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 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 in general.)
The original definition of Stirling numbers of the first kind was algebraic: they are the coefficients s(n, k) in the expansion of the falling factorial
into powers of the variable x:
For example, , leading to the values , , and .
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 or . They may be defined directly to be the number of permutations of n elements with k disjoint cycles. For example, of the permutations of three elements, there is one permutation with three cycles (the identity permutation, given in one-line notation by or in cycle notation by ), three permutations with two cycles (, , and ) and two permutations with one cycle ( and ). Thus, , and . These can be seen to agree with the previous calculation of for .
The unsigned Stirling numbers may also be defined algebraically, as the coefficients of the rising factorial:
The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources. (The square bracket notation is also common notation for the Gaussian coefficients.)
The image at right shows that : the symmetric group on 4 objects has 3 permutations of the form
- (having 2 orbits, each of size 2),
and 8 permutations of the form
- (having 1 orbit of size 3 and 1 orbit of size 1).
The signs of the (signed) Stirling numbers of the first kind are predictable and depend on the parity of n − k. In particular,
The unsigned Stirling numbers of the first kind can be calculated by the recurrence relation
for , with the initial conditions
for n > 0.
It follows immediately that the (signed) Stirling numbers of the first kind satisfy the recurrence
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.
Note that although
- , we have if n > 0
- if k > 0, or more generally if k > n.
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.
Expansions for fixed k
Since the Stirling numbers are the coefficients of a polynomial with roots 0, 1, ..., n − 1, one has by Vieta's formulas that
In other words, the Stirling numbers of the first kind are given by elementary symmetric polynomials evaluated at 0, 1, ..., n − 1. In this form, the simple identities given above take the form
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
where Hn is the harmonic number and Hn(m) is the generalized harmonic number
These relations can be generalized to give
where w(n, m) is defined recursively in terms of the generalized harmonic numbers by
For fixed these weighted harmonic number expansions are generated by the generating function
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 -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 the second-order and third-order harmonic numbers are given by
For all positive integer m and n, one has
Other related formulas involving the falling factorials, Stirling numbers of the first kind, and in some cases Stirling numbers of the second kind include the following:
Inversion relations and the Stirling transform
For any pair of sequences, and , related by a finite sum Stirling number formula given by
for all integers , we have a corresponding inversion formula for given by
The differential operators and are related by the following formulas for all integers :
The following congruence identity may be proved via a generating function-based approach:
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 we can prove that
and working modulo we can similarly prove that
More generally, for fixed integers if we define the ordered roots
then we may expand congruences for these Stirling numbers defined as the coefficients
in the following form where the functions, , denote fixed polynomials of degree in for each , , and :
Section 6.2 of the reference cited above provides more explicit expansions related to these congruences for the -order harmonic numbers and for the generalized factorial products, . In the previous examples, the notation denotes Iverson's convention.
A variety of identities may be derived by manipulating the generating function:
Using the equality
it follows that
(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:
where is the Gamma function. There also exist more complicated expressions for the zeta-functions involving the Stirling numbers. One, for example, has
This series generalizes Hasse's series for the Hurwitz zeta-function (we obtain Hasse's series by setting k=1).
The next estimate given in terms of the Euler gamma constant applies:
For fixed we have the following estimate as :
We can also apply the saddle point asymptotic methods from Temme's article to obtain other estimates for the Stirling numbers of the first kind. These estimates are more involved and complicated to state. Nonetheless, we provide an example. In particular, we define the log gamma function, , whose higher-order derivatives are given in terms of polygamma functions as
where we consider the (unique) saddle point of the function to be the solution of when . Then for and the constants
we have the following asymptotic estimate as :
Since permutations are partitioned by number of cycles, one has
can be proved by the techniques on the page Stirling numbers and exponential 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
Additionally, if we define the second-order Eulerian numbers by the triangular recurrence relation
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 :
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 :
Software tools for working with finite sums involving Stirling numbers and Eulerian numbers are provided by the 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 Sage here and here, respectively.
where The sum is a sum over all partitions of p.
Another exact nested sum expansion for these Stirling numbers is computed by elementary symmetric polynomials corresponding to the coefficients in of a product of the form . In particular, we see that
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 generalized Nielsen polylogarithm functions.
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, , we may extend this notion to define triangular recurrence relations for more general classes of products.
In particular, for any fixed arithmetic function and symbolic parameters , related generalized factorial products of the form
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 in the expansions of and then by the next corresponding triangular recurrence relation:
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, .
One special case of these bracketed coefficients corresponding to allows us to expand the multiple factorial, or multifactorial functions as polynomials in (see generalizations of the double factorial).
for integers and where whenever or . 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 (not all zero).
- See section 6.2 and 6.5 of Concrete Mathematics.
- Richard P. Stanley, Enumerative Combinatorics, volume 1 (2nd ed.). Page 34 of the online version.
- Adamchik, V. (1996). "On Stirling numbers and Euler sums" (PDF). Cite journal requires
- Flajolet and Sedgewick (1995). "Mellin transforms and asymptotics: Finite differences and Rice's integrals". Theoretical Computer Science. 144 (1–2): 101–124. doi:10.1016/0304-3975(94)00281-m.
- Schmidt, M. D. (30 Oct 2016). "Zeta Series Generating Function Transformations Related to Polylogarithm Functions and the k-Order Harmonic Numbers". arXiv:1610.09666 [math.CO].
- Schmidt, M. D. (3 Nov 2016). "Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function". arXiv:1611.00957 [math.CO].
- Mező, István (2012). "The dual of Spivey's Bell number formula". Journal of Integer Sequences. 15.
- See Table 265 (Section 6.1) of the Concrete Mathematics reference.
- Concrete Mathematics exercise 13 of section 6. Note that this formula immediately implies the first positive-order Stirling number transformation given in the main article on generating function transformations.
- Olver, Frank; Lozier, Daniel; Boisvert, Ronald; Clark, Charles (2010). "NIST Handbook of Mathematical Functions". Nist Handbook of Mathematical Functions. (Section 26.8)
- Herbert Wilf, Generatingfunctionology, Section 4.6.
- Schmidt, M. D. (2017). "Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions". J. Integer Seq. 20 (3).
- Ia. V. Blagouchine (2016). "Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1". Journal of Mathematical Analysis and Applications. 442 (2): 404–434. arXiv:1408.3902. doi:10.1016/j.jmaa.2016.04.032. arXiv
- Blagouchine, Iaroslav V. (2018). "Three Notes on Ser's and Hasse's Representations for the Zeta-functions". INTEGERS: The Electronic Journal of Combinatorial Number Theory. 18A: 1–45. arXiv:1606.02044. Bibcode:2016arXiv160602044B.
- See also some more interesting series representations and expansions mentioned in Connon's article: Connon, D. F. (2007). "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers (Volume I)". arXiv:0710.4022 [math.HO]..
- These estimates are found in Section 26.8 of the NIST Handbook of Mathematical Functions.
- Temme, N. M. "Asymptotic Estimates of Stirling Numbers" (PDF). Cite journal requires
- The first identity below follows as a special case of the Bell polynomial identity found in section 4.1.8 of S. Roman's The Umbral Calculus where , though several other related formulas for the Stirling numbers defined in this manner are derived similarly.
- A table of the second-order Eulerian numbers and a synopsis of their properties is found in section 6.2 of Concrete Mathematics. For example, we have that . These numbers also have the following combinatorial interpretation: If we form all permutations of the multiset with the property that all numbers between the two occurrences of are greater than for , then is the number of such permutations that have ascents.
- Schmidt, M. D. (2014 and 2016). "A Computer Algebra Package for Polynomial Sequence Recognition". arXiv:1609.07301 [math.CO]. Check date values in:
- Ia. V. Blagouchine (2016). "Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only". Journal of Number Theory. 158 (2): 365–396. doi:10.1016/j.jnt.2015.06.012. arXiv
- J. Malenfant, "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers"
- Schmidt, M. D. (2018). "Combinatorial Identities for Generalized Stirling Numbers Expanding f-Factorial Functions and the f-Harmonic Numbers". J. Integer Seq. 21 (Article 18.2.7): 7–8.
- Combinatorial Identities for Generalized Stirling Numbers Expanding f-Factorial Functions and the f-Harmonic Numbers (2016).
- Schmidt, Maxie D. (2010). "Generalized j-Factorial Functions, Polynomials, and Applications". J. Integer Seq. 13.
- The Art of Computer Programming
- Concrete Mathematics
- M. Abramowitz, I. Stegun, ed. (1972). "§24.1.3. Stirling Numbers of the First Kind". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (9th ed.). New York: Dover. p. 824.
- Stirling numbers of the first kind, s(n,k) at PlanetMath.org..
- Sloane, N. J. A. (ed.). "Sequence A008275 (Triangle read by rows of Stirling numbers of first kind)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.