International Journal of Mathematics and Mathematical Sciences
Volume 2018 (2018), Article ID 9096761, 8 pages
https://doi.org/10.1155/2018/9096761

Research Article

Generalized Bell Numbers and Peirce Matrix via Pascal Matrix

Eunmi Choi

Department of Mathematics, HanNam Univ., Daejeon, Republic of Korea

Correspondence should be addressed to Eunmi Choi; emc@hnu.kr

Received 20 June 2018; Accepted 29 July 2018; Published 8 August 2018

Academic Editor: Pentti Haukkanen

Copyright © 2018 Eunmi Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

With the Stirling matrix and the Pascal matrix , we show that () satisfies a type of generalized Stirling recurrence. Then, by expressing the sum of components of each row of as -Bell number, we investigate properties of -Bell numbers as well as -Peirce matrix.

1. Introduction

The Stirling number of the second kind is the number of ways to partition an elements set into nonempty subsets for any . can be expressed by , and the Stirling matrix satisfies the Stirling recurrence [1]. Some researches including [2–4] were devoted to investigating the Stirling matrix with the Pascal matrix and binomial expressions. On the other hand, the sum of numbers of the th row of is called a Bell number, so counts the number of partitions of an elements set. One effective way for generating Bell numbers is to use the Peirce matrix (often called Bell table or Aitken’s array), on which Bell numbers appear along both borders [5–7].

A main purpose of work is to study for . We show that satisfies a kind of generalized Stirling recurrence, and then by expressing the sum of components of each row of as a -Bell number we investigate -Bell numbers as well as -Peirce matrix. We discuss recurrence rules of a -Peirce matrix and then interrelationships between each -Peirce matrices.

2. Generalized -Stirling Matrix and -Bell Numbers

Throughout the work, we write for the Pascal matrix and for the th row of . And means a transpose matrix of . Let the Stirling matrix be

Theorem 1. We have the following:

(1), and .(2) and .

Proof. The first two identities in (1) are easy to observe. And

Clearly . And in [8] proves for , so we have . Moreover since

it follows that

So and .

Hence the next corollary follows immediately.

Corollary 2 (). As a generalization of , we shall consider for . Write for , and let . Clearly .

Theorem 3. satisfies the recurrence .

Proof. in Corollary 2 shows . And , give identities and , and it is easy to observe

for ; . Now we assume for some , . Since

any th component comes from the th row of and the th column of . So

and

Therefore we have

by the induction hypothesis on . But since for all , it follows immediately from Pascal’s rule that

Theorem 3 agrees with the Stirling rule if , so we may call the generalized -Stirling matrix for . Like the Bell numbers which are the sum of each row of , we will take sum of each row of the -Stirling matrix and call this the -Bell number and denote it by . The first few terms of the -Bell numbers are obtained by examining .

Obviously the (original) Bell numbers, and , so () by Theorem 1. Note that is listed in A005493-OEIS as the numbers of partitions with a distinguished block. And () are coefficients of (A005494, A045379, and A196834). On the other hand, the -Bell numbers were studied in [9] by using certain binomial expressions and . A key feature of our study is to have these -Bell numbers from multiplications of and explicitly. Hence it enables us to find relations between -Bell numbers and , as follows.

Theorem 4. and .

Proof. Since , it is clear that

And, due to Theorem 1, we have

So if we assume for some then we have

Theorem 4 says , so . Thus the expression in the proof of Theorem 1, which is the multiplication of the th row of with , is equal to multiplication of the th row of with . Note that equals the arithmetic table of the polynomial , while Pascal’s table is the arithmetic table of . Hence if we let be the th row of of then

which implies . Thus, owing to Theorem 4, it is clear to have generalizations to -Bell numbers () that

3. Matrix of General Bell Numbers

With all the -Bell numbers , we make a table in which each th column is composed of -Bell numbers. We call it a matrix of general Bell numbers and denote it by ().

Theorem 5. The entries of the th column in satisfy . So .

Proof. Observe that each th column () satisfies the following:

Suppose for some . By looking at

it is enough to prove the following two cases.
(i) For any , assume in some th column and show is true at th column.
(ii) For any , assume at every th row () and show is true at th row.
Using the th row of , the induction hypothesis in (i) with (16) implies

On the other hand, we also have

for for all . Hence identities (20) and (21) prove (i).
Now, for (ii), the induction hypothesis () in (ii) with again (16) implies

Moreover

Then we complete the proof by comparing (22) and (23).

Therefore Theorem 5 with (16) shows

As an example, the -Bell numbers and then -Bell numbers are obtained sequentially from the -Bell numbers.

The next theorem gives another way to have -Bell numbers . Let denote the -tuple .

Theorem 6. Let and be the th row of . Then for any .

Proof. Clearly and . Similarly () equals and . Thus, for instance, when , with th row of , we have , , , , and so on. Hence we can say that for .
Now assume for . Then since , Theorem 5 implies

Theorem 6 can be restated in terms of -Bell numbers.

Corollary 7. .

Hence and

. Precisely, with , we have , , , , , and , where these yield the 7th row .

4. -Peirce Matrix

The Peirce matrix ) was designed to generate Bell numbers.

where it holds a recurrence with , . So the 6th row of begins with followed by , , , and and then reach , the next Bell number of . Thus all Bell numbers are on both borders of . The matrix is often called the Bell matrix or Aitken’s array named after E. T. Bell and A. Aitken. Here we call it Peirce matrix after C. S. Peirce [10] to avoid confusion with the matrix of general Bell numbers in Section 3.

Now, for any , let be a matrix satisfying the following two rules.

(i) Each th row begins with the Bell number .

(ii)

Obviously is the Peirce matrix, and () are equal to

Notice that the left border is always comprised of Bell numbers, while the right diagonals (r.diag.) of and are of and -Bell numbers, respectively. And the second right diagonal (2nd r.diag.) of equals the r.diag. of . The next theorem shows that the r.diag. of () is composed of -Bell numbers .

Theorem 8. (1) The entries of r.diag. of are -Bell numbers; i.e., , while the entries of left border are Bell numbers.
(2) The 2nd r.diag. is composed of the -Bell numbers; i.e., .

Proof. Following the rules in (28), is equal to

When , the r.diag. of yields , , and , where these correspond to , , and .
Assume the r.diag

of equals the -Bell numbers . Then, due to (16), we have

And it is not hard to see that it corresponds to the r.diag. of

Moreover, by comparing with , we also notice that the r.diag. of equals the 2nd r.diag. of .

Owing to Theorem 8, we may call the Peirce -matrix. Write for the diagonal matrix having diagonal entries . Every -Bell number can be obtained by Bell numbers as follows.

Theorem 9. .

Proof. When , it is due to Theorem 1. And, by Theorem 8, we have

But we notice

Hence, with the th row of , the th -Bell number is equal to .

For example, with , the -Bell number equals

Data Availability

All data used to support the findings of this study are either obtained via the cited sources or included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by 2018 HanNam University Research Fund.

References

A. Tucker, Applied combinatorics, John Wiley, New York, NY, USA, 2nd edition, 1984.

P. Maltais and T. A. Gulliver, “Pascal matrices and Stirling numbers,” Applied Mathematics Letters, vol. 11, no. 2, pp. 7–11, 1998.

M. Z. Spivey and A. M. Zimmer, “Symmetric polynomials, Pascal matrices, and Stirling matrices,” Linear Algebra and its Applications, vol. 428, no. 4, pp. 1127–1134, 2008.

S.-L. Yang and H. You, “On a connection between the Pascal, Stirling and Vandermonde matrices,” Discrete Applied Mathematics: The Journal of Combinatorial Algorithms, Informatics and Computational Sciences, vol. 155, no. 15, pp. 2025–2030, 2007.

M. Abbas and S. Bouroubi, “On new identities for Bell's polynomials,” Discrete Mathematics, vol. 293, no. 1-3, pp. 5–10, 2005.

M. Griffiths, “Families of sequences from a class of multinomial sums,” Journal of Integer Sequences, vol. 15, no. 1, Article 12.1.8, 10 pages, 2012.

S. Noschese and P. E. Ricci, “Differentiation of multivariable composite functions and Bell polynomials,” Journal of Computational Analysis and Applications, vol. 5, no. 3, pp. 333–340, 2003.

J. Shallit, “A triangle for the Bell numbers,” in A collection of manuscripts related to the Fibonacci sequence, pp. 69–71, Fibonacci Assoc., Santa Clara, Calif., 1980.

I. Mező, “TThe r-Bell numbers,” Journal of Integer Sequences, vol. 14, no. 1, Article ID 11.1.1, 2011.

C. S. Peirce, “On the algebra of logic,” American Journal of Mathematics, vol. 3, no. 1, pp. 15–57, 1880.