64 research outputs found
An instance of umbral methods in representation theory: the parking function module
We test the umbral methods introduced by Rota and Taylor within the theory of
representation of symmetric group. We define a simple bijection between the set
of all parking functions of length and the set of all noncrossing
partitions of . Then we give an umbral expression of the
Frobenius characteristic of the parking function module introduced by Haiman
that allows an explicit relation between this symmetric function and the volume
polynomial of Pitman and Stanley
A symbolic method for k-statistics
Trough the classical umbral calculus, we provide new, compact and easy to
handle expressions of k-statistics, and more in general of U-statistics. In
addition such a symbolic method can be naturally extended to multivariate case
and to generalized k-statistics.Comment: Extended abstract with corrected typos and change conten
Cumulants and convolutions via Abel polynomials
We provide an unifying polynomial expression giving moments in terms of
cumulants, and viceversa, holding in the classical, boolean and free setting.
This is done by using a symbolic treatment of Abel polynomials. As a
by-product, we show that in the free cumulant theory the volume polynomial of
Pitman and Stanley plays the role of the complete Bell exponential polynomial
in the classical theory. Moreover via generalized Abel polynomials we construct
a new class of cumulants, including the classical, boolean and free ones, and
the convolutions linearized by them. Finally, via an umbral Fourier transform,
we state a explicit connection between boolean and free convolution
Natural statistics for spectral samples
Spectral sampling is associated with the group of unitary transformations
acting on matrices in much the same way that simple random sampling is
associated with the symmetric group acting on vectors. This parallel extends to
symmetric functions, k-statistics and polykays. We construct spectral
k-statistics as unbiased estimators of cumulants of trace powers of a suitable
random matrix. Moreover we define normalized spectral polykays in such a way
that when the sampling is from an infinite population they return products of
free cumulants.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1107 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The classical umbral calculus: Sheffer sequences
Following the approach of Rota and Taylor \cite{SIAM}, we present an
innovative theory of Sheffer sequences in which the main properties are encoded
by using umbrae. This syntax allows us noteworthy computational simplifications
and conceptual clarifications in many results involving Sheffer sequences. To
give an indication of the effectiveness of the theory, we describe applications
to the well-known connection constants problem, to Lagrange inversion formula
and to solving some recurrence relations
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