15 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 power series is positive when all its coefficients are nonnegative. The positivity conjecture for…

Kang–Park conjecture. For all positive integers $d$ and $n$,

Alder's conjecture. For all positive integers $d$ and $n$,

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 $D'(k,m,n)$ 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 $\overline{\mathcal{OE}}(n)$ 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

Duncan–Khunger–Tamura conjecture. If $a,d\geq 1$ and $1\leq a\leq d+2$, then for all $n\geq 1$,

Duncan–Khunger–Tamura conjecture. For all $n,d\geq 1$,

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…

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…