Title: Chain decompositions for q,t-Catalan numbers
| Speaker: | Nick Loehr |
| Affiliation: | Virginia Tech |
| Zoom: | Contact Karen Yeats |
Abstract:
The
q,t-Catalan
numbers
Cat_n(q,t)
are
polynomials
in
q
and
t
that
reduce
to
the
ordinary
Catalan
numbers
when
q=t=1.
These
polynomials
have
important
connections
to
representation
theory,
algebraic
geometry,
and
symmetric
functions.
Work
of
Garsia,
Haglund,
and
Haiman
has
given
us
combinatorial
formulas
for
Cat_n(q,t)
as
sums
of
Dyck
vectors
weighted
by
area
and
dinv.
This
talk
narrates
our
ongoing
quest
for
a
bijective
proof
of
the
notorious
symmetry
property
Cat_n(q,t)=Cat_n(t,q).
We
describe
some
structural
decompositions
of
integer
partitions
into
infinite
chains
that
can
be
paired
to
prove
the
symmetry
of
certain
coefficients
in
Cat_n(q,t).
The
chains
are
built
from
initial
objects
by
applying
an
operator
NU
that
increases
dinv
by
1
and
reduces
area
by
1.
A
remarkable
feature
of
these
chains
is
that
they
are
independent
of
n
and
explain
symmetry
for
all
n
simultaneously.
Our
chain
construction
leads
to
a
combinatorial
proof
that
for
all
k<12
and
all
n,
the
terms
in
Cat_n(q,t)
of
total
degree
n(n-1)/2-k
satisfy
the
required
symmetry
property.