Discrete Mathematics Letters www.dmlett.com Discrete Math. Lett. 1 (2019) 26–29 Border and tangent cells in bargraphs Toufik Mansour ∗ Department of Mathematics, University of Haifa, 3498838 Haifa, Israel (Received: 5 October 2018. Received in revised form: 18 November 2018. Accepted: 26 November 2018. Published online: 3 January 2019.) c  2019 the author. This is an open access article under the CC BY (International 4.0) license (https://creativecommons.org/licenses/by/4.0/). Abstract A border cell is a cell inside a bargraph that has at least one edge in common with an outside cell. A tangent cell is a cell inside a bargraph which is not a border cell and that has at least one vertex in common with an outside cell. In this paper, we study the generating function for the number of bargraphs according to the area, the number of border cells and the number of tangent cells. In particular, we find the generating function for the number of bargraphs according to the area and the inner site-perimeter (number of border cells) and tangent cells. Keywords: bargraphs; generating functions; site-perimeter. 2010 Mathematics Subject Classification: 05A18. 1. Introduction A composition of n is a word σ = σ 1 ··· σ m over alphabet of positive integers such that σ 1 + ··· + σ m = n (see [6]). The letters of σ are called parts. Any composition can be represented as a bargraph which is a column convex polyomino, where the lower edge lies on the x-axis. It is drawn on a regular planar lattice grid and made up of square cells. Thus, the size and the number of parts of σ, namely n and m, are the total number of cells and number columns, called area and width, in the representing bargraph, respectively. Moreover, the part σ i is the length of the i-th column in the representing bargraph. For instance, the bargraph of the composition 122343412 of 22 is represented on left side of Figure 1. Let B be any bargraph, the perimeter of B is the number of edges on the boundary of B, the site-perimeter b b b b b b b b b b b t b b b t b b b b Figure 1: The bargraph 122343412 and border/tangent cells. of B is the number of nearest-neighbour cells outside the boundary of B, and the inner site-perimeter is the number of cells inside B that have at least one edge in common with an outside cell. In [4, 5], the polyominoes according to the area and perimeter were enumerated, while in [3] the site-perimeter of bargraphs was considered, for staircase polygons in [5] and for directed animals in [2]. Recently, Blecher, Brennan and Knopfmacher [1] considered the inner site-perimeter in bargraphs. We refine the main result of [1] as follows. Let B be any bargraph. A border cell of B is a cell inside of B that has at least one edge in common with an outside cell of B. Clearly, the inner site-perimeter of B is the number of border cells of B.A tangent cell of B is a cell inside of B which is not a border cell of B and that has at least one vertex in common with an outside cell of B. For instance, in the right side of Figure 1, the border cells are marked by b and the tangent cells are marked by t. Our main result is to enumerate the number of bargraphs according to the area, number of border cells and number of tangent cells. 2. Main results Let C (x, y, p, q) be the generating function for the number of bargraphs according to the area, width, number of border cells and number of tangent cells marked by x, y, p and q, respectively. * E-mail address: tmansour@univ.haifa.ac.il