A Combinatorial Method for Log-Concavity of Rows of Triangular Arrays

Abstract

Recurrences of the form
\[
T(n,k)=c(n,k)\,T(n-1,k)+d(n,k)\,T(n-1,k-1)+\delta_{n,0}\delta_{k,0}
\]
arise in a wide range of classical combinatorial sequences, including the Stirling numbers of both kinds, Lah numbers, and Eulerian numbers. We provide a combinatorial interpretation for arrays \(T(n,k)\) defined by such recurrences and use this framework to study the log-concavity of rows of triangular arrays. Our approach yields new and uniform combinatorial proofs of log-concavity for many well-known combinatorial sequences. Motivated by recent generalizations of these classical arrays, we further investigate recurrences of the form
\[
T(n,k)= (\alpha n+\beta k+\gamma)^l\,T(n-1,k)
+ (\alpha'n+\beta'k+\gamma')^l\,T(n-1,k-1)
+ \delta_{n,0}\delta_{k,0},
\]
where \(l\) is a positive integer. We establish sufficient conditions under which the rows of triangular arrays defined by such recurrences are log-concave. This result implies the log-concavity of several previously unexplored combinatorial arrays and, in particular, confirms a conjecture of Žigon~Tankosič (J. Integer Seq. 26, 2 (2023), article 23.2.6, 16) on the log-concavity of generalized Lah numbers. Our main technique is to interpret the triangular array \((T(n,k))\) in terms of weighted lattice paths and to construct a weight-increasing injection that proves log-concavity. Finally, we introduce a two-parameter generalization of the Eulerian numbers, analogous to the generalized Stirling and Lah numbers. We show that this array is palindromic in \(k\) and conclude with remarks on its $\gamma$-nonnegativity and real-rootedness.