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 $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-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 $(q, d)$-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

Assessing recent increases in cash demand

Contrary to predictions that demand for cash will decline with the increased availability and use of non-cash payment means, currency demand has increased in the Euro area and the US over the past 15 years. Against this background, this short article summarizes recent findings from Jobst and Stix (2017), who provide a discussion of trends in currency demand, and presents additional descriptive evidence. In a first step, currency demand over a longer period is analyzed for the USA, Germany and the Euro area. This is helpful for understanding and assessing recent trends. In a second step, evidence from 70 economies is analyzed for the period from 2001 to 2014. This broader perspective informs us about the development for currencies that do not circulate internationally. Our descriptive account provides several insights: (i) Recent increases for the euro and the US dollar are strong even if seen over a 100 year horizon. (ii) Over the period from 2001 to 2014 currency demand has increased in many economies. (iii) In economies where currency demand increased, the increase typically happened after the start of the economic and financial crisis of 2007/08. What are the drivers of recent increases in currency demand? Jobst and Stix (2017) estimate panel money demand models, accounting for changes in GDP, interest rates and shadow economic activities. In economies with high GDP, a substantial share of the increase cannot be explained by changes in interest rates or in the size of the shadow economy. We conjecture that the unexplained component is related to increased hoarding

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