Abstract

Some combinatorial properties of -Stirling numbers are proved. Moreover, two asymptotic formulas for -Stirling numbers of the first kind derived using different methods are discussed and corresponding asymptotic formulas for the -Stirling type numbers of the first kind are obtained as corollaries.

Mathematics Subject Classification (2010). 11B73, 41A60.

1. Introduction

The r-Stirling numbers and r-Stirling type numbers are generalizations of the classical Stirling Numbers of the first kind. Introduced first by Andrei Broder ¹, the r-Stirling numbers of the first kind count the number of permutations of the set with cycles such that the first r elements are in distinct cycles. Broder denoted these numbers by Since =0 for this study considers the r-Stirling numbers of the first kind where are positive integers. These numbers satisfy the relation

(1)

A natural generalization of the r-Stirling numbers of the first kind is to introduce one more parameter to the generating function of these numbers. Thus, the r-Stirling type numbers of the first kind are defined by the relation

(2)

where When (2) reduced to (1).

The generalized Stirling numbers of the first kind as generalized by Hsu and Shuie ² denoted by satisfy the relation

(3)

where are complex numbers. Taking and (3) becomes

which is exactly (1). Thus,

(4)

Taking and (3) becomes

(5)

which is exactly (2). Thus,

(6)

In this paper, some combinatorial formulas for r-Stirling numbers are obtained. Moreover, two asymptotic formulas for these numbers derived using two different methods are mentioned and corresponding asymptotic formulas for the r-Stirling type numbers of the first kind are obtained as corollaries. These formulas may be used to compute values of these numbers when the parameters m and n are large within a certain range of m.

2. Some Combinatorial Properties

The r-Stirling numbers of the second kind, denoted by are defined by A.Z. Broder as the number of ways to partition the set into nonempty subsets such that the first r elements in must be in different subsets. The total number of partitions is defined to be the r-Bell numbers ³ denoted by That is,

(7)

If the linear order of the elements in each subset of the partition counts, then the number of ways to partition into k nonempty subsets such that the first r elements in must be in different subsets is equal to the r-Lah numbers, denoted by Motivated by the work of Feng Qi ⁴ the r-Bell numbers can also be expressed in terms of r-Lah numbers ⁵ and r-Stirling numbers of the second kind as follows,

(8)

The proof makes use of the following identity in ⁵

(9)

and inverse relation

An identity that involves r-Stirling numbers of the first kind parallel to (8) is given in the following theorem.

Theorem 2.1. For n and r positive integers, the following explicit formula holds

(10)

Proof. To establish (10), another form of inverse relation will be needed. Using the orthogonality relation

where is the Kronecker delta, one can easily prove that

(11)

Note that (9) can be expressed as

(12)

By making use of inverse relation in (11), the identity (12) with

can be expressed as

which gives

Summing up both sides over k from 0 to n yields

This is equivalent to

(13)

where

The preceding equation counts the total number of permutations of such that the first elements of are in distinct cycles. We observe that the structure of the identity (13) is analogous to (8). Thus, one may try to construct another combinatorial interpretation for -Bell numbers using (8) which may be the basis to construct another combinatorial interpretation for

The first values of the classical Stirling numbers of the first kind can be computed using the recurrence relation

(14)

and the Schlömilch formula

(15)

On the other hand, the first values of -Stirling numbers of the first kind can also be computed using the recurrence relation (see ¹)

(16)

And the Schlömilch-type formula ⁶

(17)

This explicit formula is derived in ⁶ using the following exponential generating function

(18)

and the facts that

(19)

(20)

More precisely, equations (18), (19) and (20) yield

and, by comparing the coefficients of and using the Schlömilch formula (15), the desired explicit formula for -Stirling numbers of the first kind is easily obtained.

3. Asymptotic Formulas for r-Stirling Numbers of the First Kind

Let be any closed contour enclosing Applying the Cauchy-Integral Formula to (1) gives

(21)

A modified saddle point method used in ⁷ was applied to the integral above to obtain the following asymptotic approximation:

Theorem 3.1. [C.B. Corcino, L.C. Hsu and E.L. Tan, ⁸] For positive integers and the asymptotic formula holds,

(22)

as valid uniformly with in the range where

(23)

(24)

and

(25)

The number is the unique positive solution to the equation the function is

(26)

And

Remark: The number may be obtained using mathematica.

Using the method in ⁹, Vega and Corcino ¹⁰ obtained an asymptotic formula for the generalized Stirling numbers of the first kind which is given by

(27)

as valid for in the range where is a function such that and is the gamma function, In this paper, and The that appears in (27) is

(28)

and is the unique positive solution to the equation

(29)

The constants and are given by

(30)

and

(31)

With a little modification in the computations in ¹⁰, the same formula as (27) is obtained when

(32)

and is the unique positive solution to the equation

(33)

Since [see ⁶], taking in (27), the following asymptotic formula for the -Stirling numbers of the first kind is obtained:

Theorem 3.2. (Corcino-Corcino, ¹¹) For positive integers and and as the following asymptotic formula for the -Stirling numbers of the first kind holds:

(34)

valid for in the range where is the unique positive solution to the equation

(35)

and

(36)

The corresponding constants and are as follows,

(37)

(38)

The next lemma gives the connection formula for the defined in Theorem 3.1. and the number defined in Theorem 3.2.

Lemma 3.3. (Corcino-Corcino, ¹¹) The numbers and satisfy the relation

4. r-Stirling Type Numbers of the First Kind

Applying the Cauchy Integral Formula to (2) we obtain

(39)

(40)

(41)

where

Following (17) and (41), we have the following corollary.

Corollary 4.1. The -Stirling type numbers satisfy

(42)

where

The asymptotic formula corresponding to (22) is given in Corollary 4.2.

Corollary 4.2. For positive integers and

(43)

as where

is the unique positive solution to the equation and

Proof. This follows from Theorem 3.1. and (41).

The formula corresponding to (11) is given in Corollary 4.3.

Corollary 4.3. For positive integers and

(49)

where

(50)

(51)

(52)

Proof. That follows from Lemma 3.3. With (27), where and (41), the corollary is then an immediate consequence of Theorem 3.2.

Remark. The asymptotic formulas in Theorem 3.1 and Theorem 3.2 can be shown to be asymptotically equivalent in the range of where both are valid. Proof for the equivalence is done in ¹¹. This implies the equivalence of the asymptotic formulas in Corollary 4.2 and 4.3.

References