22 problems

Let $S_{n,k}$ be the triangle defined by … for $1\leq k\leq n-2$, with $S_{n,n}=S_{n,n-1}=S_{n,n-2}$ for $n\geq 3$ and $S_{1,1}=S_{2,1}=S_{2,2}=1$. For a pair of patterns…

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…

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

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 $(\nu,\mu)$ 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 $e$,…

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…

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 … Let $\operatorname{{\bf I}}_n(\geq,\geq,-)$ be the subset of inversion sequences for which the…

Let $S_{n,k}$ be the quantity defined earlier in the paper, let $\rho$ and $\tau$ be patterns of length $4$, and let $d53e_n(\rho,\tau)$ denote the permutations in $d53e_n$ avoidin…

Ascent conjecture. The number of Schröder paths $p\in R_{n-1}$ with $k-1$ ascents is equal to the number of inversion sequences $e\in\mathbf{I}_n(021)$ with $k$ distinct values.

The Entringer numbers $d_{n,k}$ count down/up permutations of $[n+1]$ whose first entry is $k+1$, and satisfy … An Entringer number conjecture asserts that, for $n\geq 1$ and…

Let $s=(1,4,3,8,5,12,\ldots)$, with $s_{2i}=4i$ and $s_{2i-1}=2i-1$ for $i\geq 1$. Let $I_n$ be the set of $s$-inversion sequences of length $n$, and let $P_n$ be the set of signed…

Let $P^{\pm}(\{1,1,2,2,\ldots,n,n\})$ be the signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$, and let $\mathfrak I_{2n}^{(1,4,3,8,\ldots,2n-1,4n)}$ be the correspondin…