26 problems

Let … The sign patterns are coefficient patterns in the displayed order. Schlosser and Zhou's conjecture. For $\delta=1$, the coefficients of $Q_{10}(q)^{\delta}$ exhibit…

Schlosser's conjecture. If $\delta$ is real and

Borwein's conjecture. Its coefficients have a repeating sign pattern of $+--$. This conjecture is known as Borwein's conjecture and was proved by Wang in 2019 using an analytical m…

Let ${\cal A}$ be a sign pattern, and let $mr({\cal A})$ denote its minimum rank over the reals while $mr_{\mathbb Q}({\cal A})$ denotes its minimum rank over the rationals. Ration…

Let $\mathcal{P}$ be an irreducible, tridiagonal sign pattern with a $0$-diagonal. A sign pattern is consistent when it has the consistency property defined in the paper, and it re…

Let $\mathcal{A}$ be an irreducible, tridiagonal sign pattern with a $0$-diagonal. Its signed undirected graph has at most one maximal signed path with odd length. Eschenbach–Johns…

Baruah–Sarma sign conjecture. For all integers $n\geq 0$,

Wang's conjecture. For any positive integers $t$ and $m$, the sequence of coefficients $\{c_t^{(m)}(n)\}_{n\geq 0}$ is ultimately periodic in sign, with least period of sign divisi…

Bringmann–Harrison–Heim–Kane's conjecture. The coefficients satisfy

Conjectured sign pattern. For all integers $n\geq0$,

Let $\Sigma \in \operatorname{PR}_5$ be a real symmetric matrix whose principal-minor sign pattern $\operatorname{signs}_\Sigma$ agrees with $s_*$ on all subsets of sizes $1$ and…

Let $\mathcal{M}'$ be the set of completely multiplicative functions $f:\mathbb{N}\to\{+1,-1\}$. A function in $\mathcal{M}'$ has length $k$ if $k$ is the largest positive integer…

Non-eventual periodicity conjecture. The sign behaviour of this sequence is not eventually periodic. The supplied source attributes this conjecture to GU; no resolution is stated t…

Modulus-seven Borwein conjecture. For all $m$ and $n$, $d_{7m}(n)\ge0$ and $d_{7m+1}(n),d_{7m+3}(n),d_{7m+4}(n),d_{7m+6}(n)\le0$. Further, $d_{7m+5}(n)\ge0$ when…

Modulus-four Borwein conjecture. For all $m$ and $n$, $c_{4m}^{(\delta)}(n)\ge0$ and $c_{4m+2}^{(\delta)}(n)\le0$. Moreover, $c_{4m+1}^{(\delta)}(n)\le0$ for…

Ismail–Kim–Stanton conjecture. If $m\equiv \pm(2l+1)a\pmod K$ for some $l$ with $0\le l<K/2$, then $b_m\le0$; otherwise $b_m\ge0$. Thus the sign of $b_m$ is determined by $m$ modul…

Third Borwein conjecture. The sign pattern of the coefficients in the expansion of $S_n(q)$ is $+----+----+----\cdots$, with the convention that zero coefficients may be regarded a…

Cubic Borwein conjecture. The sign pattern of the coefficients in the expansion of $P_n(q)^3$ is $+--+--+--\cdots$, with the same convention concerning zero coefficients as above.…

The Second Borwein conjecture. The sign pattern of the coefficients in the expansion of $P_n(q)^2$ is $+--+--+--\cdots$, with the same convention that a zero coefficient is conside…

The First Borwein conjecture. The sign pattern of the coefficients in the expansion of $P_n(q)$ is $+--+--+--\cdots$, with a coefficient $0$ considered as both $+$ and $-$. This co…

Let $c_t^{(m)}(n)$ denote the coefficient of $q^n$ in $(q;q)_\infty^m(q^t;q^t)_\infty^{-m}$, and let a sequence be a UPS sequence when its signs are periodic with a least sign peri…

Converse characterization conjecture. The following are equivalent:

Realizability conjecture. The only pairs $(pos,neg)$ that can be non-realizable have either $pos=0$ or $neg=0$. Equivalently, every pair $(pos,neg)$ satisfying the standard conditi…

A sign pattern is a matrix whose entries are $0$, $+$, or $-$. It is spectrally arbitrary if every monic real polynomial of degree $n$ is the characteristic polynomial of an…

Realizability conjecture. For an arbitrary sign pattern ${\bar \sigma}$, the only type of pairs $(pos,neg)$ which can be non-realizable has either $pos$ or $neg$ vanishing. Equival…