Degenerate $ r $-truncated Stirling numbers

Taekyun Kim; Dae San Kim; Jin-Woo Park; Taekyun Kim; Dae San Kim; Jin-Woo Park

doi:10.3934/math.20231322

AIMS Mathematics

2023, Volume 8, Issue 11: 25957-25965. doi: 10.3934/math.20231322

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Degenerate $ r $-truncated Stirling numbers

1.
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2.
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
3.
Department of Mathematics Education, Daegu University, Gyeongsan, 38453, Republic of Korea

Received: 04 July 2023 Revised: 22 August 2023 Accepted: 31 August 2023 Published: 08 September 2023
MSC : 11B73, 11B83

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.
Citation: Taekyun Kim, Dae San Kim, Jin-Woo Park. Degenerate $ r $-truncated Stirling numbers[J]. AIMS Mathematics, 2023, 8(11): 25957-25965. doi: 10.3934/math.20231322

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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.

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