All associated Stirling numbers are arithmetical triangles

Khaled Ben Letaïef
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 3, Pages 27–34
Full paper (PDF, 156 Kb)

Details

Authors and affiliations

Khaled Ben Letaïef 
Aeronautics and aerospace high graduate engineer
16 Bd du Maréchal de Lattre, apt. 095, 21300 Chenove, France

Abstract

Associated Stirling numbers of first and second kind are usually found in the literature in various forms of stairs depending on their order r. Yet, it is shown in this note that all of these numbers can be arranged, through a linear transformation, in the same arithmetical triangle structure as the “Pascal’s triangle”.

Keywords

• Number theory
• Associated Stirling numbers
• Arithmetical triangle

AMS Classification

• 11B73
• 05A18

References

1. Comtet, L. (1970) Analyse combinatoire, PUF.
2. Hilton, P. & Pedersen, J. (1999) Two recurrence relations for Stirling factors, Bulletin Belg. Math. Soc. Simon Stevin, 6, 615–623.
3. Hilton, P., et al. (1994) On partitions, surjections and Stirling numbers, Bulletin Belg. Math. Soc. Simon Stevin, 1, 713–725.
4. Howard, F.T. (1990), Congruences for the Stirling numbers and associated Stirling numbers, Acta Arithmetica, LV, 29–41.
5. Sándor, J., & Crstici, B. (2004) Handbook of number theory II, Kluwer Acad. Publ., Chapter 5, 459–618.
6. Weisstein, Eric W., “Stirling Number of the Second Kind.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html
7. Weisstein, Eric W., “Stirling Number of the First Kind.” From MathWorld–A Wolfram Web Resource.  http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html
8. Weisstein, Eric W., “Permutation Cycle.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PermutationCycle.html