Stirling Numbers
Jane Go
Introduction to Stirling Numbers
Stirling numbers, which it was named after, was created by James Stirling himself. Stirling was a Scottish
mathematician that was born in May 1692 and passed away in December 15, 1770. His main contributions
are areas in infinite series and infinitesimal calculus. Moreover, his other works include Stirling numbers,
Stirling Permutation, Stirling’s approximation. During his mathematical his encounter with Isaac Newton
led to him proving Newton’s classification of cubics. In 1730, he published a book called the Methodus
differentialis: sive tractatus de summatione et interpolatione serierum infinitarum which ’he
most useful representation of terms was in a series of factorials, positive or negative. Manipulation often
required conversion of factorials into powers, and Stirling gave tables of the coefficients for this conversion.
He then showed that the columns of the tables gave the coefficients for the inverse expressions, of powers in
factorials; those for positive (negative) powers are now called ’Stirling Numbers of the first [or second] kind in
his honor.’ Throughout his lifetime, he wrote many other books and papers that dealt greatly with complex
mathematics and other subjects.
Stirling numbers derive from analytic and combinatoric which is the field of mathematics concerned
with problems of selection, arrangement, and operation within a finite or discrete system problems. There are two different sets of Stirling numbers called: Stirling numbers of the first kind and Stirling
numbers of the second kind. Since Stirling numbers deal with combinatoric problems, we first need to understand what binomial coefficients are which consists of combinations and permutations.
Binomial coefficients are denoted by
n
k
it is interpreted as the ’number of ways of picking K unordered outcomes from n possibilities,’ which are also
known as combinations. Binomial coefficients are usually denoted in two ways
n
k
and
n Ck
it gives the ’number of k subsets possible out of a set of n distinct items.’ For example, we would solve
3
2
by reading it as 2 subset of {1, 2, 3} and distributing it pairs in {1,2} {1,3} {2,3} ways. Therefore,
3
2
is 3. The formula to calculate binomial coefficients is
n
n!
=
n Ck =
k
k!(n − k)!
and n and k are nonnegative.
Another concept that could help us understand the Stirling numbers is binomial distribution which is
closely related to binomial coefficients and combination. Binomial coefficients, hence the root ’bi’, always deal
with two possible outcomes. Some traditional examples of these are coin tossing or rolling a die. When we
throw a coin, we may ask ’what are the chances of getting heads or tails?’ The only possible outcome is either
a head or tail. For a die, we might ask ourselves ’what is the probability of rolling an even number?’ In this
case, we will either get an even number to not an even number. Sometimes calculating these probabilities can
be too time consuming when it comes to bigger numbers, for example, what is the chance of getting five heads
in 9 coin tosses?
3
2
As a result, we have the formula
n
n!
=
k
k!(n − k)!
n is the total number and k is the number we want. We can read this as ’n choose k.’ This formula calculates
the question ’how many.’ On the other hand, when the probability is unequally likely, there are different
approaches for this. For example, if someone is selling a chocolate bar and gum, and 70% of the people have
favored chocolate and the rest have chosen gum, what is the probability of selling three chocolate bars to the
next 4 customers? For this problem we would use this equation
pk
which calculates the probability of choosing three chocolate bars and
(1 − p)n−k
which calculates the probability of selecting gum. When we multiply these together we get
pk (1 − p)n−k
which calculus the ’probability of each.’ When we multiply the ’how many’ equation and ’probability of each’
equation
n!
P (k out of n) =
pk (1 − p)n−k
k!(n − k)!
This equation is the general binomial probability formula.
In combination, order does not matter; in other words, ordering x, y, z is the same as z, y, x. No matter
in what order the object is place in, it looks the same.
C(n,k) =
n!
k!(n − k)!
From the notation, it is choosing k things from a set of n things without replacement. There are two types of
combination: combination with repetition and combination without repetition. An example of combination
with repetition is choosing from different flavors of ice cream, for example, let us say we have flavors of banana,
chocolate, lemon, strawberry, and vanilla. How many different combinations can we get from choosing three
scoops? We denote our options by {b, c, l, s, v} and some ways of combinations are (c, c, ,c) (b, l, v). Since
order does not matter in combination, we can repeat the different flavors of ice cream. Accordingly, we can
think about the flavors as arrows and circles, in other words, c, c, c would be é l l l é é é. When you
choose one flavor of ice cream, you move that arrow to the back of the list. Through this method, we can
count how many different ways to arrange circles and arrows.
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The other type of combination is without repetition; a common example of this would be the lottery.
In lotteries, one has to pick a set of numbers from a set of balls that are numbered from 1 to 50 (number
varies) and has to be the correct set of six numbers. Of course, there is not repetition in lotteries because
when one number gets chosen, that specific number is not placed back into the wheel. One way to explain
how combination without repetition works is by assuming that order does matter (which is permutation) then
thinking that order does not matter.
n!
1
n!
· =
(n − r)! r!
r!(n − r)!
This notation is derived from ’changing the permutation formula to reduce it by how many ways objects could
be in order.’
In permutation, on the other hand, order does matter. It some sense, it can get more complicated than
combination. For example, if we have x,y,z and we order these letters differently, then z,y,x is completely
different than how we had it ordered originally. Another popular example that we are more familiar to us is
the combination lock. The lock should, in fact, be called the permutation since the order in which we put
in the numbers actually matter. However, just like combination, permutation also can have repetition or no
repetition.
With permutations that allow repetition, the way we calculate it is to multiply the number of different
types that many times. For example, if we have 3 three things, we multiply
n· n·n
Since repetitions are allowed, we are essentially not ’canceling’ out anything that we have already used.
Therefore, we have ’n’ possibilities for the first choice, then ’n’ possibilities the next time, and so on. We can
write this exponentially
n · n · n(rtimes) = nr
Therefore, nr is the ’n’ is the number of things to choose from and we choose ’r’ of them while repetition is
allowed and order matters.
Next, there is permutations without repetition. For these, we are essentially eliminating the number of
available choices each time that were used up in earlier choices. For example, if we have a dice and want to in
what order the 6 numbers can be ordered, then we cannot choose the number 4 more than once. Therefore,
the equation would be
6 · 5 · 4 · 3 · 2 · 1· = 720
These numbers could be manipulated in different ways if are choosing different numbers. Instead of writing
the equation in a long form, the mathematical way to write it would be using the factorial function. Therefore,
the example above would be
6! = 720
If we want to choose 3 numbers out of 6 we can write this as
6·5·4
3·2·1
Therefore, the general formula is
n
Pk =
n!
(n − k)!
Stirling Numbers of the First Kind
Introduction of Stirling Numbers of the First Kind
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Now that we have covered combinations and permutations, we can move onto the types of Stirling numbers
which are Stirling Numbers of the First Kind and the Second Kind. Stirling numbers are also defined as the
change from rising power to ordinary power, and ordinary power to falling power. The notation for Stirring
Numbers of the First Kind is [n/k] which is interpreted as n bracket. Other notations for the First Kind
includes c(n,k) or s(n,k) which is the number of permutations of a n element set with exactly k cycles. We
write these permutations in cycle notation. For example, if we have the Stirling Numbers of the First Kind of
s(4,2) which is 11. We write this in cycle notation as
(1)(234), (1)(243), (134)(2), (142)(2), (124)(3), (142)(3), (12)(34), (13)(24), (14)(32)
There are number of ways we can denote Stirling numbers but it is important to note that we note Stirling
Numbers of the First Kind with a small case s and the Second Kind with upper case S. Moreover, the Second
Kind are never negative, while the First Kind can be negative. As a result, there are distinct notations for
these two.
Combinatorial Interpretation of Stirling Numbers of the First Kind
[n/k] denotes the number of permutations of {1,2..n} with exactly k cycles. For example [5,4] can be
graphically represented as
This can essentially be interpreted as having 5 different objects and potentially permuting (moving) them
around into each other’s places.
Number Theoretic Interpretation of Stirling Numbers of the First Kind
Another way to illustrate Stirling Numbers of the First Kind is to use the notation [n/k] which is the sum
of all products of n-k elements from {1,2..n-1}
4
=1·2·3=6
1
4
=1+2+3=6
3
4
= (1 · 2) + (1 · 3) + (2 · 3) = 11
2
5
= (1 · 2 · 3) + (1 · 2 · 4) + (1 · 3 · 4) + (2 · 3 · 4) = 50
2
Introduction of Stirling Numbers of the Second Kind
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S(n,k) is the number of partitions of an n-element set into exactly k non empty subsets. If we take out the
fact that the set can be empty, then there will be sets that are empty without any n elements. An example of
using nonempty set S(4,2) which is 7. This partition is illustrated as
1|234, 2|134, 3|124, 4|123, 12, |34, 13|23, 14|23
A formula to generate the answer is S(n+1,k) = k S(n,k) + S(n, k-1) which satisfies recursion. Therefore, the
total number of partitions of an n element set is the
n
X
S(n, k)
k+1
The Stirling Numbers of the Second Kind is also expressed by standard powers in terms of falling powers.
(write the matrix here) The notation for the Second Kind is n/k = which is read as n brace k (enter notation)
Recurrence Relation
Recurrence relation is important to mention in the topic of Stirling Numbers of the Second Kind since it
follows the rule. Recurrence relation is an equation that defines a sequence of values, most importantly, each
term in the sequence is determined by the one before except the first one that starts the sequence of numbers.
Simply, a sequence is a ordered list of elements. The function is defined by a function of sequence
an−1 , an−2
The recurrence relation that the second kind follows is
nno n n+1
=k
+
k
k
k−1
when k is greater than 0
0
=1
0
nno
and
0
0
=
=0
n
Combinatorial Interpretation of Stirling Numbers of the Second Kind
n/k is the number of partitions (breaking into sub pieces) of n objects into k nonempty subsets. For
example,
4
=7
2
and can be illustrated as
{1, 2}{3, 4}, {1, 3}{2, 4}, {1, 4}{2, 3}, {1, 2, 3}{4}, {1, 3, 4}{2}, {1, 2, 4}{3}, {2, 3, 4}{1}
Looking at this, we can ask ourselves, how many different blocks are there? The notation above is read
as 1 and 2 switching place while 3 and 4 switches place. Since there are four numbers, we need to essentially
divide these numbers into 2 subsets without repetition.
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References
https://betterexplained.com/articles/easy-permutations-and-combinations/
https://www.mathsisfun.com/combinatorics/combinations-permutations.html
http://dlmf.nist.gov/26.8i
http://www.regentsprep.org/regents/math/algtrig/ats5/lcomb.htm
https://www.youtube.com/watch?v=LB68I4mkb8Y
http://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/james-stirling
http://mathworld.wolfram.com/BinomialCoefficient.html
https://www.mathsisfun.com/data/binomial-distribution.html
http://myweb.facstaff.wwu.edu/trenees/Math566/stirling.pdf
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