2025/08/17 by Umesh Shankar, Shankar, Umesh
Computer Science · Engineering · #05A10 #05A20 #11B73 #Antenna Design and Optimization #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2508.12467
openalex publication_date 2025/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recurrences of the form T(n,k) = (αn+βk +γ) T(n-1,k) + (α'n+β'k+γ') T(n-1,k-1)+δn,0δk,0. 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 T(n,k) = (αn+βk +γ)l T(n-1,k) + (α'n+β'k+γ')l T(n-1,k-1)+δn,0δk,0. 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.