laboratoire de physique statistique
 
 
laboratoire de physique statistique

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DISCRETE MATHEMATICS 


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P U B L I C A T I O N S



 
2007
On arithmetic and asymptotic properties of up-down numbers - Brown, Francis C. S. and Fink, Thomas M. A. and Willbrand, Karen
DISCRETE MATHEMATICS 3071722-1736 (2007)

Abstract : Let sigma = (sigma(1),...,sigma(N) UN), where sigma(1) = +/- 1 and let C(sigma) denote the number of permutations pi of 1, 2,..., N + 1, whose up-down signature sign(pi(i + 1) - pi(i)) = sigma(i), for i = 1,..., N. We prove that the set of all up-down numbers C(sigma) can be expressed by a single universal polynomial Phi, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that Phi is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers C (sigma), for fixed N. We prove a concise upper bound for C (sigma), which describes the asymptotic behaviour of the up-down function C(sigma) in the limit C(sigma) << (N +1)!. (C) 2006 Elsevier B.V. All rights reserved.