From Triangular Array Progression to Near-Log-Concave State Transitions

Abstract

The paper introduces an infinite integer sequence progression that produces a triangular array where the $n$th row sums to $2^n$ for every $n\ge 0$. Every row of the triangular array can be realized as the degree sequence of a multigraph without loops. Although the first four rows coincide with those from Pascal's triangle, the proposed array progression becomes asymmetric and diverges from binomial coefficients thereafter. The triangular array hosts infinitely many unimodal near-log-concave sequences with no zeros, and its log-concavity deviation converges to $\log (4/3)$ under logarithmic scaling. The progression introduces near-log-concave random variables, produces state-transitions of an infinite Markov chain, and hosts a totally non-negative Toeplitz matrix of order two. The construct provides a sequence of probability mass functions where the mean and variance decouple, with mean diverging and variance asymptotically approaching a steady-state limit. We study the Shannon entropy dynamics of the construct, and provide a transformation of the triangular array over the Tychonoff cube to produce an infinite family of right-stochastic matrices.