Paired patterns in lattice paths
arXiv: Combinatorics(2016)
摘要
Let (mathscr {L}_n) denote the set of all paths from [0, 0] to [n, n] which consist of either unit north steps N or unit east steps E or, equivalently, the set of all words (L {E,N}^*) with n E’s and n N’s. Given (L mathscr {L}_n) and a subset A of ([n] = {1, ldots , n}), we let (ps_{L}(A)) denote the word that results from L by removing the (i^{th}) occurrence of E and the (i^{th}) occurrence of N in L for all (i [n]-A), reading from left to right. Then, we say that a paired pattern (P mathscr {L}_k) occurs in L if there is some (A subseteq [n]) of size k such that (ps_L(A) = P). In this paper, we study the generating functions of paired pattern matching in (mathscr {L}_n).
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