10 research outputs found
The Rank and Minimal Border Strip Decompositions of a Skew Partition
Nazarov and Tarasov recently generalized the notion of the rank of a
partition to skew partitions. We give several characterizations of the rank of
a skew partition and one possible characterization that remains open. One of
the characterizations involves the decomposition of a skew shape into a minimal
number of border strips, and we develop a theory of these MBSD's as well as of
the closely related minimal border strip tableaux. An application is given to
the value of a character of the symmetric group S_n indexed by a skew shape z
at a permutation whose number of cycles is the rank of z.Comment: 31 pages, 10 figure
Power sum expansion of chromatic quasisymmetric functions
The chromatic quasisymmetric function of a graph was introduced by Shareshian
and Wachs as a refinement of Stanley's chromatic symmetric function. An
explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing
the chromatic quasisymmetric function of the incomparability graph of a natural
unit interval order in terms of power sum symmetric functions, is proven. The
proof uses a formula of Roichman for the irreducible characters of the
symmetric group.Comment: Final version, 9 pages; comments by a referee incorporate
The excedances and descents of bi-increasing permutations
Starting from some considerations we make about the relations between certain
difference statistics and the classical permutation statistics we study
permutations whose inversion number and excedance difference coincide. It turns
out that these (so-called bi-increasing) permutations are just the 321-avoiding
ones. The paper investigates their excedance and descent structure. In
particular, we find some nice combinatorial interpretations for the
distribution coefficients of the number of excedances and descents,
respectively, and their difference analogues over the bi-increasing
permutations in terms of parallelogram polyominoes and 2-Motzkin paths. This
yields a connection between restricted permutations, parallelogram polyominoes,
and lattice paths that reveals the relations between several well-known
bijections given for these objects (e.g. by Delest-Viennot,
Billey-Jockusch-Stanley, Francon-Viennot, and Foata-Zeilberger). As an
application, we enumerate skew diagrams according to their rank and give a
simple combinatorial proof for a result concerning the symmetry of the joint
distribution of the number of excedances and inversions, respectively, over the
symmetric group.Comment: 36 page