27 problems

Let $\mathcal{A}_n$ be the set of ascent sequences of length $n$, and let $\mathsf{asc}$, $\mathsf{rep}$, $\mathsf{zero}$, and $\mathsf{max}$ denote the corresponding Euler–Stirlin…

Pattern-avoidance conjecture. The map $\mathrm{hat}_{\max}$ restricts to the following bijections:

Duncan–Steingrímsson conjecture. Modified ascent sequences avoiding any one of the patterns

Standardization conjecture. Using the BiSC algorithm and the transport theorem, the standardizations of primitive modified ascent sequences avoiding $312$ satisfy

Quadruple equidistribution conjecture. For every $n\geq1$,

Cerbai–Claesson–Ferrari conjecture. For every $n\geq 0$, there is a bijection between $A_n(201)$ and the permutations in $S_n$ that are $312$-machine sortable.

Let $w=(w_1,\ldots,w_n)$ be a weak ascent sequence, and suppose that … for all $i$. Pattern-avoidance enumeration conjecture. The number of such weak ascent sequences equals the nu…

Let $c_n$ be the coefficient sequence for either the $110$-avoiding or the $100$-avoiding ascent sequences. For the $100$-avoiding sequence, the paper estimates … with…

Refined asymptotic conjecture. The parameters are conjectured to satisfy

Factorial-growth constant conjecture. The constant is conjectured to be

Let $\operatorname{Ascseq}_n(1123)$ denote the ascent sequences of length $n$ that avoid the pattern $1123$, and let $C_n$ be the $n$th Catalan number. Duncan–Steingrímsson's conje…

Lin and Yan's conjecture. For $n\geq1$ and $0\leq k\leq n-1$,

Let $\mathcal{A}_n$ be the set of ascent sequences of length $n$, and let $\mathsf{asc}$, $\mathsf{rep}$, $\mathsf{zero}$, and $\mathsf{max}$ denote their ascent, repetition, zero,…

Let an ascent sequence be a sequence of nonnegative integers satisfying $s_i\leq \mathsf{asc}(s_1,\ldots,s_{i-1})+1$ for $i\geq2$, and let a $({\bf2-1})$-avoiding inversion sequenc…

Kreweras–Riordan conjecture. These two formal power series are equal.

Conjecture 3.3. The number $A_{210}(n)$ equals the number of non-3-crossing set partitions of $\{1,2,\ldots,n\}$.

Conjecture 3.2. We have

An ascent sequence of length $n$ is a nonnegative integer sequence $x=x_1x_2\ldots x_n$ with $x_1=0$ and $x_i\leq \operatorname{asc}(x_1x_2\ldots x_{i-1})+1$ for $1<i\leq n$, where…

A modified ascent sequence is obtained from an ascent sequence by the modification process described in the source. For patterns $101$, $0101$, $1021$, $1102$, $1120$, and $1210$,…

For a pattern $p$, let $A_p(n)$ be the number of ascent sequences of length $n$ avoiding $p$. Two patterns are Wilf equivalent if their avoidance sequences agree for every $n$. The…

Let $A_{0123}(n)$ be the number of ascent sequences of length $n$ avoiding $0123$. A Dyck path of semilength $n$ has height at most $5$ if its maximum vertical height is at most…

Let $A_{210}(n)$ be the number of ascent sequences of length $n$ avoiding $210$. A set partition of $\{1,2,\ldots,n\}$ is 3-noncrossing if it has no 3-crossing. The 210 counting co…

Let ${\mathcal S}_{0012}(n)$ be the set of ascent sequences of length $n$ avoiding $0012$, let $A_{0012}(n)=|{\mathcal S}_{0012}(n)|$, let $\operatorname{fwd}$ be the length of the…

Let ${\mathcal S}_{021}(n)$ be the set of ascent sequences of length $n$ avoiding $021$, and let $\operatorname{asc}$ and $\operatorname{RLmin}$ denote, respectively, the number of…

For a pattern $p$, let $A_p(n)$ denote the number of ascent sequences of length $n$ avoiding $p$. Marcus–Tardos growth conjecture. For every fixed pattern $p$, there is a constant…