In this paper, we consider two statistics on bargraphs, which are defined to be lattice paths in the first quadrant, starting at the origin and ending upon first return to the x-axis. Each bargraph is represented as a sequence of columns π 1 π 2 . . . π m such that column k contains π k cells. First we enumerate interior vertices, where naturally, interior vertex is a vertex that belongs to exactly four cells of bargraphs. Then we enumerate d-edges -edges that contain d interior vertices. More precisely, we find the generating function for the number of bargraphs with n cells and m columns according: to interior vertices and according to horizontal (vertical) d-edges. In addition we consider several special cases in detail, where we obtain asymptotic results for total number of statistics under consideration.
In this short note we prove a conjecture for the interval (0, 1), related to a logarithmically completely monotonic function, presented in [5]. Then, we extend by proving a more generalized theorem. At the end we pose an open problem on a logarithmically completely monotonic function involving q-Digamma function.
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