On shortening u-cycles and u-words for permutations
Abstract This paper initiates the study of shortening universal cycles (u-cycles) and universal words (u-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for n -permutations exist of lengths n ! + ( 1 − k ) ( n − 1 ) for k = 0 , 1 , … , ( n − 2 ) ! .
Mahonian STAT on words
In 2000, Babson and Steingrimsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006.In 2010, Burstein refined some of these results by giving a bijection between permutations with a fixed value for the major index and those with the same value for STAT , where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingrimsson's paper. Several other statistics are preserved as well by Burstein's bijection.At…
Gray coding cubic planar maps
International audience; The idea of (combinatorial) Gray codes is to list objects in question in such a way that two successive objects differ in some pre-specified small way. In this paper, we utilize beta-description trees to cyclicly Gray code three classes of cubic planar maps, namely, bicubic planar maps, 3-connected cubic planar maps, and cubic non-separable planar maps. (C) 2015 Elsevier B.V. All rights reserved.