Macdonald Symmetric Functions and Parking Functions
| Speaker: | Mike Zabrocki |
|---|---|
| Affiliation: | York University |
| Room: | Mathematics and Computer Building (MC) 5479 |
Abstract:
A
parking
function
can
be
thought
of
as
a
Dyck
path
of
length
n
where
the
vertical
edges
are
labeled
with
the
integers
1
through
n,
increasing
in
the
columns.
Haglund's
"shuffle
conjecture"
from
2005
is
a
combinatorial
formula
for
the
symmetric
function
expression
for
$\nabla(e_n)$
with
one
term
for
each
parking
function.
In
2008
Haglund,
Morse
and
myself
extended
this
conjecture
to
the
action
of
nabla
on
a
symmetric
function
indexed
by
a
composition.
The
combinatorial
formula
has
one
term
for
each
parking
function
which
touches
the
diagonal
according
to
the
composition.
In
the
last
few
years
Hicks,
Garsia,
Xin
and
myself
were
partially
able
to
prove
the
compositional
shuffle
conjecture
by
showing
algebraic
recurrences
on
coefficients
exist
and
agree
with
the
combinatorics.
In
this
talk
I'll
show
algebraic
recurrences
which
generalize
those
that
were
used
to
prove
those
results
and
draws
a
direct
connection
with
Macdonald
symmetric
functions.