A generalization of carries processes and Eulerian numbers

Nakano, Fumihiko; Sadahiro, Taizo

doi:10.1016/j.aam.2013.09.005

Cited by 10 publications

(14 citation statements)

References 7 publications

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“…A different q-analogue of Eulerian numbers considered in [25] is of the form P n ∈ E qvn + 1 − v, qv; 1 , which is of type A (1, q, 1); see also [210]. These polynomials also arise in the analysis of carries processes; see [181]. The reciprocal polynomials are of type A (q − 1, q, 1), which appeared on the webpage [156].…”

Section: General Q >mentioning

confidence: 99%

An asymptotic distribution theory for Eulerian recurrences with applications

Hwang

Chern

Duh

2020

Advances in Applied Mathematics

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We study linear recurrences of Eulerian type of the formwith P 0 (v) given, where α(v), β(v) and γ(v) are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of P n (v) for large n using the method of moments and analytic combinatorial tools under varying α(v), β(v) and γ(v), and apply our results to more than two hundred of concrete examples when β(v) = 0 and more than three hundred when β(v) = 0 that we collected from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.

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Section: General Q >mentioning

confidence: 99%

An asymptotic distribution theory for Eulerian recurrences with applications

Hwang

Chern

Duh

2020

Advances in Applied Mathematics

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“…We find explicit formulae for the left and right eigenvectors of the balanced carries chain; we show that the left eigenvectors can be identified with Miller's hyperoctahedral Foulkes characters [17], and that the right eigenvectors can be identified with hyperoctahedral Eulerian idempotents of Bergeron and Bergeron [3]. Before we finished writing this paper, the preprint [18] appeared, and there is some overlap with our work; one can deduce our formula for the left eigenvectors of K(i, j) from their paper.…”

Section: Introductionmentioning

confidence: 97%

Combinatorics of balanced carries

Diaconis

Fulman

2014

Advances in Applied Mathematics

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Abstract. We study the combinatorics of addition using balanced digits, deriving an analog of Holte's "amazing matrix" for carries in usual addition. The eigenvalues of this matrix for base b balanced addition of n numbers are found to be 1, 1/b, · · · , 1/b n , and formulas are given for its left and right eigenvectors. It is shown that the left eigenvectors can be identified with hyperoctahedral Foulkes characters, and the the right eigenvectors can be identified with hyperoctahedral Eulerian idempotents. We also examine the carries that occur when a column of balanced digits is added, showing this process to be determinantal. The transfer matrix method and a serendipitous diagonalization are used to study this determinantal process.

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“…uniformly distributed random variables on [0, 1]. This paper is the continuation of [7] ; here we study (i) the right eigenvectors of P , which yields the correlation of carries of different steps, and (ii) the equivalence to the descent process induced by the repeated generalized riffle shuffles on the colored permutation group. [8] is a review article of our results obtained so far.…”

Section: Background and Definitionmentioning

confidence: 99%

“…In what follows, we first recall the definitions of carries process and some results in [7] (subsection 1.2), and then state our results in this paper(subsection 1.3). To simplify the statements, we shall discuss the positive/negative base simultaneously.…”

Section: Background and Definitionmentioning

confidence: 99%