On arithmetic and asymptotic properties of up–down numbers

Let σ=(σ1,…,σN)σ=(σ1,…,σN), where σi=±1σi=±1, and let C(σ)C(σ) denote the number of permutations ππ of 1,2,…,N+1,1,2,…,N+1, whose up–down signature sign(π(i+1)-π(i))=σisign(π(i+1)-π(i))=σi, for i=1,…,Ni=1,…,N. We prove that the set of all up–down numbers C(σ)C(σ) can be expressed by a single universal polynomial ΦΦ, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that ΦΦ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers C(σ)C(σ), for fixed N  . We prove a concise upper bound for C(σ)C(σ), which describes the asymptotic behaviour of the up–down function C(σ)C(σ) in the limit C(σ)⪡(N+1)!C(σ)⪡(N+1)!.