222 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
Random matrices, continuous circular systems and the triangular operator
We present a Hilbert space approach to the limit joint *-distributions of
complex independent Gaussian random matrices. For that purpose, we use a
suitably defined family of creation and annihilation operators living in some
direct integral of Hilbert spaces. These operators are decomposed in terms of
continuous circular systems of operators acting between the fibers of the
considered Hilbert space direct integral. In the case of square matrices with
i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the
case of upper-triangular matrices with i.i.d. entries, we obtain the triangular
operators of Dykema and Haagerup. We apply this approach to give a bijective
proof of a formula for *-moments of the triangular operator, using the
enumeration formula of Chauve, Dulucq and Rechnizter for alternating ordered
rooted trees.Comment: 26 pages, 5 figures, 1 reference added, minor change
A second look at the toric h-polynomial of a cubical complex
We provide an explicit formula for the toric -contribution of each cubical
shelling component, and a new combinatorial model to prove Clara Chan's result
on the non-negativity of these contributions. Our model allows for a variant of
the Gessel-Shapiro result on the -polynomial of the cubical lattice, this
variant may be shown by simple inclusion-exclusion. We establish an isomorphism
between our model and Chan's model and provide a reinterpretation in terms of
noncrossing partitions. By discovering another variant of the Gessel-Shapiro
result in the work of Denise and Simion, we find evidence that the toric
-polynomials of cubes are related to the Morgan-Voyce polynomials via
Viennot's combinatorial theory of orthogonal polynomials.Comment: Minor correction
Parity Reversing Involutions on Plane Trees and 2-Motzkin Paths
The problem of counting plane trees with edges and an even or an odd
number of leaves was studied by Eu, Liu and Yeh, in connection with an identity
on coloring nets due to Stanley. This identity was also obtained by Bonin,
Shapiro and Simion in their study of Schr\"oder paths, and it was recently
derived by Coker using the Lagrange inversion formula. An equivalent problem
for partitions was independently studied by Klazar. We present three parity
reversing involutions, one for unlabelled plane trees, the other for labelled
plane trees and one for 2-Motzkin paths which are in one-to-one correspondence
with Dyck paths.Comment: 8 pages, 4 figure
Enumeration of connected Catalan objects by type
Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted
plane trees are four classes of Catalan objects which carry a notion of type.
There exists a product formula which enumerates these objects according to
type. We define a notion of `connectivity' for these objects and prove an
analogous product formula which counts connected objects by type. Our proof of
this product formula is combinatorial and bijective. We extend this to a
product formula which counts objects with a fixed type and number of connected
components. We relate our product formulas to symmetric functions arising from
parking functions. We close by presenting an alternative proof of our product
formulas communicated to us by Christian Krattenthaler which uses generating
functions and Lagrange inversion
A self-dual poset on objects counted by the Catalan numbers and a type-B analogue
We introduce two partially ordered sets, and , of the same
cardinalities as the type-A and type-B noncrossing partition lattices. The
ground sets of and are subsets of the symmetric and the
hyperoctahedral groups, consisting of permutations which avoid certain
patterns. The order relation is given by (strict) containment of the descent
sets. In each case, by means of an explicit order-preserving bijection, we show
that the poset of restricted permutations is an extension of the refinement
order on noncrossing partitions. Several structural properties of these
permutation posets follow, including self-duality and the strong Sperner
property. We also discuss posets and similarly associated with
noncrossing partitions, defined by means of the excedence sets of suitable
pattern-avoiding subsets of the symmetric and hyperoctahedral groups.Comment: 15 pages, 2 figure
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