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. AndClearly . And in [8] proves for , so we have . Moreover sinceit follows thatSo 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 . SoandTherefore we haveby 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 haveSo 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 thenwhich 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) impliesOn the other hand, we also havefor for all . Hence identities (20) and (21) prove (i).
Now, for (ii), the induction hypothesis () in (ii) with again (16) impliesMoreoverThen 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.diagof equals the -Bell numbers . Then, due to (16), we haveAnd 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.