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In this paper we enumerate the number of ways of selecting $k$ objects from $n$ objects arrayed in a line such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects and provide three different formulas when $m,pgeq 1$ and $ngeq pm(k-1)$. Also, we prove that the number of ways of selecting $k$ objects from $n$ objects arrayed in a circle such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects is given by $frac{n}{n-pk}binom{n-pk}{k}$, where $m,pgeq 1$ and $ngeq mpk+1$.
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DP-coloring is a generalization of list coloring, which was introduced by Dvov{r}{a}k and Postle [J. Combin. Theory Ser. B 129 (2018) 38--54]. Zhang [Inform. Process. Lett. 113 (9) (2013) 354--356] showed that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is 3-choosable. Liu et al. [Discrete Math. 342 (2019) 178--189] showed that every planar graph without 4-, 5-, 6- and 9-cycles is DP-3-colorable. In this paper, we show that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is DP-3-colorable, which generalizes these results. Yu et al. gave three Bordeaux-type results by showing that (i) every planar graph with the distance of triangles at least three and no 4-, 5-cycles is DP-3-colorable; (ii) every planar graph with the distance of triangles at least two and no 4-, 5-, 6-cycles is DP-3-colorable; (iii) every planar graph with the distance of triangles at least two and no 5-, 6-, 7-cycles is DP-3-colorable. We also give two Bordeaux-type results in the last section: (i) every plane graph with neither 5-, 6-, 8-cycles nor triangles at distance less than two is DP-3-colorable; (ii) every plane graph with neither 4-, 5-, 7-cycles nor triangles at distance less than two is DP-3-colorable.
Abstract separation systems are a new unifying framework in which separations of graph, matroids and other combinatorial structures can be expressed and studied. We characterize the abstract separation systems that have representations as separation systems of graphs, sets, or set bipartitions.
In this paper, we investigate statistics on alternating words under correspondence between ``possible reflection paths within several layers of glass and ``alternating words. For $v=(v_1,v_2,cdots,v_n)inmathbb{Z}^{n}$, we say $P$ is a path within $n$ glass plates corresponding to $v$, if $P$ has exactly $v_i$ reflections occurring at the $i^{rm{th}}$ plate for all $iin{1,2,cdots,n}$. We give a recursion for the number of paths corresponding to $v$ satisfying $v in mathbb{Z}^n$ and $sum_{igeq 1} v_i=m$. Also, we establish recursions for statistics around the number of paths corresponding to a given vector $vinmathbb{Z}^n$ and a closed form for $n=3$. Finally, we give a equivalent condition for the existence of path corresponding to a given vector $v$.
We show that, in an alphabet of $n$ symbols, the number of words of length $n$ whose number of different symbols is away from $(1-1/e)n$, which is the value expected by the Poisson distribution, has exponential decay in $n$. We use Laplaces method for sums and known bounds of Stirling numbers of the second kind. We express our result in terms of inequalities.
The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreover, we present grammatical descriptions of the Jacobi-Stirling numbers of both kinds.
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