110 research outputs found
Complementary Riordan arrays
Abstract Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas for the elements of many recursive matrices
Human and constructive proof of combinatorial identities: an example from Romik
International audienceIt has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on "automated'' proofs. As a meaningful example, we consider the four formulas by Romik, related to Motzkin and central trinomial numbers. We show that a proof of these identities can be obtained by using the method of coefficients, a human method only requiring hand computations
Generic MANOVA limit theorems for products of projections
We study the convergence of the empirical spectral distribution of
for orthogonal projection
matrices and , where
and
converge as , to Wachter's MANOVA law. Using free probability, we
show mild sufficient conditions for convergence in moments and in probability,
and use this to prove a conjecture of Haikin, Zamir, and Gavish (2017) on
random subsets of unit-norm tight frames. This result generalizes previous ones
of Farrell (2011) and Magsino, Mixon, and Parshall (2021). We also derive an
explicit recursion for the difference between the empirical moments
and the limiting
MANOVA moments, and use this to prove a sufficient condition for convergence in
probability of the largest eigenvalue of to
the right edge of the support of the limiting law in the special case where
that law belongs to the Kesten-McKay family. As an application, we give a new
proof of convergence in probability of the largest eigenvalue when
is unitarily invariant; equivalently, this determines the limiting operator
norm of a rectangular submatrix of size of a
Haar-distributed unitary matrix for any .
Unlike previous proofs, we use only moment calculations and non-asymptotic
bounds on the unitary Weingarten function, which we believe should pave the way
to analyzing the largest eigenvalue for products of random projections having
other distributions.Comment: 49 pages, 1 figur
Counting Dyck paths by area and rank
The set of Dyck paths of length inherits a lattice structure from a
bijection with the set of noncrossing partitions with the usual partial order.
In this paper, we study the joint distribution of two statistics for Dyck
paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the
lattice).
While area for Dyck paths has been studied, pairing it with this rank
function seems new, and we get an interesting -refinement of the Catalan
numbers. We present two decompositions of the corresponding generating
function: one refines an identity of Carlitz and Riordan; the other refines the
notion of -nonnegativity, and is based on a decomposition of the
lattice of noncrossing partitions due to Simion and Ullman.
Further, Biane's correspondence and a result of Stump allow us to conclude
that the joint distribution of area and rank for Dyck paths equals the joint
distribution of length and reflection length for the permutations lying below
the -cycle in the absolute order on the symmetric group.Comment: 24 pages, 7 figures. Connections with work of C. Stump
(arXiv:0808.2822v2) eliminated the need for 5 pages of proof in the first
draf
- …