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Register a new userOn the Refinement of Certain Statistics on Alternating Words
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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 $v$.
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A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. Safety one-way alternating automata with one register on infinite data words are considered, their nonemptiness is shown EXPSPACE-complete, and their inclusion decidable but not primitive recursive. The same complexity bounds are obtained for satisfiability and refinement, respectively, for the safety fragment of linear temporal logic with freeze quantification. Dropping the safety restriction, adding past temporal operators, or adding one more register, each causes undecidability.
Let S be a double occurrence word, and let M_S be the words interlacement matrix, regarded as a matrix over GF(2). Gauss addressed the question of which double occurrence words are realizable by generic closed curves in the plane. We reformulate answers given by Rosenstiehl and by de Fraysseix and Ossona de Mendez to give new graph-theoretic and algebraic characterizations of realizable words. Our algebraic characterization is especially pleasing: S is realizable if and only if there exists a diagonal matrix D_S such that M_S+D_S is idempotent over GF(2).
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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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