8,852 research outputs found
Canonical characters on quasi-symmetric functions and bivariate Catalan numbers
Every character on a graded connected Hopf algebra decomposes uniquely as a
product of an even character and an odd character (Aguiar, Bergeron, and
Sottile, math.CO/0310016).
We obtain explicit formulas for the even and odd parts of the universal
character on the Hopf algebra of quasi-symmetric functions. They can be
described in terms of Legendre's beta function evaluated at half-integers, or
in terms of bivariate Catalan numbers:
Properties of characters and of quasi-symmetric functions are then used to
derive several interesting identities among bivariate Catalan numbers and in
particular among Catalan numbers and central binomial coefficients
Overpartitions, lattice paths and Rogers-Ramanujan identities
We extend partition-theoretic work of Andrews, Bressoud, and Burge to
overpartitions, defining the notions of successive ranks, generalized Durfee
squares, and generalized lattice paths, and then relating these to
overpartitions defined by multiplicity conditions on the parts. This leads to
many new partition and overpartition identities, and provides a unification of
a number of well-known identities of the Rogers-Ramanujan type. Among these are
Gordon's generalization of the Rogers-Ramanujan identities, Andrews'
generalization of the G\"ollnitz-Gordon identities, and Lovejoy's ``Gordon's
theorems for overpartitions.
Identities for classical group characters of nearly rectangular shape
We derive several identities that feature irreducible characters of the
general linear, the symplectic, the orthogonal, and the special orthogonal
groups. All the identities feature characters that are indexed by shapes that
are "nearly" rectangular, by which we mean that the shapes are rectangles
except for one row or column that might be shorter than the others. As
applications we prove new results in plane partitions and tableaux enumeration,
including new refinements of the Bender-Knuth and MacMahon (ex-)conjectures.Comment: 55 pages, AmS-TeX; to appear in J. Algebr
Permutation Statistics on the Alternating Group
Let denote the alternating and the symmetric groups on
. MacMahaon's theorem, about the equi-distribution of the length and
the major indices in , has received far reaching refinements and
generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and
followers. Our main goal is to find analogous statistics and identities for the
alternating group . A new statistic for , {\it the delent number},
is introduced. This new statistic is involved with new equi-distribution
identities, refining some of the results of Foata-Schutzenberger and
Garsia-Gessel. By a certain covering map , such
identities are `lifted' to , yielding the corresponding
equi-distribution identities.Comment: 45 page
On the weighted enumeration of alternating sign matrices and descending plane partitions
We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices
and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359]
that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs)
for which the 1 of the first row is in column k+1 and there are exactly m -1's
and m+p inversions is equal to the number of descending plane partitions (DPPs)
for which each part is at most n and there are exactly k parts equal to n, m
special parts and p nonspecial parts. The proof involves expressing the
associated generating functions for ASMs and DPPs with fixed n as determinants
of nxn matrices, and using elementary transformations to show that these
determinants are equal. The determinants themselves are obtained by standard
methods: for ASMs this involves using the Izergin-Korepin formula for the
partition function of the six-vertex model with domain-wall boundary
conditions, together with a bijection between ASMs and configurations of this
model, and for DPPs it involves using the Lindstrom-Gessel-Viennot theorem,
together with a bijection between DPPs and certain sets of nonintersecting
lattice paths.Comment: v2: published versio
The blob complex
Given an n-manifold M and an n-category C, we define a chain complex (the
"blob complex") B_*(M;C). The blob complex can be thought of as a derived
category analogue of the Hilbert space of a TQFT, and as a generalization of
Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice
formal properties, including a higher dimensional generalization of Deligne's
conjecture about the action of the little disks operad on Hochschild cochains.
Along the way, we give a definition of a weak n-category with strong duality
which is particularly well suited for work with TQFTs.Comment: 106 pages. Version 3 contains many improvements following suggestions
from the referee and others, and some additional materia
The generalized Borwein conjecture. II. Refined q-trinomial coefficients
Transformation formulas for four-parameter refinements of the q-trinomial
coefficients are proven. The iterative nature of these transformations allows
for the easy derivation of several infinite series of q-trinomial identities,
and can be applied to prove many instances of Bressoud's generalized Borwein
conjecture.Comment: 36 pages, AMS-LaTe
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