In order to solve the following tutorial sheet, it helps to solve/understand and then apply the following beautiful cyclic relations or identities:
(Note if these look new to you, then you need to check the truth of all them; if all are v v familiar to you, just go ahead and crack the tutorial sheet below):
Core Identities in Cyclic Expressions:
Solve or simplify the following:
We shall now complicate the situation a little bit. Let us ask for the combinations of the symbols with repetitions of each symbol allowed once more in the combinations. For example, let there be only two symbols . Let us look for combinations of the form:
, , , , , , ,
where, in each combination, each symbol may occur once, twice, or not at all. The OGF for this can be constructed by reasoning as follows: the choices for are not-, once, twice. This is represented by the factor . Similarly, the possible choices for correspond to the factor . So, the required OGF is
On expansion, this gives :
Note that if we omit the term 1 (which corresponds to not choosing any ), the other 8 terms correspond to the 8 different combinations listed in (*). Also, observe that the exponent r of the tells us that the coefficient of has the list or inventory of the r-combinations (under the required specification — in this case, with the restriction on repetitions of symbols) in it:
In the light of the foregoing discussion, let us now take up the following question again: in how many ways, can a total of 16 be obtained by rolling 4 dice once?; the contribution of each die to the total is either a “1” or a “2” or a “3” or a “4” or a “5” or a “6”. The contributions from each of the 4 dice have to be added to get the total — in this case, 16. So, if we write:
as the factor corresponding to the first die, the factors corresponding to the other three dice are exactly the same. The product of these factors would be:
Each term in the expansion of this would be a power of t, and the exponent k of such a term is nothing but the total of the four contributions which went into it. The number of times a term can be obtained is exactly the number of times k can be obtained as a total on a throw of the four dice. So, if is the coefficient of in the expansion, is the answer for the above question. Further, since (*) simplifies to , it follows that the answer for the above question tallies with the coefficient specified in the following next question: calculate the coefficient of in .6
Now, consider the following problem: Express the number of ways of obtaining a total of n by rolling p dice, as a certain coefficient in a suitable product of binomial expansions in powers of t. [ this in turn, is related to the observation that the number of ways a total of 16 can be obtained by rolling 4 dice once is the same as the coefficient of in ]:
So, we get that coefficient of in
Let us take an example from a graphical enumeration:
A is a set V of vertices a, b, c, …, together with a set of If is considered the same as , we say the graph is . Otherwise, the graph is said to be , and we say ‘ has a direction from a to b’. The edge is called a loop. The graph is said to be of order .
If the edge-set E is allowed to be a multiset, that is, if an edge is allowed to occur more than once, (and, this may be called a ‘multiple edge’), we refer to the graph as a general graph.
If and denote the numbers of undirected (respectively, directed) loopless graphs of order 5, with n edges, none of them a multiple edge, find the series and .
Applying our recently developed techniques to the above question, a graph of 5 specified vertices is uniquely determined once you specify which pairs of vertices are ‘joined’. Suppose we are required to consider only graphs with 4 edges. This would need four pairs of vertices to be selected out of the total of equal to 10 pairs that are available. So selection of pairs of vertices could be made in ways. Each such selection corresponds to one unique graph, with the selected pairs being considered as edges. More informally, having selected a certain pairs of vertices, imagine that the vertices are represented by dots in a diagram and join the vertices of each selected pair by a running line. Then, the “graph” becomes a “visible” object. Note that the number of graphs is just the number of selections of pairs of vertices. Hence, .
Or, one could approach this problem in a different way. Imagine that you have a complete graph on 5 vertices — the “completeness” here means that every possible pair of vertices has been joined by an edge. From the complete graph which has 10 edges, one has to choose 4 edges — any four, for that matter — in order to get a graph as required by the problem.
On the same lines for a directed graph, one has a universe of 10 by 2, that is, 29 edges to choose from, for, each pair x,y gives rise to two possible edges and . Hence,
Thus, the counting series for labelled graphs on 5 vertices is
and the counting series for directed labelled graphs on 5 vertices is
Finally, the OGF for increasing words on an alphabet with is
The corresponding OE is which is nothing but (this explains the following problem: Verify that the number of increasing words of length 10 out of the alphabet with is the coefficient of in ).
We will continue this detailed discussion/exploration in the next article.
Until then aufwiedersehen,
Thanks Dr. Gowers’. These are invaluable insights into basics. Thanks for giving so much of your time.
Reference: Nordic Mathematical Contest 1987, R. Todev:
Let f be a function, defined for natural numbers, that is strictly increasing, such that values of the function are also natural numbers and which satisfies the conditions and for all natural numbers m and n. Define the smallest possible value of a.
Since, is a function satisfying the conditions of the problem, the smallest possible a is at most 4. Assume that . It is easy to prove by induction that for all . So, taking into account that f is strictly increasing, we get
as well as .
So, we arrive at . But, this is not possible, since is an integer. So, .
If , find the least value of .
will be minimum when will be minimum.
Now, ….call this equation I.
Hence, z will be maximum when is maximum but is the product of two factors whose sum is .
Hence, will be maximum when both these factors are equal, that is, when
. From equation I, maximum value of . Hence, the least value of .
Some basics related to maximum and minimum:
Let a and b be two positive quantities, S their sum and P their product; then, from the identity:
, we have
Hence, if S is given, P is greatest when ; and if P is given, S is least when . That is, if the sum of two positive quantities is given, their product is greatest when they are equal; and, if the product of two positive quantities is given, their sum is least when they are equal.
To find the greatest value of a product the sum of whose factors is constant.
Let there be n factors , and suppose that their sum is constant and equal to s.
Consider the product , and suppose that a and b are any two unequal factors. If we replace the two unequal factors a and b by the two equal factors , the product is increased, while the sum remains unaltered; hence, so long as the product contains two unequal factors it can be increased without altering the sum of the factors; therefore, the product is greatest when all the factors are equal. In this case, the value of each of the n factors is , and the greatest value of the product is , or .
Corollary to Basic 2:
If are unequal, ;
that is, .
By an extension of the meaning of the terms arithmetic mean and geometric mean, this result is usually stated as follows: the arithmetic mean of any number of positive quantities is greater than the geometric mean.
To find the greatest value of when is constant; m,n, p, ….being positive integers.
Solution to Basic 3:
Since m,n,p, …are constants, the expression will be greatest when is greatest. But, this last expression is the product of factors whose sum is , or , and therefor constant. Hence, will be greatest when the factors are all equal, that is, when
Thus, the greatest value is .
Some examples using the above techniques:
Show that where r is any real number.
that is, , which is the desired result.
Find the greatest value of for any real value of x numerically less than a.
The given expression is greatest when is greatest; but, the sum of the factors of this expression is , that is, ; hence, is greatest when , that is, . Thus, the greatest value is .
The determination of maximum and minimum values may often be more simply effected by the solution of a quadratic equation than by the foregoing methods. For example:
Divide an odd integer into two integral parts whose product is a maximum.
Let an odd integer be represented as ; the two parts by x and ; and the product by y; then ; hence,
but the quantity under the radical sign must be positive, and therefore y cannot be greater than , or, ; and since y is integral its greatest value must be ; in which case , or n; thus, the two parts are n and .
Sometimes we may use the following method:
Find the minimum value of .
Put ; then the expression
which in turn equals
Hence, the expression is a a minimum when the square term is zero; that is when .
Thus, the minimum value is , and the corresponding value of x is .
Problems for Practice:
- Find the greatest value of x in order that may be greater than .
- Find the minimum value of , and the maximum value of .
- Show that and .
- Find the maximum value of when x lies between 7 and -2.
- Find the minimum value of .
If , , be the roots of the cubic equation . Prove that the equation in y whose roots are is obtained by the transformation . Hence, form the equation with above roots.
Given that , , are the roots of the equation:
…call this equation I.
By relationships between roots and co-efficients, (Viete’s relations), we get
and , and
Now, , that is,
, or …call this equation II.
Subtracting Equation II from Equation I, we get
which is the required transformation.
Now, , that is,
Putting this value of x in Equation I, we get
, that is,
, which is the required equation.
Some classic examples are presented below to illustrate the use of mathematical induction to prove inequalities:
Prove the inequality for all positive integers n.
Let be the proposition that .
is true, because . This completes the basis step.
We first assume the inductive hypothesis that is true for the positive integer k. That is, the inductive hypothesis is the statement that . To complete the inductive step, we need to show that if is true, then , which is the statement that is also true. That is, we need to show that if , then . To show that this conditional statement is true for the positive integer k, we first add 1 to both sides of and then note that . This tells us that
This shows that is true; namely, that , based on the assumption that is true. The induction step is complete.
Therefore, because we have completed both the basis step and the inductive step, by the principle of mathematical induction we have shown that is true for all positive integers n. QED.
Prove that for every positive integer n with . (Note that this inequality is false for ).
Let be the proposition that
To prove the inequality for requires that the basis step be . Note that is true because .
For the inductive step, we assume that is true for the positive integer k with . That is, we assume that with . We must show that under this hypothesis, is also also true. That is, we must show that if for the positive integer , then . We have
(by definition of exposition)
which (by the inductive hypothesis)
which equals (by definition of factorial function).
This shows that is true when is true. This completes the inductive step of the proof.
We have completed the basis step and the inductive step. Hence, by mathematical induction is true for all integers n with . That is, we have proved that is true for all integers n with . QED.
An important inequality for the sum of the reciprocals of a set of positive integers will be proved in the example below:
An inequality for Harmonic Numbers
The harmonic numbers are defined by .
For instance, . Use mathematical induction to show that , whenever n is a nonnegative integer.
To carry out the proof, let be the proposition that .
is true because .
The inductive hypothesis is the statement that is true, that is, , where k is a nonnegative integer. We must show that if is true, then , which states that , is also true. So, assuming the inductive hypothesis, it follows that
(by the definition of harmonic number)
which equals (by the definition of the harmonic number)
(by the inductive hypothesis)
(because there are terms each greater than or equal to )
(canceling a common factor of in second term), which in turn, equals .
This establishes the inductive step of the proof.
We have completed the basis step and the inductive step. Thus, by mathematical induction is true for all nonnegative integers. That is, the inequality for the harmonic number is valid for all non-negative integers n. QED.
The inequality established here shows that the harmonic series is a divergent infinite series. This is an important example in the study of infinite series.
Reference: Discrete Mathematics and its Applications by Kenneth H. Rosen, Seventh Edition.