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The site-perimeter of words | ||
| Transactions on Combinatorics | ||
| مقاله 8، دوره 6، شماره 2، شهریور 2017، صفحه 37-48 اصل مقاله (255.96 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2017.21465 | ||
| نویسندگان | ||
| Aubrey Blecher1؛ Charlotte Brennan* 2؛ Arnold Knopfmacher1؛ Toufik Mansour1 | ||
| 1University of the Witwatersrand | ||
| 21 Jan Smuts Avenue | ||
| چکیده | ||
| We define $[k]=\{1, 2, 3,\ldots,k\}$ to be a (totally ordered) {\em alphabet} on $k$ letters. A {\em word} $w$ of length $n$ on the alphabet $[k]$ is an element of $[k]^n$. A word can be represented by a bargraph which is a family of column-convex polyominoes whose lower edge lies on the $x$-axis and in which the height of the $i$-th column in the bargraph equals the size of the $i$-th part of the word. Thus these bargraphs have heights which are less than or equal to $k$. We consider the site-perimeter, which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the site-perimeter of words is obtained explicitly. From a functional equation we find the average site-perimeter of words of length $n$ over the alphabet $[k]$. We also show how these statistics may be obtained using a direct counting method and obtain the minimum and maximum values of the site-perimeters. | ||
| کلیدواژه‌ها | ||
| words؛ bargraphs؛ site-perimeter؛ generating functions | ||
| مراجع | ||
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[8] S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Discrete Mathematics and its applications, CRC press, Taylor and Francis group, 2010.
[9] N. Sieb en, Polyomino es with minimum site-p erimeter and full set achievement games, Europ. J. Combin., 29 (2008) 108-117.
[10] D. Stauffer and A. Aharony, An Introduction to Percolation Theory, 2nd edition, Taylor and Francis, London, 1992. | ||
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