28 problems

For a permutation class $\mathcal{C}$ and an integer $n\geq 1$, say that $\mathcal{C}$ is $n$-well-quasi-ordered (or $n$-wqo) if the set of permutations in $\mathcal{C}$ labeled by…

Let $\mathcal{P}$ be the permutation class consisting of all proper pin sequences and their subpermutations. Its generating function is … where $\mathcal{P}_n$ is the set of member…

Let $\mathcal{C}$ be a permutation class. It is strongly algebraic if $\mathcal{C}$ and every subclass of $\mathcal{C}$ have algebraic generating functions, and it is well-quasi-or…

Let $\mathcal{C}$ be a permutation class and let … be its generating function. A generating function may be rational, algebraic, or D-finite. Zeilberger's characterization problem.…

Let $\mathcal{C}$ be a permutation class, meaning a downset under pattern containment. It is well-quasi-ordered if it contains no infinite antichain, and its generating function is…

Let $M$ be a partial multiplication matrix, meaning a matrix whose nonzero entries are monotone classes, and suppose its cell graph is cyclic. Let $\mathcal X$ be a permutation cla…

A monotone grid class is a permutation class defined by a finite grid whose nonempty cells contain monotone permutations. Huczynska–Vatter's conjecture. Every monotone grid class i…

Let $\mathcal{C}$ be a permutation class, and let $\langle\mathcal{C}\rangle$ denote its substitution closure. The substitution-closure conjecture. If $\langle\mathcal{C}\rangle$ i…

Let $\mathcal{C}$ be a permutation class, and let $\mathcal{C}^{+1}$ denote its one-point extension. A class is lwqo when it is well-quasi-ordered under labeling by arbitrary wqo l…

Geometric griddability and finite-letter-graph conjecture. $X$ is geometrically griddable if and only if ${\cal G}_X$ is a class of $k$-letter graphs for some finite value of $k$.

Let an upper growth rate of a permutation class mean a real number that occurs as the upper exponential growth rate of that class, and let a sum closed class be a permutation class…

Let $\mathcal{G}_\gamma$ denote the cell class with growth rate parameter $\gamma$, and let $\varphi=(1+\sqrt{5})/2$ be the golden ratio. Conjecture on concentration thresholds. Th…

Deflatability conjecture. The class $\operatorname{Av}(pi)$ is deflatable.

Finite-basis conjecture for monotone grid classes. Every monotone grid class has a finite basis. The source notes that only limited cases are known, including skew-merged permutati…

Eventual-interval conjecture. At some point, the set of growth rates of permutation classes contains all subsequent real numbers. Albert and Linton constructed an uncountable set o…

Atkinson's conjecture.

Strongly algebraic characterization conjecture. A permutation class is strongly algebraic if and only if it is wpo. The source proves that every strongly algebraic class is wpo and…

Growth-rate limit conjecture. For every permutation class $\mathcal{C}$, $\overline{\mathrm{gr}}(\mathcal{C})=\underline{\mathrm{gr}}(\mathcal{C})$. The equality is not known in ge…

Let $\operatorname{Av}(321)$ denote the permutation class consisting of permutations avoiding the pattern $321$, and let a proper subclass mean a subclass strictly contained in…

A permutation class $\mathcal{C}$ is broadly rational if every finitely based subclass of $\mathcal{C}$ has a rational generating function, and it is strongly rational if it and al…

Let $\mathcal{C}$ be a permutation class, with lower and upper growth rates … and … If these quantities coincide, their common value is called the growth rate of $\mathcal{C}$. Eve…

A permutation class is a set of permutations closed under taking patterns, and it is finitely based if it can be specified by avoiding a finite set of permutations. Its generating…

For a permutation class $\mathcal{C}$, let … Let $\lambda$ be the unique positive root of … so that $\lambda\approx2.48$. Vatter's lower-bound conjecture. The number $\lambda$ is b…

Stanley–Wilf conjecture for permutation classes. Every proper permutation class has finite growth rate.

Let $\xi\approx 2.30524$ be the unique positive root of $x^5-2x^4-x^2-x-1$, and let the accumulation points from above of a set of real numbers be limits approached by larger eleme…