论文信息 - Signed excedance enumeration via determinants

Signed excedance enumeration via determinants

We show a simple connection between determinants and signed-excedance enumeration of permutations. This gives us an alternate proof of a result of Mantaci about enumerating signed excedances in permutations. The connection also gives an alternate proof of a result of Mantaci and Rakotondrajao about enumerating signed excedances over derangements.Motivated by this connection, we define several excedance-like statistics on permutations and show interesting values for their signed enumerator. In some cases, we also obtain the signed excedance-like statistic enumerator with respect to positive integral weights.

Sivaramakrishnan Sivasubramanian

[1] J.-L. Loday,et al. Opérations sur l'homologie cyclique des algèbres commutatives , 1989 .

[2] Jacques Désarménien,et al. The signed Eulerian numbers , 1992, Discret. Math..

[3] Toufik Mansour,et al. Combinatorics of Compositions and Words: Solutions Manual , 2009 .

[4] P. A. Macmahon,et al. The Indices of Permutations and the Derivation Therefrom of Functions of a Single Variable Associated with the Permutations of any Assemblage of Objects , 1913 .

[5] Roberto Mantaci,et al. Exceedingly deranging! , 2003, Adv. Appl. Math..

[6] S. Tanimoto. A STUDY OF EULERIAN NUMBERS FOR PERMUTATIONS IN THE ALTERNATING GROUP , 2006 .

[7] R. Brualdi. Introductory Combinatorics , 1992 .

[8] The signed Eulerian numbers on involutions , 2008 .

[9] Roberto Mantaci. Binomial Coefficients and Anti-exceedances of Even Permutations: A Combinatorial Proof , 1993, J. Comb. Theory, Ser. A.

[10] David Garber,et al. On the excedance number of colored permutation groups , 2005, math/0505436.

[11] Jiang Zeng,et al. TWO INVOLUTIONS FOR SIGNED EXCEDANCE NUMBERS , 2002 .

[12] D. Foata. ON THE NETTO INVERSION NUMBER OF A SEQUENCE , 1968 .

[13] Michelle L. Wachs,et al. An involution for signed Eulerian numbers , 1992, Discret. Math..