35 problems

There are $7^3=343$ triples of binary relations $(\rho_1,\rho_2,\rho_3)$, where each relation belongs to $\{<,>,\leq,\geq,=,\neq,-\}$. Two triples are equivalent if they have the s…

Let $\mathcal{I}(010,102)$ be the class of inversion sequences avoiding the patterns $010$ and $102$, and let $F(x)$ be its ordinary generating function. Pantone's algebraicity con…

Let $\mathcal{I}(000,102)$ be the class of inversion sequences avoiding the patterns $000$ and $102$, and let $F(x)$ be its ordinary generating function. Pantone's algebraicity con…

A sequence $e=(e_1,e_2,\dots,e_n)$ is an inversion sequence of length $n$ if $0\leq e_i<i$ for every $i\in[n]$. An inversion sequence avoids a pattern when it contains no occurrenc…

Let $\textbf{I}(010,110,120,210)$, $\textbf{I}(010,100,120,210)$, and $\textbf{I}(011,201)$ denote the classes of inversion sequences avoiding the displayed patterns. Martinez and…

Quintuple equidistribution conjecture. There is a bijection $\Omega:\mathsf{I}_n\to\mathsf{I}_n$ such that, for every $s\in\mathsf{I}_n$,

Hong and Li's conjecture. The value $a(n)$ counts the inversion sequences of length $n$ that avoid the pattern $0021$; equivalently, $\mathcal{A}(x)$ generates…

Let $n$ be a positive integer. Let $D\mathcal{P}^{(2)}_{n,2n}(NE,EN)$ be the set of Delannoy paths from $(0,0)$ to $(n,2n)$ consisting of north-steps $N=(0,1)$, east-steps…

Inversion-number conjecture. The expected value is the rational function

For $n\geq 1$, let $\mathbf{I}_n(\geq,\neq,>)$ and $\mathbf{I}_n(>,\neq,\geq)$ be the two classes of inversion sequences avoiding the indicated relation triples. For an inversion s…

Let $\mathfrak{S}_n(4231,42513)$ denote the permutations in $\mathfrak{S}_n$ avoiding the patterns $4231$ and $42513$, and let $\mathbf{I}_n(>,-,>)$, $\mathbf{I}_n(\geq,\neq,>)$, a…

Let $(\nu,\mu)$ be a pair of patterns of length four. For a permutation $\pi$, let $\operatorname{asc}(\pi)$ denote its number of ascents and let $\operatorname{last}(\pi)$ denote…

Let $\mathcal{I}_n$ be the set of inversion sequences of length $n$, let $\mathfrak{S}_n$ be the symmetric group, and let $b:\mathfrak{S}_n\to\mathcal{I}_n$ be the bijection due to…

An inversion sequence of length $n$ is a sequence $e=e_1e_2\cdots e_n$ such that $0\le e_i\le i-1$ for $1\le i\le n$; let $\mathbf{I}_n(0012)$ denote the set of inversion sequences…

Let $cmathfrak{S}_n(\underline{23}14)$ be the permutations avoiding the vincular pattern $\underline{23}14$, and let $cmathbf{I}_n(110)$ be the inversion sequences avoiding $110$.…

Let $cmathbf{I}_n$ be the set of inversion sequences of length $n$, and let $cmathcal{PA}_n$ be the set of primitive ascent sequences of length $n$. For an inversion sequence $e$,…

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

Let $\mathbf I_n(0012)$ denote the set of inversion sequences of length $n$ avoiding the pattern $0012$. Jun Ma and Lin's conjecture. For $n\geq 1$, we have … In other words, the u…

For a tuple of patterns, write $(\rho_1,\ldots,\rho_k)\thicksim(\sigma_1,\ldots,\sigma_\ell)$ when the corresponding inversion-sequence avoidance classes are Wilf-equivalent, meani…

Let $\mathbf I_n(\rho_1,\rho_2,\rho_3)$ denote the set of inversion sequences of length $n$ avoiding the indicated patterns or relations. Martinez and Savage's conjecture. For…

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…

Let $\mathbf{I}_{n}(>,\leq,-)$ denote the inversion sequences of length $n$ avoiding the classical relation pattern $(>,\leq,-)$. Let $\mathcal{R}'_{n}$ be the set of underdiagonal…

An inversion sequence of length $n$ is a sequence $x=x_1x_2\cdots x_n$ satisfying $0\leq x_i<i$ for all $1\leq i\leq n$. A set partition of $[n]$ is enhanced 3-nonnesting (respecti…

For each $n\geq1$, let $\operatorname{{\bf I}}_n$ be the set of inversion sequences … An inversion sequence $e=(e_1,e_2,\ldots,e_n)$ is an $\mathcal{AW}$-inversion sequence if, for…

For each $n\geq1$, let $\operatorname{{\bf I}}_n$ be the set of inversion sequences … Let $\operatorname{{\bf I}}_n(\geq,\geq,-)$ be the subset of inversion sequences for which the…