376 research outputs found

    On Universal Cycles for new Classes of Combinatorial Structures

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    A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, matroids, restricted multisets, chains of subsets, multichains, and lattice paths. For subsets, we show that a u-cycle exists for the kk-subsets of an nn-set if we let kk vary in a non zero length interval. We use this result to construct a "covering" of length (1+o(1))(1+o(1))(nk)n \choose k for all subsets of [n][n] of size exactly kk with a specific formula for the o(1)o(1) term. We also show that u-cycles exist for all nn-length words over some alphabet Ξ£,\Sigma, which contain all characters from RβŠ‚Ξ£.R \subset \Sigma. Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets

    Sperner systems with restricted differences

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    Let F\mathcal{F} be a family of subsets of [n][n] and LL be a subset of [n][n]. We say F\mathcal{F} is an LL-differencing Sperner system if ∣Aβˆ–B∣∈L|A\setminus B|\in L for any distinct A,B∈FA,B\in\mathcal{F}. Let pp be a prime and qq be a power of pp. Frankl first studied pp-modular LL-differencing Sperner systems and showed an upper bound of the form βˆ‘i=0∣L∣(ni)\sum_{i=0}^{|L|}\binom{n}{i}. In this paper, we obtain new upper bounds on qq-modular LL-differencing Sperner systems using elementary pp-adic analysis and polynomial method, extending and improving existing results substantially. Moreover, our techniques can be used to derive new upper bounds on subsets of the hypercube with restricted Hamming distances. One highlight of the paper is the first analogue of the celebrated Snevily's theorem in the qq-modular setting, which results in several new upper bounds on qq-modular LL-avoiding LL-intersecting systems. In particular, we improve a result of Felszeghy, Heged\H{u}s, and R\'{o}nyai, and give a partial answer to a question posed by Babai, Frankl, Kutin, and \v{S}tefankovi\v{c}.Comment: 22 pages, results in table 1 and section 6.1 improve
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