Matrix factorization of the \(n \times n\) shift Bell matrix. (English) Zbl 1418.05014
Summary: Let \(B_n = [B_{n, k}]_{n, k \geqslant 0}\) be the Bell matrix. Define the \(n \times n\) shift Bell matrix \(P_{n, k}\) by \((P_{n, k})_{i, j} = B_{k + i - 1, k + j - 1}\) for \(i, j = 1, 2, \cdots n\) and \(k = 0, 1, 2 \cdots\). In this paper, matrix factorizations of the \(n \times n\) shift Bell matrix and the \(n \times n\) generalized Riordan matrix are studied. As a result, many lower triangular matrices related to Bell polynomials can be factorized by the corresponding matrices and some identities are derived from the matrix representations. In addition, some harmonic number identities are obtained from the Riordan array method.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A05 | Permutations, words, matrices |
| 15B36 | Matrices of integers |
| 15A06 | Linear equations (linear algebraic aspects) |
| 05A19 | Combinatorial identities, bijective combinatorics |
Keywords:
Bell polynomials; \(n \times n\) shift Bell matrix; \(n \times n\) shift iteration matrix; generalized Riordan arrayReferences:
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