A combinatorial proof of the log-convexity of sequences in Riordan arrays

A Riordan array R = [ r n , k ] n , k ≥ 0 can be characterized by two sequences A = ( a n ) n ≥ 0 and Z = ( z n ) n ≥ 0 such that r 0 , 0 = 1 , r 0 , k = 0 ( k ≥ 1 ) and r n + 1 , 0 = ∑ j ≥ 0 z j r n , j , r n + 1 , k + 1 = ∑ j ≥ 0 a j r n , k + j for n , k ≥ 0 . Using an algebraic approach, Chen, Liang and Wang showed that the sequence ( r n , 0 ) n ≥ 0 is log-convex if the production matrix [ ζ , A ] is TP 2 , where ζ = [ z 0 , z 1 , z 2 , … ] ′ and A = [ a n - k ] n , k ≥ 0 is the Toeplitz matrix of the A -sequence. In this paper, we present an injective proof of this result from the point of view of weighted Łukasiewicz paths, which gives a combinatorial proof of the log-convexity of many combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the Schröder numbers, the central binomial coefficients, and the restricted hexagonal numbers, in a unified approach. This method also present an injective proof of the strong q -log-convexity of many well-known combinatorial polynomials.