33 research outputs found
The combinatorics of Jeff Remmel
We give a brief overview of the life and combinatorics of Jeff Remmel, a mathematician with successful careers in both logic and combinatorics
Signed Mahonians
A classical result of MacMahon gives a simple product formula for the
generating function of major index over the symmetric group. A similar
factorial-type product formula for the generating function of major index
together with sign was given by Gessel and Simion. Several extensions are given
in this paper, including a recurrence formula, a specialization at roots of
unity and type analogues.Comment: 23 page
Labeled Partitions with Colored Permutations
In this paper, we extend the notion of labeled partitions with ordinary
permutations to colored permutations in the sense that the colors are endowed
with a cyclic structure. We use labeled partitions with colored permutations to
derive the generating function of the indices of colored
permutations. The second result is a combinatorial treatment of a relation on
the q-derangement numbers with respect to colored permutations which leads to
the formula of Chow for signed permutations and the formula of Faliharimalala
and Zeng [10] on colored permutations. The third result is an involution on
permutations that implies the generating function formula for the signed
q-counting of the major indices due to Gessel and Simon. This involution can be
extended to signed permutations. In this way, we obtain a combinatorial
interpretation of a formula of Adin, Gessel and Roichman.Comment: 14 page
Advanced Determinant Calculus: A Complement
This is a complement to my previous article "Advanced Determinant Calculus"
(S\'eminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the
present article, I share with the reader my experience of applying the methods
described in the previous article in order to solve a particular problem from
number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12
(2003), 441-456). Moreover, I add a list of determinant evaluations which I
consider as interesting, which have been found since the appearance of the
previous article, or which I failed to mention there, including several
conjectures and open problems.Comment: AmS-LaTeX, 85 pages; Final, largely revised versio