publication . Preprint . 2006

A New Approach to Signed Eulerian Numbers

Tanimoto, Shinji;
Open Access English
  • Published: 13 Feb 2006
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the recurrence relation for signed Eulerian numbers is deduced, which was obtained by an analytic method. Our approach is straightforward and enables us to deduce other properties including divisibility by prime powers.
arXiv: Mathematics::Combinatorics
free text keywords: Mathematics - Combinatorics, 05A05, 20B30
Download from

1. J. D´esarm´enien and D. Foata, The signed Eulerian numbers, Discrete Math. 99 (1992) 49-58. [OpenAIRE]

2. D. Foata, and M.-P. Schu¨tzenberger, Th´eorie G´eom´etrique des Polynˆomes Eul´eriens, Lecture Notes in Mathematics, Vol. 138, Springer-Verlag, Berlin, 1970.

3. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 1989.

4. A. Kerber, Algebraic Combinatorics Via Finite Group Actions. BI-Wissenschaftsverlag, Mannheim, 1991.

5. D. E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching. AddisonWesley, Reading, 1973.

6. L. Lesieur and J.-L. Nicolas, On the Eulerian numbers Mn = max1≤k≤n A(n, k), European J. Combin. 13 (1992) 379-399.

7. J.-L. Loday, Op´erations sur l'homologie cyclique des alg`ebres commutatives, Invent. Math. 96 (1989) 205-230.

8. R. Mantaci, Binomial coefficients and anti-excedances of even permutations: A combinatorial proof, J. of Comb. Theory (A) 63 (1993) 330-337. [OpenAIRE]

9. S. Tanimoto, An operator on permutations and its application to Eulerian numbers, European J. Combin. 22 (2001) 569-576. [OpenAIRE]

10. S. Tanimoto, A study of Eulerian numbers by means of an operator on permutations, European J. Combin. 24 (2003) 34-44.

11. S. Tanimoto, On the numbers of orbits of permutations under an operator related to Eulerian numbers, Annals of Combin. 8 (2004) 239-250.

Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue