On the Number of Nilpotents in the Partial Symmetric Semigroup

Abstract

In this note, we obtain and discuss formulae for the total number of nilpotent partial and nilpotent partial one–one transformations of a finite set.

Key Words:

• Semigroup
• Idempotent
• Nilpotent
• Symmetric semigroup
• Symmetric inverse semigroup
• Partial symmetric semigroup
• Stirling numbers
• Lah numbers
• Generalized Laguerre polynomials

2000 AMS Subject Classification:

• 20M18
• 20M20
• 05A10
• 05A15

Acknowledgments

The authors gratefully acknowledge the support of King Fahd University of Petroleum and Minerals. They also thank the referee and David Easdown for their helpful suggestions.

Notes

^#Communicated by D. Easdown.