We collect open problems in permutation patterns on four themes: rank-unimodality in the permutation pattern poset, Wilf-equivalence and shape-Wilf-equivalence, the enumeration of derangements in permutation classes, and sorting by stacks in series, generalized stacks, and restricted containers (C-machines).
This paper accompanies a talk I gave at the pre-conference workshop for early career researchers at Permutation Patterns 2025, the 23rd year of the conference series, held at the University of St Andrews and organized by Christian Bean and Ruth Hoffmann. It is not a list of the best-known or most difficult open problems in permutation patterns, but simply an assortment of problems on a few themes: problems I have encountered, wondered about, or been asked about. Many are of the "someone really ought to..." variety. A few are folklore, and some may be trivial. This is not meant as a roadmap for the field; describing what I feel are the deepest or most important open problems would require a different approach and considerably more buildup. Think of this instead as a collection of interesting attractions that do not take too long to reach.
The best-known open problem in the field is surely the enumeration of 1324-avoiding permutations. This one is easy to reach, but many have looked and progress has been hard to come by. Asked why, Zeilberger replied, “Because life is hard. The few combinatorial objects that we can count exactly are the trivial ones” [57] (emphasis in original). There are, in any case, plenty of places to read more about this conundrum [14, 15, 21, 22, 32-34, 48, 60]. We have nothing new to add about this problem, and so turn to areas that seem more tractable, or at least where we have less evidence of intractability.
The primary venue for results in permutation patterns, and for the exchange of open problems, has been the Permutation Patterns conference series, founded in 2003 by Michael Albert and Michael Atkinson at the University of Otago in Dunedin, New Zealand. Atkinson, who introduced Albert to the subject [58], drew many of the field’s current researchers into permutation patterns; his retirement was honored at Permutation Patterns 2013. Albert and Atkinson, and the conference series they founded, have shaped the research agenda considerably, and the conference’s tradition of open problems sessions has been particularly influential. For his part, Albert became a central figure in permutation patterns, known for his computational and collaborative approach to the subject, and retired in 2024. In this spirit, many of the problems here beg for computation, either because data is the goal, or just to better illuminate the path forward.
There are earlier collections of open problems that the reader may also find valuable. Wilf, widely regarded as a founder of the field and the plenary speaker at the inaugural Permutation Patterns conference in 2003, wrote a survey in 1999 [89] at an early stage in its development; I particularly enjoyed reading it as a graduate student. In that survey, Wilf expressed doubt about the Stanley-Wilf conjecture due to recent results of Alon and Friedgut [3]; his doubt was proved wrong a few years later when Marcus and Tardos [59] proved it. Wilf also recounted Stanley’s skepticism about the Noonan-Zeilberger conjecture [63] that every finitely based permutation class has a D-finite generating function; that skepticism was vindicated many years later when Garrabrant and Pak [43] disproved the conjecture.
Another valuable resource is Steingrímsson’s 2013 survey [80], which covers a broad range of topics, including the Möbius function of the permutation pattern poset, topological properties of intervals, vincular and mesh patterns, and the structure of growth rates. Steingrímsson is himself a leader in the field who has trained many of its current researchers; he delivered the plenary address at Permutation Patterns 2009.
For readers seeking broader introductions to permutation patterns, several general references are available. Kitaev’s Patterns in Permutations and Words [50] is a comprehensive compendium of the field. Bóna’s undergraduate textbook A Walk Through Combinatorics [23] contains a very accessible introduction to the area, while his monograph Combinatorics of Permutations [24] treats the subject in greater depth. Finally, I can’t help but recommend my own survey [85] that appears in the Handbook of Enumerative Combinatorics.
The remainder of the paper is organized as follows. Section 2 concerns rank-unimodality of intervals in the permutation pattern poset. Section 3 discusses symmetries and Wilf-equivalence. Section 4 addresses the enumeration of derangements in permutation classes. Section 5 looks at sorting machines. Below, we collect the basic definitions and offer some remarks on permutations as relational structures.
The basic notions of permutation patterns are easily stated. With apologies to those who view permutations as elements of a Coxeter group (Tenner [83]), we use the term length to mean the number of entries in a permutation, which we denote by |π|. We identify a permutation π with its plot: the set of points {(i, π(i))} in the plane. When we speak of an entry being to the left or right of, or above or below, another entry, we r
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