2023/08/09 by Yu-Jie Cui, Bao-Xuan Zhu, Cui, Yu-Jie +1
Computer Science · Engineering · Mathematics · #05A15 #05A20 #15B05 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2308.05167
openalex publication_date 2023/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The main purpose of this paper is to study total positivity of a matrix M=[Mn,k]n,k generated by the weighted lattice paths in ℕ2 from the origin (0,0) to the point (k,n) consisting of types of steps: (0,1) and (1,t+i) for 0≤ i≤ ℓ, where each step (0,1) from height~n-1 gets the weight~bn(y) and each step (1,t+i) from height~n-t-i gets the weight an(i)(x). Using an algebraic method, we prove that the x-total positivity of the weight matrix [ai(i-j)(x)]i,j implies that of M. Furthermore, using the Lindström-Gessel-Viennot lemma, we obtain that both M and the Toeplitz matrix of each row sequence of M with t≥1 are x-totally positive under the following three cases respectively: (1) ℓ=1, (2) ℓ=2 and restrictions for an(i), (3) general ℓ and both a(i)n and bn are independent of n. In addition, for the case (3), we show that the matrix M is a Riordan array, present its explicit formula and prove total positivity of the Toeplitz matrix of the each column of M. In particular, from the results for Toeplitz-total positivity, we also obtain the Pólya frequency and log-concavity of the corresponding sequence. Finally, as applications, we in a unified manner establish total positivity and the Toeplitz-total positivity for many well-known combinatorial triangles, including the Pascal triangle, the Pascal square, the Delannoy triangle, the Delannoy square, the signless Stirling triangle of the first kind, the Legendre-Stirling triangle of the first kind, the Jacobi-Stirling triangle of the first kind, the Brenti's recursive matrix, and so on.