Ratsimandresy Yeriel Fiandrianana, Purwanto Purwanto, I. Made Sulandra
Let [n] denote the set {1, 2, ..., n}, t be a permutation of [4] and p=B1|B2| ... |Bk be a partition of [n], in the standard sense, where the blocks be arranged such that the first entries from each block be in increasing order, and entries in each block be also in increasing order. A flattened partition f of [n] is the permutation of [n] obtained by erasing the symbol which separates each block in p; f avoids the pattern t, or f is t-avoiding, if there is no subsequence of f which is order-isomorphic to t. A run in f is a subsequence of the form fifi+1 ... fi+p with fi<fi+1<...<fi+p, where fi<fi-1 and fi+p>fi+p+1; fi is called the starting point of the run. Pattern avoidance in flattened partitions is an open and active area of research, and so far only few related works have been published. In this paper, we give a formula which counts the number of flattened partitions of [n] avoiding the pattern 1234 and 1243, for n=1, by considering the number of runs in it and using a simple yet powerful principle, namely the pigeonhole principle. One of our results is related to one of the sequences in Online Encyclopedia of Integers Sequences. © 2024 Author(s).
Department of Mathematics, Universitas Negeri Malang, Jalan Semarang 5, Malang, 65145, Indonesia