Stirling Numbers Of The First Kind

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the unsigned Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one).

The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers.

Definitions

Definition by algebra

The Stirling numbers of the first kind are the coefficients $s(n,k)$ in the expansion of the falling factorial

:$(x)_n = x(x-1)(x-2)\cdots(x-n+1)$

into powers of the variable $x$:

:$(x)_n = \sum_^n s(n,k) x^k,$

For example, $(x)_3 = x(x-1)(x - 2) = x^3 - 3x^2 + 2x$, leading to the values $s(3, 3) = 1$, $s(3, 2) = -3$, and $s(3, 1) = 2$.

The unsigned Stirling numbers may also be defined algebraically as the coefficients of the rising factorial:
:$x^ = x(x+1)\cdots(x+n-1)=\sum_^n \left$rightx^k.

The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources; the square bracket notation $\left$right/math> is also common notation for the Gaussian coefficients.

Definition by permutation

Subsequently, it was discovered that the absolute values $, s(n,k),$ of these numbers are equal to the number of permutations of certain kinds. These absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted $c(n,k)$ or $\left$right/math>. They may be defined directly to be the number of permutations of $n$ elements with $k$ disjoint cycles.

For example, of the $3! = 6$ permutations of three elements, there is one permutation with three cycles (the identity permutation, given in one-line notation by $123$ or in cycle notation by $(1)(2)(3)$), three permutations with two cycles ($132 = (1)(23)$, $213 = (12)(3)$, and $321 = (13)(2)$) and two permutations with one cycle ($312 = (132)$ and $231 = (123)$). Thus $\left$right= 1, $\left$right= 3, and $\left$right= 2. These can be seen to agree with the previous algebraic calculations of $s(n, k)$ for $n = 3$.
For another example, the image at right shows that $\left$right= 11: the symmetric group on 4 objects has 3 permutations of the form

:$(\bullet \bullet)(\bullet \bullet)$ (having 2 orbits, each of size 2),

and 8 permutations of the form

:$(\bullet\bullet\bullet)(\bullet)$ (having 1 orbit of size 3 and 1 orbit of size 1).
These numbers can be calculated by considering the orbits as conjugacy classes. Alfréd Rényi observed that the unsigned Stirling number of the first kind $\left$right/math> also counts the number of permutations of size $n$ with $k$ left-to-right maxima.

Signs

The signs of the signed Stirling numbers of the first kind depend only on the parity of .
:$s(n,k) = (-1)^ \left$right.

Recurrence relation

The unsigned Stirling numbers of the first kind follow the recurrence relation

: $\left$right= n \leftright+ \leftright/math>

for $k > 0$, with the boundary conditions

:$\left$right= 1 \quad\mbox\quad \leftright\leftright0
for $n>0$.

It follows immediately that the signed Stirling numbers of the first kind satisfy the recurrence

: $s(n + 1, k) = - n \cdot s(n, k) + s(n, k - 1)$.

Table of values

Below is a triangular array of unsigned values for the Stirling numbers of the first kind, similar in form to Pascal's triangle. These values are easy to generate using the recurrence relation in the previous section.

Properties

Simple identities

Using the Kronecker delta one has,

:$\left$right= \delta_n

and

:$\left$right= 0 if $k > 0$, or more generally $\left$right= 0 if ''k'' > ''n''.

Also

:$\left$right= (n-1)!,
\quad
\leftright= 1,
\quad
\leftright= ,

and

:$\left$right= \frac \quad\mbox\quad\leftright= .

Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials. Many relations for the Stirling numbers shadow similar relations on the binomial coefficients. The study of these 'shadow relationships' is termed umbral calculus and culminates in the theory of Sheffer sequences. Generalizations of the Stirling numbers of both kinds to arbitrary complex-valued inputs may be extended through the relations of these triangles to the Stirling convolution polynomials.

Note that all the combinatorial proofs above use either binomials or multinomials of $n$.

Therefore if $p$ is prime, then:

$p\ , \left$right/math> for 1.

Expansions for fixed ''k''

Since the Stirling numbers are the coefficients of a polynomial with roots 0, 1, ..., , one has by Vieta's formulas that

\left \begin n \\ n - k \end \right= \sum_ i_1 i_2 \cdots i_k.

In other words, the Stirling numbers of the first kind are given by elementary symmetric polynomials evaluated at 0, 1, ..., . In this form, the simple identities given above take the form

\left \begin n \\ n - 1 \end \right = \sum_^ i = \binom,

\left \begin n \\ n - 2 \end \right = \sum_^\sum_^ ij = \frac \binom,

\left \begin n \\ n - 3 \end \right = \sum_^\sum_^\sum_^ ijk = \binom \binom,

and so on.

One may produce alternative forms for the Stirling numbers of the first kind with a similar approach preceded by some algebraic manipulation: since

:$(x+1)(x+2) \cdots (x+n-1) = (n-1)! \cdot (x+1) \left(\frac+1\right) \cdots \left(\frac+1\right),$

it follows from Newton's formulas that one can expand the Stirling numbers of the first kind in terms of generalized harmonic numbers. This yields identities like

:$\left$right= (n-1)!\; H_,

:$\left$right= \frac (n-1)! \left (H_)^2 - H_^ \right/math>

:$\left$right= \frac(n-1)! \left (H_)^3 - 3H_H_^+2H_^ \right

where ''H''_''n'' is the harmonic number $H_n = \frac + \frac + \ldots + \frac$ and ''H''_''n''^(''m'') is the generalized harmonic number
$H_n^ = \frac + \frac + \ldots + \frac.$

These relations can be generalized to give
:\frac \left begin n \\ k+1 \end \right= \sum_^ \sum_^ \cdots \sum_^ \frac = \frac
where ''w''(''n'', ''m'') is defined recursively in terms of the generalized harmonic numbers by
:$w(n, m) = \delta_ + \sum_^ (1-m)_k H_^ w(n, m-1-k).$
(Here ''δ'' is the Kronecker delta function and $(m)_k$ is the Pochhammer symbol.)

For fixed $n \geq 0$ these weighted harmonic number expansions are generated by the generating function

:\frac \left begin n+1 \\ k \end \right= ^kexp\left(\sum_ \frac x^m\right),

where the notation ^k/math> means extraction of the coefficient of $x^k$ from the following formal power series (see the non-exponential Bell polynomials and section 3 of ).

More generally, sums related to these weighted harmonic number expansions of the Stirling numbers of the first kind can be defined through generalized zeta series transforms of generating functions.

One can also "invert" the relations for these Stirling numbers given in terms of the $k$-order harmonic numbers to write the integer-order generalized harmonic numbers in terms of weighted sums of terms involving the Stirling numbers of the first kind. For example, when $k = 2, 3$ the second-order and third-order harmonic numbers are given by

:(n!)^2 \cdot H_n^ = \left \begin n+1 \\ 2 \end \right2 - 2 \left \begin n+1 \\ 1 \end \right\left \begin n+1 \\ 3 \end \right/math>

:(n!)^3 \cdot H_n^ = \left \begin n+1 \\ 2 \end \right3 - 3 \left \begin n+1 \\ 1 \end \right\left \begin n+1 \\ 2 \end \right\left \begin n+1 \\ 3 \end \right+ 3 \left \begin n+1 \\ 1 \end \right2 \left \begin n+1 \\ 4 \end \right

More generally, one can invert the Bell polynomial generating function for the Stirling numbers expanded in terms of the $m$-order harmonic numbers to obtain that for integers $m \geq 2$

:H_n^ = -m \times ^mlog\left(1+\sum_ \left \begin n+1 \\ k+1 \end \right\frac\right).

Finite sums

Since permutations are partitioned by number of cycles, one has
:$\sum_^n \left$right= n!
The identities
:$\sum_^n \left$right^k = n!\binom,\,u>0

and

:$\sum_^ = \left$right/math>
can be proved by the techniques at
Stirling numbers and exponential generating functions#Stirling numbers of the first kind and Binomial coefficient#Ordinary generating functions.

The table in section 6.1 of ''Concrete Mathematics'' provides a plethora of generalized forms of finite sums involving the Stirling numbers. Several particular finite sums relevant to this article include

:
\begin
\left \right& = \sum_^n \left \right\binom (-1)^ \\
\left \right&= \sum_^n \left \right\frac \\
\left \right& = \sum_^m (n+k) \leftright\\
\left \right\binom & = \sum_k \left \right\left \right\binom.
\end

Additionally, if we define the ''second-order'' Eulerian numbers by the triangular recurrence relation

:$\left\langle \!\! \left\langle \right\rangle \!\! \right\rangle = (k+1) \left\langle \!\! \left\langle \right\rangle \!\! \right\rangle + (2n-1-k) \left\langle \!\! \left\langle \right\rangle \!\! \right\rangle,$

we arrive at the following identity related to the form of the Stirling convolution polynomials which can be employed to generalize both Stirling number triangles to arbitrary real, or complex-valued, values of the input $x$:

:$\left$right= \sum_^n \left\langle \!\! \left\langle \right\rangle \!\! \right\rangle \binom.

Particular expansions of the previous identity lead to the following identities expanding the Stirling numbers of the first kind for the first few small values of $n := 1, 2, 3$:

:
\begin
\left begin x \\ x-1 \end \right& = \binom \\
\left begin x \\ x-2 \end \right& = \binom + 2 \binom \\
\left begin x \\ x-3 \end \right& = \binom + 8 \binom + 6 \binom.
\end

Software tools for working with finite sums involving Stirling numbers and Eulerian numbers are provided by th
RISC Stirling.m package
utilities in ''Mathematica''. Other software packages for ''guessing'' formulas for sequences (and polynomial sequence sums) involving Stirling numbers and other special triangles is available for both Mathematica and Sagebr>here
an
here
respectively.

Congruences

The following congruence identity may be proved via a generating function-based approach:

:
\begin
\left begin n \\ m \end \right& \equiv
\binom =
^m\left(
x^ (x+1)^
\right) && \pmod,
\end

More recent results providing Jacobi-type J-fractions that generate the single factorial function and generalized factorial-related products lead to other new congruence results for the Stirling numbers of the first kind.
For example, working modulo $2$ we can prove that

:
\begin
\left begin n \\ 1 \end \right& \equiv
\frac \geq 2+ = 1&& \pmod \\
\left begin n \\ 2 \end \right& \equiv
\frac (n-1) \geq 3+
= 2&& \pmod \\
\left begin n \\ 3 \end \right& \equiv
2^ (9n-20) (n-1) \geq 4+
= 3&& \pmod \\
\left begin n \\ 4 \end \right& \equiv
2^ (3n-10) (3n-7) (n-1) \geq 5+
= 4&& \pmod
\end

Where /math> is the Iverson bracket.

and working modulo $3$ we can similarly prove that

:
\begin
\left begin n \\ m \end \right& \equiv ^m(x^ (x+1)^ (x+2)^ && \pmod \\
& \equiv \sum_^m \binom \binom 2^ && \pmod
\end

More generally, for fixed integers $h \geq 3$ if we define the ordered roots

:$\left(\omega_\right)_^ := \left\,$

then we may expand congruences for these Stirling numbers defined as the coefficients

:\left begin n \\ m \end \right= ^mR(R+1) \cdots (R+n-1),

in the following form where the functions, $p_^(n)$, denote fixed
polynomials of degree $m$ in $n$ for each $h$, $m$, and $i$:

:\left begin n \\ m \end \right=
\left(\sum_^ p_^(n) \times \omega_^
\right) > m+ = m\qquad \pmod,

Section 6.2 of the reference cited above provides more explicit expansions related to these congruences for the $r$-order harmonic numbers and for the generalized factorial products, $p_n(\alpha, R) := R(R+\alpha)\cdots(R+(n-1)\alpha)$.

Generating functions

A variety of identities may be derived by manipulating the generating function (see change of basis):

:$H(z,u)= (1+z)^u = \sum_^\infty z^n = \sum_^\infty \frac \sum_^n s(n,k) u^k = \sum_^\infty u^k \sum_^\infty \frac s(n,k).$

Using the equality
:$(1+z)^u = e^ = \sum_^\infty (\log(1 + z))^k \frac,$
it follows that
:$\sum_^\infty s(n,k) \frac = \frac$
and
:\sum_^\infty \left \right\frac = \frac.

This identity is valid for formal power series, and the sum converges in the complex plane for , ''z'', < 1.

Other identities arise by exchanging the order of summation, taking derivatives, making substitutions for ''z'' or ''u'', etc. For example, we may derive:arXiv
/ref>
:$\frac = m!\sum_^\infty \frac ,\qquad m=1,2,3,\ldots\quad , z, <1$
or
:$\sum_^\infty \frac = \zeta(i+1),\qquad i=1,2,3,\ldots$
and
:$\sum_^\infty \frac = \zeta(i+1,v),\qquad i=1,2,3,\ldots\quad \Re(v)>0$
where $\zeta(k)$ and $\zeta(k,v)$ are the Riemann zeta function and the Hurwitz zeta function respectively, and even evaluate this integral
:$\int_0^1 \frac \, dx = \frac\sum_^ s(k-1,r)\sum_^r \binom(k-2)^ \zeta(z+1-m) ,\qquad \Re(z) > k-1 ,\quad k=3,4,5,\ldots$
where $\Gamma(z)$ is the gamma function. There also exist more complicated expressions for the zeta-functions involving the Stirling numbers. One, for example, has
:$\zeta(s,v) = \frac\sum_^\infty \frac\left$rightsum_^\!(-1)^l \binom (l+v)^,\quad k=1, 2, 3,\ldots
This series generalizes Helmut Hasse, Hasse's series for the Hurwitz zeta-function (we obtain Hasse's series by setting ''k''=1).

Asymptotics

The next estimate given in terms of the Euler–Mascheroni constant, Euler gamma constant applies:

:\left \begin n+1 \\ k+1 \end \right\underset \frac \left(\gamma+\ln n\right)^k,\ \text k = o(\ln n).

For fixed $n$ we have the following estimate :

:$\left[ \begin n+k \\ k \end \right] \underset \frac.$

Explicit formula

It is well-known that we don't know any one-sum formula for Stirling numbers of the first kind. A two-sum formula can be obtained using one of the Stirling number#Symmetric formulae, symmetric formulae for Stirling numbers in conjunction with the explicit formula for Stirling numbers of the second kind.
: \left \right= \sum_^ \binom \binom \sum_^ \frac

As discussed earlier, by Vieta's formulas, one get$\left[ \begin n \\ k \end \right] = \sum_ i_1 i_2 \cdots i_.$The Stirling number ''s(n,n-p)'' can be found from the formula
:$\begin s(n,n-p) &= \frac \sum_ (-1)^K \frac , \end$
where $K =k_1 + \cdots + k_p.$ The sum is a sum over all Partition (number theory), partitions of ''p''.

Another exact nested sum expansion for these Stirling numbers is computed by elementary symmetric polynomials corresponding to the coefficients in $x$ of a product of the form $(1+c_1 x) \cdots (1+c_x)$. In particular, we see that

:
\begin
\left \right& = [x^k] (x+1)(x+2) \cdots (x+n-1)
= (n-1)! \cdot [x^k] (x+1) \left(\frac+1\right) \cdots \left(\frac+1\right) \\
& = \sum_ \frac.
\end

Newton's identities combined with the above expansions may be used to give an alternate proof of the weighted expansions involving the generalized harmonic numbers already #Expansions for fixed k, noted above.

Relations to natural logarithm function

The ''n''th derivative of the ''μ''th power of the natural logarithm involves the signed Stirling numbers of the first kind:

$=x^\sum_^\mu^s(n,n-k+1)(\ln x)^,$

where $\mu^\underline$ is the falling factorial, and $s(n,n-k+1)$ is the signed Stirling number.

It can be proved by using mathematical induction.

Other formulas

Stirling numbers of the first kind appear in the formula for Gregory coefficients and in a finite sum identity involving Bell number, Bell numbers

$n! G_n = \sum_^n \frac$

$\sum_^n \binom B_j k^ = \sum_^k \left$rightB_ (-1)^

Infinite series involving the finite sums with the Stirling numbers often lead to the special functions. For example

$\ln\Gamma(z) = \left(z-\frac\right)\!\ln z -z +\frac\ln2\pi + \frac \sum_^\infty\frac\!\sum_^\! \frac \left[\right]$

and

$\Psi(z) = \ln z - \frac - \frac \sum_^\infty\frac\!\sum_^\! \frac \left[\right]$

or even

$\gamma_m=\frac\delta_+\frac \!\sum_^\infty\frac \! \sum_^\frac \left$right\leftright \,

where ''γ''_''m'' are the Stieltjes constants and ''δ''_''m'',0 represents the Kronecker delta function.

Notice that this last identity immediately implies relations between the polylogarithm functions, the Stirling number exponential generating functions given above, and the Stirling-number-based power series for the generalize
Nielsen polylogarithm
functions.

Generalizations

There are many notions of ''generalized Stirling numbers'' that may be defined (depending on application) in a number of differing combinatorial contexts. In so much as the Stirling numbers of the first kind correspond to the coefficients of the distinct polynomial expansions of the single factorial function, $n! = n (n-1) (n-2) \cdots 2 \cdot 1$, we may extend this notion to define triangular recurrence relations for more general classes of products.

In particular, for any fixed arithmetic function $f: \mathbb \rightarrow \mathbb$ and symbolic parameters $x, t$, related generalized factorial products of the form

:$(x)_ := \prod_^ \left(x+\frac\right)$

may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of $x$ in the expansions of $(x)_$ and then by the next corresponding triangular recurrence relation:

:$\begin \left[\begin n \\ k \end \right]_ & = [x^] (x)_ \\ & = f(n-1) t^ \left[\begin n-1 \\ k \end \right]_ + \left[\begin n-1 \\ k-1 \end \right]_ + \delta_ \delta_. \end$

These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind as well as recurrence relations and functional equations related to the ''f-harmonic numbers'', $F_n^(t) := \sum_ t^k / f(k)^r$.

One special case of these bracketed coefficients corresponding to $t \equiv 1$ allows us to expand the multiple factorial, or Double factorial#Generalizations, multifactorial functions as polynomials in $n$.

The Stirling numbers of both kinds, the binomial coefficients, and the first and second-order Eulerian numbers are all defined by special cases of a triangular ''super-recurrence'' of the form

:$\left, \begin n \\ k \end \ = (\alpha n + \beta k + \gamma) \left, \begin n-1 \\ k \end \ + (\alpha^ n + \beta^ k + \gamma^) \left, \begin n-1 \\ k-1 \end \ + \delta_ \delta_,$

for integers $n, k \geq 0$ and where $\left, \begin n \\ k \end \ \equiv 0$ whenever $n < 0$ or $k < 0$. In this sense, the form of the Stirling numbers of the first kind may also be generalized by this parameterized super-recurrence for fixed scalars $\alpha, \beta, \gamma, \alpha^, \beta^, \gamma^$ (not all zero).

See also

* Stirling numbers
* Stirling numbers of the second kind
* Stirling polynomials
* Random permutation statistics

References

* The Art of Computer Programming
* Concrete Mathematics
*
* .
*

{{Classes of natural numbers

Permutations
Factorial and binomial topics
Triangles of numbers
Operations on numbers

pl:Liczby Stirlinga#Liczby Stirlinga I rodzaju