On arithmetic and asymptotic properties of up-down numbers

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