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Combinatorial parameters on bargraphs of permutations | ||
| Transactions on Combinatorics | ||
| مقاله 1، دوره 7، شماره 2، شهریور 2018، صفحه 1-16 اصل مقاله (271.72 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2017.102359.1483 | ||
| نویسندگان | ||
| Toufik Mansour1؛ Mark Shattuck* 2 | ||
| 1Department of Mathematics, University of Tennessee, Knoxville, TN, USA | ||
| 2Mathematics Department, University of Tennessee, Knoxville, TN, USA | ||
| چکیده | ||
| In this paper, we consider statistics on permutations of length $n$ represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we find the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new refined count of permutations of a given length. To do so, we consider the distribution of the parameters on permutations of a more general multiset of which $\mathcal{S}_n$ is a subset. In addition to finding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit expressions for the sign balance of these statistics, from which the comparable results on permutations follow as special cases. | ||
| کلیدواژه‌ها | ||
| combinatorial statistic؛ $q$-generalization؛ bargraph؛ permutations | ||
| مراجع | ||
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