Counting subwords in flattened permutations

Abstract: In this paper, we consider the number of occurrences of descents, ascents, 123-subwords, 321-subwords, peaks and valleys in flattened permutations, which were recently introduced by Callan in his study of finite set partitions. For descents and ascents, we make use of the kernel method and obtain an explicit formula (in terms of Eulerian polynomials) for the distribution on $\mathcal{S}_n$ in the flattened sense. For the other four patterns in question, we develop a unified approach to obtain explicit formulas for the comparable distributions. We find that the formulas so obtained for 123- and 321-subwords can be expressed in terms of the Chebyshev polynomials of the second kind, while those for peaks and valleys are more related to the Eulerian polynomials. We also provide a bijection showing the equidistribution of descents in flattened permutations of a given length with big descents in permutations of the same length in the usual sense.