4) with cycle index x41.
(iii) The permutation induced by rotating the square clockwise through 180° is g3 = (1 3) (2 4), with cycle index x22.
(iv) The permutation induced by rotating the square clockwise through 270° is g4 = (1 4 3 2), with cycle index x41.
(v) The permutation induced by reflection in the line joining the midpoints of 1 2 and 3 4 is g5 =(1 2) (3 4), with cycle index x22.
(vi) The permutation induced by reflection in the line joining the midpoints of 1 4 and 2 3 is g6 = (1 4) (2 3) with cycle index x22.
(vii) The permutation induced by reflection in the diagonal joining corners 2 and 4 is g7 = (2) (4) (1 3), with cycle index x12 x
21.
(viii) The permutation induced by reflection in the diagonal joining corners 1 and 3 is g8 = (1) (3) (2 4), with cycle index x12 x
21.
The cycle index of G is therefore
Z (G ; x1, x2, x3, x4) = 1
8 (x14 + 2x12 x2 + 3x22 + 2x4). 1.8.7 Coloring and equivalent w.r.t. group of permutation
A function f from a finite set X to a finite set of colors Y is called a coloring of X. Two colorings f and φ in the set C of all colorings of X are said to be equivalent (indistinguishable) with respect to a group G of permutations of X if there exists a permuation π in G such that f(x) = φ(π(x)) for all x in X.
In other words, if we attach names to the elements of X, so that G may be considered a group of
‘renamings’, then we do not distinguish between 2 colorings of X that become identical under some renaming in G.
Clearly, the relation of indistinguishability is reflexive, symmetric, and transitive, i.e., an equiva- lence relation.
1.8.8 Pattern
The equivalence classes into which C partitioned by the indistinguishability relation are called the patterns in C (with respect to the group G).
For example, If G = {e} then any 2 colorings are distinguishable, so that the number of patterns
is the number of colorings. Because
Z(G ; x1, x2, ... xn) = Z(e ; x1, x2, ... xn) = x1n.
1.8.9 Pattern Inventory
Let the weight function w map Y into a set of r colors, {w(y1), w(y2), ... w(yr)}. The pattern inventory (of C) with respect to G is the multinomial.
PI (G ; w(y1), w(y2), ... w(yr))
≡ 1 2 1 2 1 2 1 2 .... 0 ( , , ...., ) [ ( )] [ ( )] ...[ ( )] r r i n n n r r n n n n n n n n w y w y w y + + + = ≥ τ
∑
The coefficient τ (n1, n2, ..., nr) gives the number of distinguishable (with respect to G) colorings (= numebr of patterns) that assign color w(y1) to n1 elements of X ; color w(y2) to n2 elements ; ..., color w(yr) to nr elements. The summation is over the sizes of the color classes into which X is divided, the sum consists of C(n + r – 1, r – 1) terms.
1.8.10 Isomorphic group
Groups (G, o) and (G′, o′) are isomorphic (identical in structure) if there exists a one-to-one correspondence f between G and G′ such that f (x o y) = f (x) o′ f (y), for all x and y in G.
1.8.11 Cyclic group
If x is an element in a group (G, o), we write x o x as x2, x o x2 = x2 o x as x3, and so on.
Similary, x– 1 o x– 1 is written as x– 2, x– 1 o x– 2 as x– 3 etc.
Thus the kth power, xk, of the element x is well defined when k is any non-zero integer, we make the natural definition x° ≡ e. The group G is said to be a cyclic group if it contains an element x such that every element of G is a power of x. In this case we say that G is generated by x, and we write G = < x >. If x generates G and if the powers of x are all distinct, G is an infinite cyclic group.
1.8.12 Abelian group
A group (G, o) is abelian if x o y = y o x for every x and y in G.
1.8.13 Order of an element
If x is an element of (G, o) and if there exists a positive integer m such that xm is the identity element e in G, then x is said to be of finite order. If x is of finite order, the smallest positive integer k such that xk = e is the order of x in G.
1.8.14 Direct product
If G and G′ are two groups, the direct product of G and G′ is the set of all ordered pairs G × G′ = {g, g′} : g ∈ G, g′ ∈ G′}
endowed with the binary operation defined by (g1, g1′) (g2, g2′) = (g1g2, g1′g2′).
1.8.15 Left and right coset
If H is a subgroup of G and x is an element of G, the set xH ≡ {xh : h ∈ H} is called the left coset of H with respect to x, the right coset of H with respect to x is Hx ≡ {hx : h ∈ H}.
1.8.16 Conjugate (Permutation)
Two permutations f and g of X are said to be conjugate if there exists a permutation h of X such that hf = gh.
1.8.17 Regular Icosahedron
A regular polytope (a solid in which all faces are congurent polygons and each vertex is incident with the same number of faces) with 12 vertices, 20 faces (congruent equilateral triangles) and 30 edges is called a regular icosahedron.