Searcharxiv⌕ Search

arXiv subjects

Sergey Kitaev

Publications and source records attributed to Sergey Kitaev.

131 records · Page 8Linked to original sources
arXiv2003

On unavoidable sets of word patterns

We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern overlaps introduced in this paper, which is a subgraph of the de Bruijn graph and which we prove to be Hamiltonian. In other cases we reduce a problem under consideration to known facts on unavoidable sets of words. We also give a relation between our problem and intensively studied universal cycles, and prove there exists a universal cycle for word patterns of any length over any alphabet. Keywords: pattern, word, (un)avoidability, de Bruijn graph, universal cycles

math.CO↗
arXiv2003

Independent sets in certain classes of (almost) regular graphs

We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some cases, construct the corresponding bijections.

math.CO↗
arXiv2002

Generalized pattern avoidance with additional restrictions

Babson and Steingr\'ımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider n-permutations that avoid the generalized pattern 1-32 and whose k rightmost letters form an increasing subword. The number of such permutations is a linear combination of Bell numbers. We find a bijection between these permutations and all partitions of an $(n-1)$-element set with one subset marked that satisfy certain additional conditions. Also we find the e.g.f. for the number of permutations that avoid a generalized 3-pattern with no dashes and whose k leftmost or k rightmost letters form either an increasing or decreasing subword. Moreover, we find a bijection between n-permutations that avoid the pattern 132 and begin with the pattern 12 and increasing rooted trimmed trees with n+1 nodes.

math.CO↗
arXiv2002

There are no iterated morphisms that define the Arshon sequence and the $σ$-sequence

Berstel proved that the Arshon sequence cannot be obtained by iteration of a morphism. An alternative proof of this fact is given here. The $σ$-sequence was constructed by Evdokimov in order to construct chains of maximal length in the n-dimensional unit cube. It turns out that the $σ$-sequence has a close connection to the Dragon curve. We prove that the $σ$-sequence can not be defined by iteration of a morphism.

math.CO↗