87 problems

Let $s$ and $t$ be positive integers with $\gcd(s,t)=1$, and let $f_t(s)$ be the polynomial appearing in Fayers' formula for the number of $(s,s+t,s+2t,\dots)$-core partitions. Fay…

Let $L_r(n)$ be the vertices of local simplex dimension $r$ in the partition graph, let $\tau(r)$ be the least $n$ for which $L_r(n)$ is nonempty, and let…

For positive integers $d$, define $h_d^{(a)}(n)$ as the number of partitions of $n$ into parts that are $d$-distinct and at least $a$, and define $\ell_m^{(m_1,m_2)}(n)$ as the num…

A rectair is the rectair family used in CRIM, and $\mathcal G$ denotes its Sprague–Grundy value. Near-square rectair conjecture. For $r\geq7$, … In addition, … All other rectairs h…

Higher-order modified Alder conjecture. For all $d,n\geq 1$,

Kang and Park's conjecture. For all $d,n\geq 1$,

The partition identity conjecture. For all integers $n,r,s e 0$, the following equivalent identities hold:

The Wide Partition Conjecture. A partition $\lambda$ satisfies Rota's conjecture if and only if it is wide.

The Wide Partition Conjecture for free matroids. An integer partition $\lambda$ is wide if and only if there exists a tableau of shape $\lambda$ such that (1) for every $i$, the en…

Deficient-sequence characterization conjecture. A partition $\lambda$ is complete if and only if, for every $\lambda'\subseteq\lambda$, the set $\mathfrak{S}^2(\lambda')$ has no de…

Complete-partition classification conjecture. A partition $\lambda$ is complete if and only if

Let $\lambda'$ be a two-row partition, and let $\mathrm{prs}_{\lambda'}$ be the associated skew Schur partition function, defined on those partitions for which the function is defi…

Let $\lambda$ be a losing partition and let $\Gamma_{r,c}$ be a losing hook. Write … when the stack operation $\lhd$ is defined. Losing-hook stack conjecture. If…

A thick hook is the thick-hook family of partitions used in CRIM, and a meld is the partition operation defined in the paper. A losing position is a $\mathcal P$-position. Thick-ho…

A partition is a Young diagram, and its Conway pair is the pair of game-theoretic invariants used in the Conway–Gurvich–Ho classification. An even-by-even hook is a hook with an ev…

A rectair is the rectair family used in CRIM, and $\mathcal G$ denotes its Sprague–Grundy value. Let $r$ and $k$ be integers with $0\leq k<r$. Square-rectair conjecture. … The conj…

A partition is a Young diagram, and its rank is the relevant rank statistic on the partition used in the CRIM analysis; a position is a $\mathcal P$-position when the next player l…

Modified Ballantine–Beck–Merca conjecture. For $k\geq 3$, $pre_k$ is injective on partitions of $n$ with $j$ parts for each $j\geq k+1$.

Let $\overline{P}(n,m)$ denote the number of overpartitions of $n$ with $m$ missing integers. Define $\overline{\mathcal{M}}_e(n)$ and $\overline{\mathcal{M}}_o(n)$ to be the numbe…

Let $P(n,m)$ denote the number of partitions of $n$ with $m$ missing integers, and define … Let $q(n)$ denote the number of partitions of $n$ into distinct parts. Distinct-part bou…

Let $P(n,m)$ denote the number of partitions of $n$ with $m$ missing integers. Define … Thus, $\mathcal{M}_e(n)$ and $\mathcal{M}_o(n)$ count partitions of $n$ with an even and odd…

Andrews' Conjecture 3. The coefficients satisfy

Let $p(3,n)$ denote the number of partitions of a $3\times n$ rectangle into permitted rectangular blocks, as defined in the paper. Asymptotic form conjecture. There exist constant…

For positive integers $i,s,k$, let $\llbracket i+(k-1)s,\ldots,i+s,i\rrbracket$ denote the partition whose parts form a decreasing arithmetic progression with initial part…

Let $FO_{j,k}(n)$ be the number of partitions of $n$ with exactly $j$ parts divisible by $k$, and let $FD_{j,k}(n)$ be the number of partitions of $n$ with exactly $j$ parts appear…