1,304 research outputs found
The height of multiple edge plane trees
Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane
trees where multiple edges are allowed. We first show that -ary multi-edge
trees where the out-degrees are bounded by are in bijection with classical
-ary trees. This allows us to analyse parameters such as the height.
The main part of this paper is concerned with multi-edge trees counted by
their number of edges. The distribution of the number of vertices as well as
the height are analysed asymptotically
Analysis of Carries in Signed Digit Expansions
The number of positive and negative carries in the addition of two
independent random signed digit expansions of given length is analyzed
asymptotically for the -system and the symmetric signed digit
expansion. The results include expectation, variance, covariance between the
positive and negative carries and a central limit theorem.
Dependencies between the digits require determining suitable transition
probabilities to obtain equidistribution on all expansions of given length. A
general procedure is described to obtain such transition probabilities for
arbitrary regular languages.
The number of iterations in von Neumann's parallel addition method for the
symmetric signed digit expansion is also analyzed, again including expectation,
variance and convergence to a double exponential limiting distribution. This
analysis is carried out in a general framework for sequences of generating
functions
A precise description of the p-adic valuation of the number of alternating sign matrices
Following Sun and Moll, we study v_p(T(N)), the p-adic valuation of the
counting function of the alternating sign matrices. We find an exact analytic
expression for it that exhibits the fluctuating behaviour, by means of Fourier
coefficients. The method is the Mellin-Perron technique, which is familiar in
the analysis of the sum-of-digits function and related quantities
Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal\u27s Rhombus
The summatory function of a q-regular sequence in the sense of Allouche and Shallit is analysed asymptotically. The result is a sum of periodic fluctuations multiplied by a scaling factor. Each summand corresponds to an eigenvalues of absolute value larger than the joint spectral radius of the matrices of a linear representation of the sequence. The Fourier coefficients of the fluctuations are expressed in terms of residues of the corresponding Dirichlet generating function. A known pseudo Tauberian argument is extended in order to overcome convergence problems in Mellin-Perron summation.
Two examples are discussed in more detail: The case of sequences defined as the sum of outputs written by a transducer when reading a q-ary expansion of the input and the number of odd entries in the rows of Pascal\u27s rhombus
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