Introduction to Partially Ordered Patterns
Sergey Kitaev
Abstract
We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others. We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and Pell numbers, as well as generalizations of few results in the literature including the descent distribution. Moreover, we discuss q-analogue for a result on non-overlapping segmented POPs. Finally, we suggest several open problems for further research.
Create a lesson
Related papers
Graded Ehrhart theory for hypersimplices
Nathaniel Libman, Weston Miller
Closing the gap and settling the problem of queens on an n× n board, each attacking at most one other
Kristina Ago, Bojan Bašić, Radojka Ciganović
Leading term strandings for webs
Michael Bo, Madelyn Burns, Junyang Chen et al.
Refutation of the Non-Cancelling-Intersections Conjecture
Hermann Wilhelm
Asymptotic Bounds for Online Ramsey Numbers of Stars versus Long Paths and Cycles
Sam Beilis, Israel R. Curbelo, Elizabeth R. Koizumi
The Erdos--Gallai bound for consecutive even cycle lengths
Yaobin Chen, Hong Liu, Xia Wang et al.