Degenerate r-truncated Stirling numbers

Abstract

For any positive integer $ r $, the $ r $-truncated (or $ r $-associated) Stirling number of the second kind $ S_{2}^{(r)}(n, k) $ enumerates the number of partitions of the set $ \{1, 2, 3, \dots, n\} $ into $ k $ non-empty disjoint subsets, such that each subset contains at least $ r $ elements. We introduce the degenerate $ r $-truncated Stirling numbers of the second kind and of the first kind. They are degenerate versions of the $ r $-truncated Stirling numbers of the second kind and of the first kind, and reduce to the degenerate Stirling numbers of the second kind and of the first kind for $ r = 1 $. Our aim is to derive recurrence relations for both of those numbers.

Keywords

• degenerate $ r $-associated stirling numbers of the second kind
• degenerate $ r $-truncated stirling numbers of the first kind
• degenerate $ r $-truncated bell polynomials