12 problems

Let $D'(k,m,n)$ be defined by … where $k$ and $m$ are positive integers. A series is eventually positive when all coefficients are nonnegative from some index onward. The eventual…

Let $D'(k,m,n)$ be defined by … where $k$ and $m$ are positive integers. A coefficient is negative when it is less than zero. The exceptional negative coefficients conjecture for…

Let $\lambda$ and $\mu$ be partitions, and let $\widetilde G_\lambda(\mathbf x)$ denote the stable Grothendieck polynomial. Write $\lambda\vee\mu$ and $\lambda\wedge\mu$ for the jo…

Let $\lambda$ and $\mu$ be partitions, let $\mathbf x_k=(x_1,\ldots,x_k)$, and let $s_\lambda(\mathbf x_k;\mathbf y)$ denote the double Schur polynomial. Write $\lambda\vee\mu$ and…

Let $\overline{\mathcal{OE}}(n)$ and $\overline{\mathcal{EO}}(n)$ denote the paper's parity-separated partition-counting functions, and let $p_0(n)$ and $p_1(n)$ be the associated…

Inagaki–Tamura shift identity conjecture. If $d\geq 12$ and $n\geq d+2$, then

Alternating inequality conjecture. For $n,k\geqslant 0$, one has

Generalized double-exclusion Alder conjecture. Let $a,d$ be positive integers with $1\leq a\leq d+2$. Then for all $n\geq 1$,

For a positive integer $L$, let $G_{L,2}(q)$ be the difference of the generating series for partitions with smallest part $2$ and largest-minus-smallest part at most $L$, and for p…

For positive integers $L$ and $s$ with $L\geq s+1$, let $C^{*}_{L,s}$ be the set of partitions whose smallest part is $s$, whose parts are at most $L+s$, and in which $L$ does not…

Let $sc_9(n)$ denote the number of self-conjugate $9$-core partitions of $n$, and let $n$ be a non-negative integer. Numerical inequalities for self-conjugate 9-core partitions. Th…

Three-residue coefficient conjecture. If $z$ does not divide $y$, then all coefficients $a(L,y,n,x,z,m)$ are non-negative; if $z$ divides $y$, then only finitely many of these coef…