132 problems

Beck's conjecture. The quantity $e(n)$ equals the total number of distinct parts of all partitions of $2n+2$ with rank $n+1$.

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…

A Sprague–Grundy value is the nimber assigned to a position in an impartial combinatorial game. Welter's game is an impartial game whose positions can be represented by integer par…

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…

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

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

Berkovich–Uncu conjecture. The series $H_{L,s,k}(q)$ is eventually positive for all positive integers $L$, $s$, and $k$ satisfying $L\geq 3$ and $k\geq s+1$.

Let $M(m,n)$ and $N(m,n)$ denote respectively the number of partitions of $n$ with crank $m$ and rank $m$. Rank and crank log-concavity conjecture. The inequalities … hold for…

Let $a$ and $b$ be coprime positive integers. An $(a,b)$-core partition is a partition that is simultaneously an $a$-core and a $b$-core, meaning that no cell in its Ferrers diagra…

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

For a positive integer $k$, let $\mathrm{pre}_k$ be the elementary symmetric partition function, and let $\mathcal{P}_{>k}(n)$ denote the set of partitions of $n$ with length stric…

Let $n$ and $k$ be positive integers, and let $\mathrm{pre}_k$ map a partition with at least $k$ parts to the summands of the elementary symmetric polynomial $e_k$ evaluated at its…

A partition $\lambda$ is parity self-conjugate if $\lambda_k\equiv\lambda'_k\pmod 2$ for every $k$, where $\lambda'$ is the conjugate partition. An odd self-conjugate hook is a sel…

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…

The families $S_n$ and $\mathrm{PS}_n$ are rectair families, and $SG$ denotes their Sprague–Grundy value. Rectair-family Sprague–Grundy conjecture. For $S_n$, … … For…

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…

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 $j\geq k+1$.