14 research outputs found
Reduction of -Regular Noncrossing Partitions
In this paper, we present a reduction algorithm which transforms -regular
partitions of to -regular partitions of .
We show that this algorithm preserves the noncrossing property. This yields a
simple explanation of an identity due to Simion-Ullman and Klazar in connection
with enumeration problems on noncrossing partitions and RNA secondary
structures. For ordinary noncrossing partitions, the reduction algorithm leads
to a representation of noncrossing partitions in terms of independent arcs and
loops, as well as an identity of Simion and Ullman which expresses the Narayana
numbers in terms of the Catalan numbers
Enumeration of -noncrossing partitions
A set partition is said to be -noncrossing if it avoids the pattern
. We find an explicit formula for the ordinary generating
function of the number of -noncrossing partitions of
when .Comment: 9 pages, 1 tabl
Identities involving Narayana polynomials and Catalan numbers
We first establish the result that the Narayana polynomials can be
represented as the integrals of the Legendre polynomials. Then we represent the
Catalan numbers in terms of the Narayana polynomials by three different
identities. We give three different proofs for these identities, namely, two
algebraic proofs and one combinatorial proof. Some applications are also given
which lead to many known and new identities.Comment: 13 pages,6 figure
Heisenberg characters, unitriangular groups, and Fibonacci numbers
Let \UT_n(\FF_q) denote the group of unipotent upper triangular
matrices over a finite field with elements. We show that the Heisenberg
characters of \UT_{n+1}(\FF_q) are indexed by lattice paths from the origin
to the line using the steps , which are
labeled in a certain way by nonzero elements of \FF_q. In particular, we
prove for that the number of Heisenberg characters of
\UT_{n+1}(\FF_q) is a polynomial in with nonnegative integer
coefficients and degree , whose leading coefficient is the th Fibonacci
number. Similarly, we find that the number of Heisenberg supercharacters of
\UT_n(\FF_q) is a polynomial in whose coefficients are Delannoy numbers
and whose values give a -analogue for the Pell numbers. By counting the
fixed points of the action of a certain group of linear characters, we prove
that the numbers of supercharacters, irreducible supercharacters, Heisenberg
supercharacters, and Heisenberg characters of the subgroup of \UT_n(\FF_q)
consisting of matrices whose superdiagonal entries sum to zero are likewise all
polynomials in with nonnegative integer coefficients.Comment: 25 pages; v2: material significantly revised and condensed; v3: minor
corrections, final versio
Actions and identities on set partitions
A labeled set partition is a partition of a set of integers whose arcs are
labeled by nonzero elements of an abelian group . Inspired by the action of
the linear characters of the unitriangular group on its supercharacters, we
define a group action of on the set of -labeled partitions of an
-set. By investigating the orbit decomposition of various families of
set partitions under this action, we derive new combinatorial proofs of Coker's
identity for the Narayana polynomial and its type B analogue, and establish a
number of other related identities. In return, we also prove some enumerative
results concerning Andr\'e and Neto's supercharacter theories of type B and D.Comment: 28 pages; v3: material revised with additional final sectio