In mathematics, the Genocchi numbers Gn, named after Angelo Genocchi, are a sequence of integers that satisfy the relation

The first few Genocchi numbers are 0, 1, 1, 0, 1, 0, 3, 0, 17 (sequence A226158 in the OEIS), see OEIS: A001469.

Properties

Combinatorial interpretations

The exponential generating function for the signed even Genocchi numbers (1)nG2n is

They enumerate the following objects:

  • Permutations in S2n1 with descents after the even numbers and ascents after the odd numbers.
  • Permutations π in S2n2 with 1  π(2i1)  2n2i and 2n2i  π(2i)  2n2.
  • Pairs (a1,,an1) and (b1,,bn1) such that ai and bi are between 1 and i and every k between 1 and n1 occurs at least once among the ai's and bi's.
  • Reverse alternating permutations a1 < a2 > a3 < a4 >>a2n1 of [2n1] whose inversion table has only even entries.

See also

References

  • Weisstein, Eric W. "Genocchi Number". MathWorld.
  • Richard P. Stanley (1999). Enumerative Combinatorics, Volume 2, Exercise 5.8. Cambridge University Press. ISBN 0-521-56069-1
  • Gérard Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Seminaire de Théorie des Nombres de Bordeaux, Volume 11 (1981-1982)
  • Serkan Araci, Mehmet Acikgoz, Erdoğan Şen, Some New Identities of Genocchi Numbers and Polynomials
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.