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In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as A 050446 on OEIS. This number is exactly the two variable array T (n , m) defined by the recursion T (n , m) = T (n , m − 1) + ∑ k = 0 ⌊ n − 1 2 ⌋ T (2 k , m − 1) T (n − 1 − 2 k , m) , where T (n , 0) = T (0 , m) = 1 for all nonnegative integers m , n. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to both the row and column generating function of T (n , m). In this paper, we give explicit formula of the column generating function, which is also the generating function F (n , x) studied by Bóna, Ju, and Yoshida. We also get trig function representations by using Chebyshev polynomials of the second kind. This allows us to prove all these conjectures. [ABSTRACT FROM AUTHOR]

• We give a combinatorial proof of the parity unimodality of the (3, n) rational Catalan polynomials Q 3 , n (q). The idea is that we construct a poset whose rank generating function is Q 3 , n (q). Then we find that this poset has an identical domino chain decomposition, which implies the parity unimodality of Q 3 , n (q). A sequence a 0 , a 1 , a 2 , ⋯ is parity unimodal if it exists k 1 and k 2 such that a 0 ≤ a 2 ≤ ⋯ ≤ a k 1 ≥ a k 1 + 2 ≥ ⋯ and a 1 ≤ a 3 ≤ ⋯ ≤ a k 2 ≥ a k 2 + 2 ≥ ⋯. A polynomial f (q) = ∑ i = 0 n a i q i is parity unimodal if its coefficient sequence a 0 , a 1 , ⋯ , a n is parity unimodal. Recently, Xin and Zhong conjectured that the (m , n) -rational q -Catalan polynomial is parity unimodal. They showed that this conjecture holds for m ≤ 5 using the constant term method and generating functions. In this paper, we give a combinatorial proof of the m = 3 case. [ABSTRACT FROM AUTHOR]