Log-concavity of rows of triangular arrays satisfying a certain super-recurrence

Recurrences of the form equation* T (n, k) = (αn+βk +γ) \ T (n-1, k) + (α'n+β'k+γ') \ T (n-1, k-1) +δ₍, ₀δ₊, ₀. equation* show up as the recurrence for many well-studied combinatorial sequences such as the Stirling numbers of first and second kind, the Lah numbers, Eulerian numbers etc. Recently, many of these sequences have received generalisations that obey a recurrence of the form equation* T (n, k) = (αn+βk +γ) ˡ \ T (n-1, k) + (α'n+β'k+γ') ˡ\ T (n-1, k-1) +δ₍, ₀δ₊, ₀. equation* where l is a positive integer. Many of these generalised sequences also satisfy properties such as unimodality, log-concavity, gamma-nonnegativity, real-rootedness that the original sequences satisfy. In this article, we give sufficient conditions for rows of triangular arrays, arising from the recurrence stated above, to be log-concave. We show that our sufficient condition is satisfied by many of the classical examples, thereby giving a new unified approach to proving their log-concavity. This sufficient condition also confirms a conjecture of Tankosic about the log-concavity of generalised Lah numbers. Our main technique will be to interpret the triangular array (T (n, k) ) as weighted lattice paths and produce an injection that is increasing in weight. Finally, we introduce a two-parameter generalisation of the Eulerian numbers analogous to the generalised Stirling and Lah counterparts. We prove that this sequence is palindromic and make some remarks about their gamma-nonnegativity and real-rootedness.