Toufik Mansour and Armend Sh. Shabani

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 2, Pages 181-189

DOI: 10.7546/nntdm.2019.25.2.181-189

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## Details

### Authors and affiliations

Toufik Mansour

*Department of Mathematics, University of Haifa
3498838 Haifa, Israel
*

Armend Sh. Shabani

*Department of Mathematics, University of Prishtina
10000 Prishtine, Republic of Kosovo*

### Abstract

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.

### Keywords

- Bargraphs
- Generating functions
- Interior vertices
- Interior edges

### 2010 Mathematics Subject Classification

- 05A18

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## Cite this paper

APAMansour, T. & Shabani, Armend Sh. (2019). Interior vertices and edges in bargraphs Notes on Number Theory and Discrete Mathematics, 25(2), 181-189, doi: 10.7546/nntdm.2019.25.2.181-189.

ChicagoMansour, T. and Armend Sh. Shabani. “Interior vertices and edges in bargraphs.” Notes on Number Theory and Discrete Mathematics 25, no. 2 (2019): 181-189, doi: 10.7546/nntdm.2019.25.2.181-189.

MLAMansour, T. and Armend Sh. Shabani. “Interior vertices and edges in bargraphs.” Notes on Number Theory and Discrete Mathematics 25.2 (2019): 181-189. Print, doi: 10.7546/nntdm.2019.25.2.181-189.