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We report here on the computational verification of Zhi-Wei Sun's "1-3-5 conjecture" for all natural numbers up to 105 103 560 126. This, together with a result of two of the authors, completes the proof of that conjecture. Furthermore, the computations made in the verification process of the 1-3-5 conjecture revealed a refinement, which we state as a separate conjecture at the end of the paper. [ABSTRACT FROM AUTHOR]

A sequence e = e 1 e 2 ⋯ e n of natural numbers is called an inversion sequence if 0 ≤ e i ≤ i − 1 for all i ∈ { 1 , 2 , ... , n }. Recently, Martinez and Savage initiated an investigation of inversion sequences that avoid patterns of relation triples. Let ρ 1 , ρ 2 and ρ 3 be among the binary relations { < , > , ≤ , ≥ , = , ≠ , − }. Martinez and Savage defined I n (ρ 1 , ρ 2 , ρ 3) as the set of inversion sequences of length n such that there are no indices 1 ≤ i < j < k ≤ n with e i ρ 1 e j , e j ρ 2 e k and e i ρ 3 e k. In this paper, we will prove a curious identity concerning the ascent statistic over the sets I n (> , ≠ , ≥) and I n (≥ , ≠ , >). This confirms a recent conjecture of Zhicong Lin. [ABSTRACT FROM AUTHOR]